Geometric Complexity Theory: Introduction

Geometric Complexit y Theory: In tro duction Dedicated to Sri Ramakrishna Ketan D. Mulm ule y 1 The Univ ersi ty of Chicago Milind Sohoni I.I.T., Mum bai T ec hnical Rep ort TR-2007-1 6 Computer Science Departmen t The Univ ersi ty of Chicago Septem ber 2007 August 2, 2014 1 Part of the work on GCT w as done while the ﬁrst a uthor was vis itin g I.I.T. Mum bai to whic h he is grateful for its hospitalit y F orew ord These are lectures n ote s for the in tro ductory graduate courses on geo- metric complexit y theory (GCT) in the computer science departmen t, the univ ersit y of Ch icag o. P art I consists of the lecture note s for the course giv en b y the ﬁrst author in the sp ring quarter, 2007 . It gi ve s in tro duction to th e basic stru cture of GCT. Pa rt I I consists o f the lecture notes for the course gi v en b y the second author in the spring quarter, 2003. It giv es in- tro duction to inv ariant theory with a view to w ards GCT. No b ac kground in algebraic geometry or r ep r esen tation theory is assumed. These lecture notes in conjunction with the article [GCTﬂ ip1], whic h describ es in detail the basic plan of G CT based on the principle cal led the ﬂip , should pro vide a high lev el picture of GCT assuming familiarit y with only basic notions of algebra, suc h as groups, rings, ﬁelds etc. Man y of the theorems in th ese lecture notes are stated without pr oofs, but after giving enough motiv ation so that they can b e tak en on faith. F or the readers interest ed in f urther study , Figure 1 shows logical dep endence among the v arious pap ers of GCT and a suggested r eading sequence. The ﬁr st author is grateful to P aolo Co denotti, Josh ua Gro c ho w, Soura v Chakrab ort y and Hari Naray anan for taking notes for h is lect ures. 1 GCT abs | ↓ GCTﬂip1 | ↓ These lecture notes −− → GCT3 | | ↓ | GCT1 | | | ↓ | GCT2 | | | ↓ ↓ GCT6 ← −− GCT5 | ↓ GCT4 −− → GCT9 | ↓ GCT7 | ↓ GCT8 | ↓ GCT10 | ↓ GCT11 | ↓ GCTﬂip2 Figure 1: Logical d ep en d ence a mong the GCT pap ers 2 Con ten ts I The basic st ructure of GCT By Ketan D. Mulm uley 8 1 Ov erview 9 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The G¨ odelian Flip . . . . . . . . . . . . . . . . . . . . . . . . 1 2 1.3 More details of the GCT appr oac h . . . . . . . . . . . . . . . 13 2 Represen t ation theory o f reductiv e groups 16 2.1 Basics of Represen tation Theory . . . . . . . . . . . . . . . . 16 2.1.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . 1 6 2.1.2 New representa tions from old . . . . . . . . . . . . . . 1 7 2.2 Reductivit y of ﬁnite group s . . . . . . . . . . . . . . . . . . . 19 2.3 Compact Groups and GL n ( C ) are reductive . . . . . . . . . . 20 2.3.1 Compact groups . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 W eyl’s unitary tric k and GL n ( C ) . . . . . . . . . . . . 21 3 Represen t ation theory o f reductiv e groups (con t ) 22 3.1 Pro jection F orm ula . . . . . . . . . . . . . . . . . . . . . . . . 2 3 3.2 The charac ters of ir reducible represen tations form a basis . . 25 3.3 Extending to Inﬁnite C omp act Groups . . . . . . . . . . . . . 27 4 Represen t ations of t he symm etric group 29 4.1 Represen tations and c haracters of S n . . . . . . . . . . . . . . 3 0 4.1.1 First Construction . . . . . . . . . . . . . . . . . . . . 30 4.1.2 Second Constru cti on . . . . . . . . . . . . . . . . . . . 31 4.1.3 Third Constru ctio n . . . . . . . . . . . . . . . . . . . . 32 4.1.4 Character of S λ [F rob enius c haracter formula] . . . . . 32 4.2 The ﬁr st decisio n problem in GCT . . . . . . . . . . . . . . . 33 3 5 Represen t ations of GL n ( C ) 35 5.1 First Approac h [Deruyts] . . . . . . . . . . . . . . . . . . . . 35 5.1.1 Highest we igh t vec tors . . . . . . . . . . . . . . . . . . 38 5.2 Second Appr oac h [W eyl] . . . . . . . . . . . . . . . . . . . . . 39 6 Deciding non v anishing of Littlewoo d-Ric hardson co eﬃcien ts 41 6.1 Littlew o o d-Ric h ard son co eﬃcien ts . . . . . . . . . . . . . . . 41 7 Littlewoo d-Ric hardson co eﬃcien t s (con t) 46 7.1 The stretc hing function . . . . . . . . . . . . . . . . . . . . . 47 7.2 O n ( C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8 Deciding non v anishing of Littlewoo d-Ric hardson co eﬃcien ts for O n ( C ) 52 9 The pleth ysm problem 56 9.1 Littlew o o d-Ric h ard son Problem [GCT 3,5] . . . . . . . . . . . 57 9.2 Kronec k er Problem [GCT 4,6] . . . . . . . . . . . . . . . . . . 57 9.3 Pleth ysm Problem [GCT 6,7] . . . . . . . . . . . . . . . . . . 58 10 Saturat e d and p ositiv e integer pro gramming 61 10.1 Saturated, p ositiv e in teger programming . . . . . . . . . . . . 61 10.2 Application to the p let hysm problem . . . . . . . . . . . . . . 63 11 Basic algebraic geometry 64 11.1 Algebraic geometry deﬁnitions . . . . . . . . . . . . . . . . . 64 11.2 Orbit closures . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 11.3 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12 The class v arieties 70 12.1 Class V arieties in GCT . . . . . . . . . . . . . . . . . . . . . . 70 13 Obstructions 73 13.1 Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.1.1 Wh y are the class v arieties exceptional? . . . . . . . . 75 14 Group theoretic v arieties 78 14.1 Represen tation theoretic data . . . . . . . . . . . . . . . . . . 79 14.2 The second fundamental theorem . . . . . . . . . . . . . . . . 80 14.3 Wh y should obstructions exist? . . . . . . . . . . . . . . . . . 81 4 15 The ﬂip 82 15.1 The ﬂip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 16 The Grassmanian 86 16.1 The second fundamental theorem . . . . . . . . . . . . . . . . 87 16.2 The Borel-W eil theorem . . . . . . . . . . . . . . . . . . . . . 88 17 Quantum group: basic deﬁnitions 90 17.1 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 18 Standard quan tum group 97 19 Quantum unitary group 103 19.1 A q -analogue of the unitary group . . . . . . . . . . . . . . . 103 19.2 Prop erties of U q . . . . . . . . . . . . . . . . . . . . . . . . . . 105 19.3 Irreducible Representa tions of G q . . . . . . . . . . . . . . . . 106 19.4 Gelfand-Tsetlin basis . . . . . . . . . . . . . . . . . . . . . . . 106 20 T o wards p ositivity hypotheses via quan tum groups 108 20.1 Littlew o o d-Ric h ardson ru le via standard quan tum groups . . 108 20.1.1 An emb edding of the W eyl m o du le . . . . . . . . . . . 109 20.1.2 Crystal op erators and crystal bases . . . . . . . . . . . 110 20.2 Explicit decomp osition of the tensor pr odu ct . . . . . . . . . 112 20.3 T o wa rds nonstandard quantum groups for the Kronec ker and pleth ysm pr oblems . . . . . . . . . . . . . . . . . . . . . . . . 113 I I In v arian t theory with a view tow ards GCT By Milind Sohoni 116 21 Finite Groups 117 21.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 21.2 The ﬁn ite group action . . . . . . . . . . . . . . . . . . . . . . 118 21.3 The Sy m metric Group . . . . . . . . . . . . . . . . . . . . . . 122 22 The Group S L n 124 22.1 The Canonical Represen tation . . . . . . . . . . . . . . . . . . 124 22.2 The Diagonal Represen tation . . . . . . . . . . . . . . . . . . 125 22.3 Other Repr esentat ions . . . . . . . . . . . . . . . . . . . . . . 128 22.4 F ull Reducibilit y . . . . . . . . . . . . . . . . . . . . . . . . . 130 5 23 Inv ariant Theory 131 23.1 Algebraic Group s and aﬃne actions . . . . . . . . . . . . . . 131 23.2 Orbits and In v ariant s . . . . . . . . . . . . . . . . . . . . . . . 132 23.3 The Nagata Hyp othesis . . . . . . . . . . . . . . . . . . . . . 136 24 Orbit-closures 139 25 T ori in S L n 142 26 The Null-cone and the Destabilizing ﬂag 147 26.1 Characters and the h alf-space criterion . . . . . . . . . . . . . 147 26.2 The destabilizing ﬂag . . . . . . . . . . . . . . . . . . . . . . . 149 27 Stability 154 6 7 P art I The basic structure of GCT By Ketan D. Mulm uley 8 Chapter 1 Ov erview Scrib e: Joshua A. Gro c how Goal: An o v erview of GCT. The purp ose of this course is to giv e an introdu ctio n to Geometric C om- plexit y Theory (GCT), wh ic h is an approac h to pr o ving P 6 = NP via al- gebraic geometry a nd represen tation theory . A basic p lan of this app roac h is describ ed in [GCTﬂ ip 1, GCTﬂip2]. I t is partially implemen ted in a se- ries of articles [GCT1]-[GCT11]. The p aper [GCTconf] is a conference an- nouncemen t of GCT . Th e p ap er [Ml] giv es an unconditional lo w er b ound in a PRAM mo del without bit op erations based on elemen tary algebraic geometry , and wa s a starting p oin t f or the GCT in v estigation via algebraic geometry . The only mathematical prerequisites for this course are a basic kno wl- edge of abstract algebra (groups, ring, ﬁelds, etc.) and a kno wledge of computational complexit y . In the ﬁ rst mon th we plan to co v er the repr esen- tation theory of ﬁnite groups, the sym m etric group S n , and GL n ( C ), and enough algebraic geometry s o that in the remaining lectures we ca n co ve r basic GCT. Most of the bac kground r esults will only b e sk etc hed or omitted. This lect ure u ses sligh tly more algebraic geometry and represen tation theory than the reader is assumed to kno w in ord er to giv e a more complete picture of GCT. As the cour s e cont inues, we will co v er this material. 1.1 Outline Here is an outli ne of the GCT app roac h. Consider the P vs. NP questi on in c haracteristic 0; i.e ., o v er in tegers. So b it op erations are not allo w ed, and 9 basic op erations on in tegers are considered to tak e constan t time. F or a sim- ilar approac h in nonzero c haracteristic (c haracteristic 2 b eing the cla ssical case fr om a computational complexit y point of view), see GCT 11. The basic principle of GCT is the called the ﬂip [GCTﬂip1]. It “reduces” (in essence, not formally) the lo wer b ound pr oblems su ch as P vs. NP in c h aract eristic 0 to upp er b ound problems: sho wing that certain decision problems in a lgebraic geometry and represen tation theory b elong to P . Eac h of these decision problems is of the form: is a gi v en (nonnegat iv e) structural constan t asso ciated to some algebro-geo metric or represen tation theoretic ob ject nonzero? T h is is akin to the decision problem: giv en a matrix, is its permanent nonzero? (W e kno w ho w to solve this partic ular problem in p olynomial time via reduction to the p erfect matc hing problem.) Next, the preceding upp er boun d pr oblems are reduced to pur e ly math- ematical p ositivit y h yp otheses [GCT6]. Th e goal is to sh ow that these and other auxilliary st ructural constan ts ha v e p ositiv e formulae . By a p ositiv e form ula w e mean a formula that do es not in v olv e any alternating signs like the usual p ositiv e formula for the p ermanent; in con trast the usu al fo rmula for the determinant in v olv es a lternating signs. Finally , these p ositivit y hypotheses are “reduced” to c onjectures i n the theory of quan tum groups [GCT6, GCT 7, GCT8, GCT10] in timately related to the Riemann hypothesis o v er ﬁnite ﬁelds pro ved in [Dl2], and the rela ted w orks [BBD, KL2, Lu2]. A pictorial su m mary of the GCT approac h is sho wn in Figure 1.1, w here the arro w s repr esen t redu cti ons, r ather than implications. T o recap: w e m ov e from a negativ e hyp othesis in complexit y theory (that there do es not exi st a p olynomial time algorithm for an NP -complete problem) to a p ositiv e hyp otheses in complexit y theory (that there exist p olynomial-ti me algorithms for certain decision pr oblems) to p ositiv e hy- p otheses in mathematics (that certain structural constan ts ha v e p ositiv e form ulae) to conjectures on quan tum groups relate d to the Riemann hy- p othesis o v er ﬁnite ﬁelds, the related works and their p ossible extensions. The ﬁrst redu ction here is the ﬂip : we redu ce a question ab out lo wer b oun ds, whic h are notoriously diﬃcult, to the one ab out upp er b ounds, which we ha v e a m uc h b etter handle on. Th is ﬂip from n egativ e to p ositiv e is a lready presen t in G¨ odel’s w ork: to sho w something is imp ossible it suﬃces to sho w that something else is p ossible. T h is was one of the motiv ations for the GCT approac h. The G¨ odelian ﬂip w ould not w ork for the P vs. NP problem b e- cause it relativizes. W e can think of GCT as a form of nonr elativizable (and non-naturalizable, if reader kno ws w h at that means) diagonaliz ation. In summary , this approac h v ery roughly “reduces” the lo wer b ound p rob- 10 P vs. NP c h ar. 0 Flip = ⇒ Decision problems in alg. geo m. & rep. th y . = ⇒ Sho w certain constan ts in alg. geom. and repr. theory h av e p ositiv e formulae Lo w er b ounds (Neg. h yp othesis in complexit y thy .) Upp er b ounds (P os. h yp otheses in complexit y thy .) P os. h yp otheses in mathematics = ⇒ Conjectures on quan tum group s related to RH o v er ﬁnite ﬁ elds Figure 1.1: The b asic approac h of GCT 11 lems suc h a s P vs. NP in charact eristic ze ro to as-y et-unpro v ed quan tum- group-conjectures rel ated to th e Riemann Hypothesis o v er ﬁnite ﬁ elds. As with the cl assical RH, there is exp erimenta l evi dence to su gg est these con- jectures h old – which indirectly suggests that certain generalizatio ns of th e Riemann h yp othesis o v er ﬁnite ﬁelds also hold – and there are hint s on h o w the problem migh t b e attac k ed. S ee [GCTﬂ ip1 , GCT6, GCT7, GCT8] for a more d eta iled exp osition. 1.2 The G¨ odelian Flip W e no w re-visit G¨ odel’s original ﬂip in mo dern languag e to get the ﬂa v or of the GCT ﬂip. G¨ odel set out to answ er the question: Q: Is truth p ro v able? But wh at “truth” and “pro v able” means here is not so obvious a priori . W e start b y setting the s tage: in an y mathematical theory , we ha v e the syntax (i.e. the language used) and the seman tics (the domain of discussion). In this case, w e ha v e: Synt ax (language) Seman tics (domain) First order logic ( ∀ , ∃ , ¬ , ∨ , ∧ , . . . ) Constan ts 0,1 V ariables x, y , z , . . . Basic Pr ed icates > , < , = F unctions +, − , × ,exp onen tiatio n Axioms Axioms of the natural num b ers N Univ erse: N A sen tence is a v alid f orm ula with all v ariables quant iﬁed, and by a truth w e mean a sent ence that is true in the domain. By a pro of we mean a v alid d eduction based on sta ndard r ules of inference and the axioms o f th e domain, w h ose ﬁnal result is the d esired statemen t. Hilb ert’s program ask ed for an algorithm that, giv en a sen tence in n um- b er theory , decides whether it is true or f alse. A sp ecial case of this is Hilb ert’s 10th problem, whic h asked for an alg orithm to decide whether a Diophan tine equation (equation with only inte ger co eﬃcien ts) h as a nonzero in teger solution. G¨ odel show ed that Hilb ert’s general program w as not 12 ac h iev able. The ten th problem remained un resolv ed u n til 197 0, at whic h p oin t Matiy asevic h sho w ed its imp ossibilit y as well. Here is the main idea of G¨ odel’s pro of, re-cast in mo dern language. F or a T uring Mac hine M , whether the empt y string ε is in the language L ( M ) recognized b y M is un decidable. T he idea is to reduce a question of the form ε ∈ L ( M ) to a qu esti on in num b er theory . If there w ere an algorithm for deciding th e truth of n umber-theoretic statemen ts, it w ould giv e an algorithm for the ab o ve T uring mac hine problem, w hic h we know do es not exist. The basic idea of the reduction is similar to the one in Cook’s pro of that SA T is NP -complete . Namely , ε ∈ L ( M ) iﬀ there is a v alid computation of M which acce pts ε . Usin g Co ok’s idea, we can u se this to get a Bo olea n form ula: ∃ m ∃ a v alid computation of M with conﬁgurations of size ≤ m s.t. the computation accepts ε. Then we u se G¨ odel num b ering – whic h assigns a unique num b er to eac h sen tence in num b er theory – to trans late this formula to a sen tence in num b er theory . The d eta ils of this should b e familiar. The key p oint here is: to s h o w that truth is un decidable in num b er theory (a negativ e statemen t), we sho w that there exists a computable reduction from ε ? ∈ L ( M ) to num b er theory (a p ositiv e statemen t). This is the essence of the G¨ odelian ﬂip, which is analogous to – and in f ac t was the original motiv ation for – the GCT ﬂip. 1.3 More details of the GCT approac h T o b egin with, GCT asso ciates to eac h complexit y class such as P and NP a pro jectiv e alge braic v ariet y χ P , χ N P , etc. [GCT1]. In fact, it a sso ciates a family of v arietie s χ N P ( n, m ): one for eac h input length n and circuit size m , but for simplicit y w e suppress this here. The languages L in the asso cia ted complexit y class will b e p oin ts on these v arieties, and the set of suc h p oin ts is dense in the v ariet y . These v arietie s are thus called class varieties . T o sho w that NP * P in c h aract eristic zero, it suﬃces to sho w that χ N P cannot b e imb edded in χ P . These c lass v arieties are in f act G -v arieties. That is, they hav e an action of the group G = GL n ( C ) on them. This action induces an actio n on the homogeneous co ordinate ring of the v ariet y , giv en b y ( σ f )( x ) = f ( σ − 1 x ) for all σ ∈ G . Th us t he coordinate rings R P and R N P of χ P and χ N P are 13 G -alge bras, i.e., algebras with G -acti on. Their degree d -comp onen ts R P ( d ) and R N P ( d ) are th us ﬁnite dimensional G -represen tations. F or the sak e of contradic tion, supp ose NP ⊆ P in c haracteristic 0. Then there m ust b e an em b edding of χ N P in to χ P as a G -subv ariet y , wh ic h in turn giv es rise (by standard algebraic geometry argumen ts) to a sur- jection R P ։ R N P of the co ordinate rings. This implies (b y standard represen tation-theoreti c a rguments) that R N P ( d ) can b e em b edded as a G - sub-representa tion of R P ( d ). The follo win g diagram summarizes the i mpli- cations. complexit y classes class v arieties co ord inate rings represen tations of GL n ( C ) N P  _   / / /o /o /o /o /o /o /o /o /o /o χ N P  _   / / /o /o /o /o /o /o /o /o /o /o R N P / / /o /o /o /o /o /o /o /o /o R N P ( d )  _   P / / /o /o /o /o /o /o /o /o /o /o /o χ P / / /o /o /o /o /o /o /o /o /o /o /o R P O O O O / / /o /o /o /o /o /o /o /o /o /o R P ( d ) W eyl’s theorem–that all ﬁn ite- dimensional r ep r esen tations of G = GL n ( C ) are c ompletely reducible, i.e. can b e writt en as a direct sum o f irreducible represen tations–implies that b oth R N P ( d ) and R P ( d ) can b e w ritten as d i- rect sum s of irreducible G -represen tations. An obstruction [GCT2] of degree d is deﬁned to b e an irreducible G -representat ion o ccuring (as a subrepr e- sen tation) in R N P ( d ) b ut n ot in R P ( d ). Its existence implies that R N P ( d ) cannot b e em b edded as a subrepr esentat ion of R P ( d ), and hence, χ N P can- not b e em b edded in χ P as a G -sub v ariet y; a con tradiction. W e act ually ha ve a family of v arieties χ N P ( n, m ): one for eac h input length n and circuit size m . Thus if an obs tru ctio n of some degree exists for all n → ∞ , assuming m = n log n (sa y), th en NP 6 = P in c haracteristic zero. Conjecture 1.1. [GCTﬂip1] Ther e is a p olynom ial-time algorithm for c on- structing such obstructio ns. This is the GCT ﬂip : to show that no p olynomial-time algorithm exists for an NP - complete problem, we h ope to sho w that there is a p olynomial time algorithm for ﬁnd ing obstru ctio ns. This task then is fur ther reduced to ﬁnding p olynomial time algo rithms for other decision problems in alge braic geometry and representa tion theory . Mere existence of an obstru ctio n for all n wo uld actually s uﬃce h ere. F or this, it suﬃces to sho w that there is an al gorithm wh ich, g iv en n , outputs 14 an obstru ctio n sho wing that χ N P ( n, m ) cannot b e im b edded in χ P ( n, m ), when m = n log n . But the c onjecture is not ju st that there is an algorithm, but that there is a p olynomial-time algorithm. The basic principle here is that th e complexit y of the pr oof of existence of an ob ject (in this case, a n obstru ction) is v ery closed tied to the computa- tional complexit y o f ﬁnding that ob ject, and hence, tec hn iques un derneath an easy (i.e. p olynomial time) time algorithm for deciding existence ma y yield an easy ( i.e. feasible) pro of of existence. T his is supp orted b y m uch anecdotal evidence: • An obstruction to planar embedd ing (a f orbidden Kur oto wski minor) can b e found in p olynomial, in fact, linear time by v arian ts of the usual pla narit y testing a lgorithms, and the un derlying tec h niques, in retrosp ect, yiel d an algorithmic proof of Kuroto wski’s th eo rem that ev ery n onplanar graph con tains a forbidd en minor. • Hall’s marriage theorem, which charact erizes the existence of p er- fect matc hings, in retrosp ect, follo ws from the tec hniques und erlying p olynomial-ti me algorithms for ﬁn ding p erfect matc hings. • T he pro of that a graph is Eulerian iﬀ all vertice s h a v e eve n degree is, essen tially , a p olynomial- time algorithm for ﬁndin g an Eulerian circuit. • In con trast, w e kno w of no Hall -t yp e theorem for Hamil tonians paths, essen tially , b ecause ﬁnd ing suc h a p ath is computationally diﬃcult ( NP -complete ). Analogously the goal is to ﬁnd a polynomial time algorithm for deciding if th ere exists an obstru ctio n for giv en n and m , and then use the un derlying tec hn iques to sh o w that an obstruction al wa ys exists for ev ery large enough n if m = n log n . The m ain mathematical work in GCT tak es steps to w ards this goal. 15 Chapter 2 Represen tation theory of reductiv e groups Scrib e: Paolo Co deno tti Goal: Basic notions in repr esen tation theory . R efer enc es: [FH, F] In this lecture w e review the basic represen tation theory of red u ctiv e groups as needed in this course. Most of the pro ofs will b e omitted, or just sk etc h ed . F or complete p roofs, see the b o oks by F ulton and Harris, and F ulton [FH , F]. The underlying ﬁeld throughout this course is C . 2.1 Basics of Re presen tation T heory 2.1.1 Deﬁnitions Deﬁnition 2.1. A represen tation of a gr oup G , als o c al le d a G -mo dule, is a ve ctor sp ac e V with an asso c iate d homom orphism ρ : G → GL ( V ) . We wil l r efe r to a r e pr esentation by V . The map ρ ind uces a natural action of G on V , deﬁ n ed by g · v = ( ρ ( g ))( v ). Deﬁnition 2.2. A map ϕ : V → W is G - equiv arian t if the fol lowing dia- gr am c ommutes: V ϕ − − − − → W   y g   y g V ϕ − − − − → W 16 That is, if ϕ ( g · v ) = g · ϕ ( v ) . A G -e q uivariant map is also c al le d G -invariant or a G -homo morphism. Deﬁnition 2.3 . A subsp ac e W ⊆ V is said to b e a s ubrepresen tation , or a G -submo dule of a r epr esentation V over a gr oup G if W is G - equiv arian t , that is if g · w ∈ W for al l w ∈ W . Deﬁnition 2.4. A r epr esentation V of a gr oup G is said to b e irr ed u cible if i t has no pr op er non-zer o G -subr epr esentations. Deﬁnition 2.5. A gr oup G is c al le d reductiv e if eve ry ﬁnite dimensional r epr esentation V of G is a dir e ct sum of irr e ducible r epr e sentation. Here are some examples of reductiv e groups: • ﬁ nite groups; • the n -dimensional torus ( C ∗ ) n ; • linear groups: – th e general linear group GL n ( C ), – th e sp ecial linear group S L n ( C ), – th e orthogonal group O n ( C ) (linear transf ormatio ns that preserve a symmetric form), – an d the symplectic group S p n ( C ) (linear transform ati ons that preserv e a sk ew symmetric form); • Exceptional Lie Group s Their r eductivit y is a n on trivial fact. It w ill b e pr o ve d later in this lecture for ﬁnite groups, and the general and s p eci al linear groups. In some sense, the list ab o v e is complete : all reductiv e groups can b e constructed b y basic op erations from the comp onen ts wh ic h are either in this list or are related to them in a simple wa y . 2.1.2 New represen t ations from old Giv en represen tations V and W of a group G , we can construct new repr e- sen tations in seve ral wa ys, some of wh ich are describ ed b elo w. • T ensor pro duct: V ⊗ W . g · ( v ⊗ w ) = ( g · v ) ⊗ ( g · w ). • Direct sum: V ⊕ W . 17 • S ymmetric tensor repr esentat ion: The sub space S y m n ( V ) ⊂ V ⊗ · · · ⊗ V spanned by elemen ts of the f orm X σ ( v 1 ⊗ · · · ⊗ v n ) · σ = X σ v σ (1) ⊗ · · · v σ ( n ) , where σ ranges o v er all p erm utations in the symmetric group S n . • Exterior tensor represen tation: The sub space Λ n ( V ) ⊂ V ⊗ · · · ⊗ V spanned by elemen ts of the form X σ sg n ( σ )( v 1 ⊗ · · · ⊗ v n ) · σ = X σ sg n ( σ ) v σ (1) ⊗ · · · v σ ( n ) . • Let V and W b e represen tations, then Hom( V , W ) is also a rep r esen- tation, where g · ϕ is d eﬁned so that the follo wing diagram comm u tes: V ϕ − − − − → W   y g   y g V g · ϕ − − − − → W More precisely , ( g · ϕ )( v ) = g · ( ϕ ( g − 1 · v )) . • In particular, V ∗ : V → C is a represen tation, and is calle d the dual r epr esentation . • Let G b e a ﬁnite group. Let S b e a ﬁnite G -set (that is, a ﬁnite set with an asso ciated action of G o n its element s). W e construct a v ector space o v er an y ﬁeld K (w e will b e m ostl y concerned with the case K = C ), with a basis vect or asso ciate d to eac h elemen t in S . More sp eciﬁcally , consider the set K [ S ] of formal sum s P s ∈ S α s e s , where α s ∈ K , and e s is a ve ctor associated with S ∈ s . Note that this set has a v ector s p ace structure ov er K , and there is a natural indu ced action of G on K [ S ], d eﬁ n ed b y: g · X s ∈ S α s e s = X s ∈ S α s e g · s . This action giv es rise to a r epresen tation of G . • In particular, G is a G -set under the act ion of le ft m ultiplication. The represen tation w e obtain in the manner describ ed ab ov e from this G - set is called the r e gular r epr esentation . 18 2.2 Reductivit y of ﬁnite groups Prop osition 2.1. L e t G b e a ﬁnite gr oup. If W is a subr epr esentation of a r epr esentation V , then ther e exists a r epr esentation W ⊥ s.t. V = W ⊕ W ⊥ . Pr o of. Cho ose an y Hermitian form H 0 of V , and constru ct a new Hermitian form H deﬁned as: H ( v , w ) = X g ∈ G H o ( g · v , g · w ) . Av eraging is a useful tric k that is us ed very often in repr ese nta tion th eory , b ecause it ensures G -in v ariance. In f ac t, H is G -in v arian t, that is, H ( v , w ) = X g ∈ G H o ( g · v , g · w ) = H ( h · v , h · w ) Let W ⊥ b e the p erp end icular complemen t to W with resp ect to the Her- mitian form H . Then W ⊥ is also G -in v arian t, and therefore it is a G - submo dule. Corollary 2.1. Every r epr esentation of a ﬁnite gr oup is a dir e ct sum of irr e ducible r epr esentations. Lemma 2.1. (Schur) If V and W ar e irr e ducible r e pr esentations over C , and ϕ : V → W is a homo morphism (i.e. a G -invariant m ap), then: 1. E ither ϕ is an isomorphism or ϕ = 0 . 2. If V = W , ϕ = λI for some λ ∈ C . Pr o of. 1. S in ce Ker( ϕ ), and Im ϕ are G -submo dules, either Im( ϕ ) = V or Im( ϕ ) = 0. 2. Let ϕ : V → V . Since C algebraicall y closed, there exists an eigen v alue λ of ϕ . Lo ok at the map ϕ − λI : V → V . By (1), ϕ − λI = 0 (it can’t b e an isomorph ism b eca use something maps to 0). So ϕ = λI . Corollary 2.2. Eve ry r epr esentation is a unique dir e ct sum of irr e ducible r epr esentations. Mor e pr e cisely, giv e n two de c omp osition s into irr e ducible r epr esentations, V = M V a i i = M W b j j , ther e is a one to one c orr esp ondenc e b etwe en the V i ’s and W j ’s, and the multiplicities c orr esp ond. Pr o of. exercise (follo ws fr om Sc hur’s lemma). 19 gu u R gR Figure 2.1: Example of a left Haar measure f or the circ le ( U 1 ( C )). Left action by a group element g on a small region R aro und u does not c ha nge the area. 2.3 Compact Groups and GL n ( C ) are reductiv e No w we prov e reductivit y of compact groups. 2.3.1 Compact groups Examples of compact groups: • U n ( C ) ⊆ GL n ( C ), the unitary groups (all ro ws are normal and orthog- onal). • S U n ( C ) ⊆ S L n ( C ), the sp ecial unitary group . Giv en a co mpact group, a left-invariant Haar me asur e is a measure that is inv ariant under the left action of the group . In other wo rds, mult iplication b y a group elemen t d oes not c hange the area of a small region (i.e., th e group action is an isometry , see ﬁgure 2.1). Theorem 2.1. Comp act gr oups ar e r e ductive Pr o of. W e use the av eraging tric k again. In fact the p roof is the same as in the ca se of ﬁ nite groups, usin g inte gration instead of su mmati on for the a veragi ng tric k. Let H 0 b e an y Hermitian form on V. Then deﬁne H as: H ( v , w ) = Z G H ( g v , g w ) dG 20 where dG is a left-in v arian t Haar measure. Note that H is G -inv ariant. Let W ⊥ b e the p erp endicular complement to W . Then W ⊥ is G -inv ariant. Hence it is a G -subm odu le. The same pro of as b efore then giv es us Sc hur’s lemma for compact groups, from whic h follo ws: Theorem 2.2. If G is c omp act, then every ﬁnite dimensional r epr esentation of G is a uniq ue dir e ct sum of irr e ducible r epr esentations. 2.3.2 W eyl’s unitary trick and GL n ( C ) Theorem 2.3. (Weyl) GL n ( C ) is r e ductive Pr o of. (general idea) Let V b e a representa tion of GL n ( C ). Then GL n ( C ) acts on V : GL n ( C ) ֒ → V . But U n ( C ) is a subgroup of GL n ( C ). Th erefore w e ha v e a n i ndu ced action of U n ( C ) on V , and w e can lo ok at V as a represen tation of U n ( C ). As a represen tation of U n ( C ), V breaks in to irreducible representat ions of U n ( C ) b y the theorem ab o v e. T o s u mmarize, w e hav e: U n ( C ) ⊆ GL n ( C ) ֒ → V = ⊕ i V i , where the V i ’s are irreducible r epresen tations of U n ( C ). W eyl’s unitary tric k uses Lie algebra to sho w that ev ery ﬁnite dimensional represen tation of U n ( C ) is also a r ep r esen tation of GL n ( C ), a nd irreducible represent ations of U n ( C ) corresp ond to irred u cible represent ations of GL n ( C ). Hence eac h V i ab o ve is an irr educible represen tation of GL n ( C ). Once w e kno w these groups are r ed u ctiv e, the goal is to construct and classify their irreducible ﬁn ite dimensional represent ations. This w ill b e done in the next lectures: Sp ec h t mo dules for S n , and W eyl m odu les for GL n ( C ). 21 Chapter 3 Represen tation theory of reductiv e groups (con t) Scrib e: Paolo Co deno tti Goal: Basic representa tion theory , contin ued from the last lecture. In this lecture w e con tin ue our introdu ctio n to repr esentat ion theory . Again we r efer the reader to the b o ok by F ulton a nd Ha rris for full d eta ils [FH]. Let G b e a ﬁnite group, and V a ﬁn ite -dimensional G -represen tation giv en b y a homomorph ism ρ : G → GL ( V ). W e deﬁ n e the char acter of the represen tation V (denoted χ V ) by χ V ( g ) = T r ( ρ ( g )). Since T r ( A − 1 B A ) = T r ( B ), χ V ( hgh − 1 ) = χ V ( g ). This means c h arac- ters are constan t on conjugacy classes (sets of the f orm { hgh − 1 | h ∈ G } , for an y g ∈ G ). W e call such fun cti ons class functions . Our goal for this lecture is to prov e the follo wing t wo facts: Goal 1 A ﬁnite dimensional represen tation is completely determined b y its c h aract er. Goal 2 The space of class functions is sp anned b y the charact ers of irred u cible represen tations. I n fact, these charac ters form an orthonormal basis of this sp ace . First, we pro v e some u seful lemmas ab out charact ers. Lemma 3.1. χ V ⊕ W = χ V + χ W Pr o of. Let g ∈ G , and let ρ, σ b e homomorphisms from G into V and W , resp ectiv ely . Let λ 1 , . . . , λ r b e the eigen v alues of ρ ( g ), and µ 1 , . . . , µ s the 22 eigen v alues of σ ( g ). Then ( ρ ⊕ σ )( g ) = ( ρ ( g ) , σ ( g )), so the eigen v alues of ( ρ ⊕ σ )( g ) are just the eigen v alues of ρ ( g ) tog ether with the eigen v alues of σ ( g ). Then χ V ( g ) = P i λ i , χ W ( g ) = P i µ i , and χ V ⊕ W = P i λ i + P i µ i . Lemma 3.2. χ V ⊗ W = χ V χ W Pr o of. Let g ∈ G , and let ρ, σ b e homomorphisms into V and W , resp ec- tiv ely . Let λ 1 , . . . , λ r b e the e igen v alues of ρ ( g ), and µ 1 , . . . , µ s the eigen v al- ues of σ ( g ). Then ( ρ ⊗ σ )( g ) is the Kronec k er pr odu ct of the matrices ρ ( g ) and σ ( g ). So its eigen v alues are all λ i µ j where 1 ≤ i ≤ r , 1 ≤ j ≤ s . Then, T r(( ρ ⊗ σ )( g )) = P i,j λ i µ j = ( P i λ i )  P j µ j  , whic h is equal to T r( ρ ( g ))T r ( σ ( g )). 3.1 Pro jection F orm ula In this sectio n, we deriv e a pro jection form ula needed for Goal 1 that allo ws us to determine the m ultiplicit y of an ir reducible represent ation in another represen tation. Give n a G -modu le V , let V G = { v |∀ g ∈ G, g · v = v } . W e will call these elemen ts G -inv ariant. Let φ = 1 | G | X g ∈ G g ∈ End( V ) , (3.1) where eac h g , via ρ is considered an elemen t o f E n d( V ). Lemma 3.3. The map φ : V → V is a G -homomorph ism; i.e., φ ∈ H om G ( V , V ) = ( H om ( V , V )) G . Pr o of. The set End( V ) is a G -mo dule, as w e sa w in last class, via the fol- lo w in g commuta tiv e d iag ram: for any π ∈ En d( V ), and h ∈ G : V π − − − − → V   y h   y h V h · π − − − − → V . Therefore π ∈ H om G ( V , V ) (i.e., π is a G -equiv ariant m orphism) iﬀ h · π = π for all h ∈ G . When φ is deﬁ n ed as in equ ation (3.1) ab o v e, h · φ = 1 | G | X g hg h − 1 = 1 | G | X g g = φ. 23 Th us h · φ = φ, ∀ h ∈ G, and φ : V → V is a G -equiv arian t morphism, i.e. φ ∈ H om G ( V , V ). Lemma 3.4. The map φ is a G -e q u ivarian t pr oje ction of V onto V G Pr o of. F or every w ∈ W , let v = φ ( w ) = 1 | G | X g ∈ G g · w . Then h · v = h · φ ( w ) = 1 | G | X g ∈ G hg · w = v , for an y h ∈ G. So v ∈ V G . That is, Im ( φ ) ⊆ V G . But if v ∈ V G , then φ ( v ) = 1 | G | X g ∈ G g · v = 1 | G | | G | v = v . So V G ⊆ Im( φ ), and φ is the iden tity on V G . This means that φ is the pro j ect ion on to V G . Lemma 3.5. dim( V G ) = 1 | G | X g ∈ G χ V ( g ) . Pr o of. W e ha v e: dim ( V G ) = T r( φ ), b ecause φ i s a pro jection ( φ = φ | V G ⊕ φ | K e r ( φ ) ). Also, T r( φ ) = 1 | G | X g ∈ G T r V ( g ) = 1 | G | X g ∈ G χ V ( g ) . This giv es us a form ula for the m ultiplicit y of the trivial representa tion (i.e., dim ( V G )) inside V . Lemma 3.6. L et V , W b e G -r epr esentations. If V is irr e ducible, di m ( Hom G ( V , W )) is the multiplicity of V inside W . If W is irr e ducible, d im ( Hom G ( V , W )) is the multiplicity of W inside V . Pr o of. By S ch u r ’s Lemma. 24 Let C class ( G ) b e the space of class fu nctions on ( G ), and let ( α, β ) = 1 | G | P g α ( g ) β ( g ) b e the Hermitian form on C class Lemma 3.7. If V and W ar e irr e ducible G r epr esentations, th en ( χ V , χ W ) = 1 | G | X g ∈ G χ V ( g ) χ W ( g ) = ( 1 i f V ∼ = W 0 i f V ≇ W. (3.2) Pr o of. Since Hom( V , W ) ∼ = V ∗ ⊗ W , χ Hom ( V ,W ) = χ V ∗ χ W = χ V χ W . Now the r esu lt follo ws fr om Lemmas 3.5 and 3.6. Lemma 3.8. The char acters of the irr e ducible r e pr esentations form an or- thonorm al set. Pr o of. F ollo ws from Lemma 3.7. If V , W are irredu cible, then h χ V , χ W i is 0 if V 6 = W and 1 otherwise. This imp lie s that: Theorem 3.1 (Goal 1) . A r epr esentation i s determine d c ompletely by its char acter. Pr o of. Let V = L i V ⊕ a i i . So χ V = P i a i χ V i , and a i = ( χ V , χ V i ). T h is giv es u s a form u la for the m ultiplicit y of an irreducible represen tation in another represen tation, solely in terms of their c haracters. Therefore, a represen tation is completely determined by its charac ter. 3.2 The c haracters of irreducible represen tations form a b asis In this section, we address Goal 2. Let R b e the regular representa tion of G , V an irredu cible repr esentat ion of G . Lemma 3.9. R = M V E nd ( V , V ) , wher e V r anges o ver al l irr e ducible r epr esentations of G . 25 Pr o of. χ R ( g ) is 0 if g is not the id entit y and | G | otherwise. ( χ R , χ V ) = 1 | G | X g ∈ G χ R ( g ) χ V ( g ) = 1 | G | | G | χ V ( e ) = χ V ( e ) = d im ( V ) Let α : G → C . F or any G -mo dule V , let φ α,V = P g α ( g ) g : V → V Exercise 3.1. φ α,V is G e quivariant (i.e. a G -homomorphism ) iﬀ α is a class fu nction. Prop osition 3.1. Supp ose α : G → C is a cla ss function, and ( α, χ V ) = 0 for a l l irr e ducible r epr esentations V . Then α is identic al ly 0 . Pr o of. If V is irr educible, then, b y Sc h ur’s lemma, since φ α,V is a G -homomorphism, and V is irreducible, φ α,V = λ Id, where λ = 1 n T r( φ α,V ), n = dim ( V ). W e ha v e: λ = 1 n X g α ( g ) χ V ( g ) = 1 n | G | ( α, χ V ∗ ) . No w V is irredu cible iﬀ V ∗ is irreducible. So λ = 1 n | G | 0 = 0. Therefore, φ α,V = 0 for an y irredu cible representat ion, and hence for an y representa- tion. No w let V b e the regular represen tation. Since g as endomorphisms of V are linearly indep end ent, φ α,V = 0 imp lie s that α ( g ) = 0. Theorem 3.2. Char acters form an orthonorm al b asis for the sp ac e of class functions. Pr o of. F ollo ws from Prop osition 3.1, and Lemma 3.8 If V = L i V ⊕ a i i , and π i : V → V ⊕ a i i is the pro jection op erato r. W e ha ve a formula π = 1 | G | P g g for the trivial representa tion. Analogo usly: Exercise 3.2. π i = dimV i | G | P g χ V i ( g ) g . 26 3.3 Extending to Inﬁnite Compact Groups In this section, we extend t he preceding resu lts to inﬁnite compact groups. W e must tak e some facts as give n, s ince these theorems are m uch more complicate d t han those for ﬁn ite groups. Consider compact G , speciﬁcally U n ( C ), the unitary subgroup of G ( C ). U 1 ( C ) is the circle group. Since U 1 ( C ) is ab elian, all its represen tations are one-dimensional. Since the group G is inﬁnite, w e can n o longer sum o v er it. The idea is to replace the s um 1 | G | P g f ( g ) in the previous sett ing with R G f ( g ) dµ , where µ is a left-in v ariant Haar measure on G . I n this fashion, w e can deriv e analogues of the preceding resu lts for compact groups. W e need to normalize, so we set R G dµ = 1 . Let ρ : G → GL ( V ), w here V is a ﬁnite dimensional G -represen tation. Let χ V ( g ) = T r( ρ ( g )). Let V = L i V a i i b e the complete decomp ositio n of V in to irreducible repr esentat ions. W e can agai n create a pro jection op erator π : V → V G , by letting π = R G ρ ( g ) dµ . Lemma 3.10. We have: dim ( V G ) = Z G χ V ( g ) dµ. Pr o of. This r esu lt is analogous to Lemma 3.5 for ﬁnite group s. F or class f unctions α, β , deﬁne an inner pro duct ( α, β ) = Z G α ( g ) β ( g ) dµ. Lemma 3.10 applied to Hom G ( V , W ) giv es ( χ V , χ W ) = Z G χ V χ W dµ = d im (Hom G ( V , W )) . Lemma 3.11. If V , W ar e irr e ducible, ( χ V , χ W ) = 1 if V and W ar e iso- morphic, and ( χ V , χ W ) = 0 o therwise. Pr o of. This r esu lt is analogous to Lemma 3.7 for ﬁnite group s. Lemma 3.12. The i rr e ducible r epr esentations ar e ortho normal, j ust as in L emma 3 .8 in the c ase of ﬁnite gr oups. 27 If V is reducible, V = L i V ⊕ a i i , then a i = ( χ V , χ V i ) = Z G χ V χ V i dµ. Hence Theorem 3.3. A ﬁnite dimensional r epr esentation is c ompletely determine d by its char acter. This ac hiev es Go al 1 for compact group s. Goal 2 is m uc h harder: Theorem 3.4 (Pe ter-W eyl Theorem) . (1) The char acters of the irr e ducible r epr esentations of G sp an a dense subset of the sp ac e of c ontinuous class functions. (2) The c o or dinate functions of al l irr e ducib le matrix r e pr esentations of G sp an a dense su b set of al l c ontinuous functions o n G . By a co ordinate fu nction of a represen tation ρ : G → GL ( V ), w e mean the function on G corresp onding to a ﬁxed en try of the matrix form of ρ ( g ). F or G = U 1 ( C ), (2) giv es the F ourier series exp ans ion o n th e circle . Hence, the Pet er-W eyl theorem constitutes a far reac hing generaliza tion of the h armonic analyis from the circle to general U n ( C ). 28 Chapter 4 Represen tations of the symmetric group Scrib e: Sour a v Chak rab ort y Goal: T o determine the irreducible representat ions of th e Symm etric group S n and their c haracters. Reference: [FH, F] Recall Let G b e a r eductiv e group. Then 1. Ev ery ﬁn ite dimensional represent ation of G is c ompletely reducible, that is, can b e w ritte n as a direct su m of irreducible represen tations. 2. Ev ery irreducible r epresen tation is d ete rmined by its c haracter. Examples of reductiv e groups: • C on tinuous: algebraic torus ( C ∗ ) m , ge neral linear group GL n ( C ), sp e- cial linear group S l n ( C ), symplectic group S p n ( C ), o rthogonal group O n ( C ). • Finite: alternating group A n , symm etric group S n , Gl n ( F p ), simple li e groups of ﬁnite t yp e. 29 4.1 Represen tations and c haracters of S n The num b er of irreducible representa tions of S n is the same as the the n umber of conjugacy classes in S n since the irreducible c haracters form a basis of the space of class functions. Eac h p erm utation can b e w ritte n uniquely as a pro du ct of disjoin t cycle s. T he collec tion of le ngths of the cycles in a p erm utation is called the cycle type of the p ermutati on. So a cycle t yp e of a p erm utation on n elemen ts is a p artit ion of n . And in S n eac h conjugacy class is d ete rmined by the cycle t yp e, whic h , in turn, is determined b y the partition of n . So the n umber of conjugacy class is same as the num b er of partitions of n . Hence: Num b er of irreducible repr esen tations of S n = Numb er of partitions of n (4.1) Let λ = { λ 1 ≥ λ 2 ≥ . . . } b e a partition o f n ; i.e ., the size | λ | = P λ i is n . Th e Y oung d iagram corresp ond in g to λ is a table sh o wn in Fig ure 1. It is like an inv erted staircase. Th e top row has λ 1 b o x es, the second ro w has λ 2 b o x es and s o on. There are exactly n b oxes. row 1 row 5 row 6 row 2 row 3 row 4 Figure 4.1: Ro w i h as λ i n umber of b oxe s F or a giv en p artiti on λ , we w an t to construct an irreducible rep r esen ta- tion S λ , called the Sp ec h t-mo dule of S n for the partition λ , and calculate the charac ter of S λ . W e sh all giv e three constructions of S λ . 4.1.1 First Construction A n umb erin g T of a Y oun g diagram is a ﬁ lling of the b o xes in its table with distinct n um b ers from 1 , . . . , n . A n umbering of a Y oung diagram is 30 also called a tableau. It is called a standard tableaux if the n um b ers are strictly in creasing in eac h ro w and column. By T ij w e m ean the v alue in the tableaux at i -th row and j -th column. W e associate with eac h tableaux T a p olynomial in C [ X 1 , X 2 , . . . , X n ]: f T = Π j Π i 0 . Conjecture 4.1 (GCT6) . This c an b e done in p olynomial time; i.e. in time p olynomia l in the bit lengths o f the inputs λ , α and β . 33 55 34 Chapter 5 Represen tations of GL n ( C ) Scrib e: Joshua A. Gro c how Goal: T o d etermine the ir reducible represen tations of GL n ( C ) and their c h aract ers. R efer enc es: [FH, F] The goal of tod a y’s lecture is to classify all irredu cible r epresen tations of GL n ( C ) and co mpu te th eir c haracters. W e will go ov er t wo appr oac hes, the ﬁ rst due to Deruyts and the second due to W eyl. A p olynomial r epr esentation of GL n ( C ) is a represen tation ρ : GL n ( C ) → GL ( V ) s u c h that eac h entry in the matrix ρ ( g ) is a p olynomial in th e en tries of the matrix g ∈ GL n ( C ). The m ain result is that the p olynomial irr educible represen tations of GL n ( C ) are in b iject iv e correspon d ence with Y oung diagrams λ of heigh t at most n , i.e. λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0. Because of the imp ortance of W eyl’s constru cti on (similar constructions can b e used on man y other L ie groups besides GL n ( C )), the irredu cible representa tion corresp onding to λ is kn o wn as the Weyl mo dule V λ . 5.1 First App roac h [Deruyts] Let X = ( x ij ) b e a generic n × n matrix with v ariable en tries x ij . Consider the p olynomial ring C [ X ] = C [ x 11 , x 12 , . . . , x nn ]. Then GL n ( C ) acts on C [ X ] by ( A ◦ f )( X ) = f ( A T X ) (it is easily chec k ed that this is in fac t a left action). Let T b e a tableau of shap e λ . T o eac h col umn C of T of length r , w e asso cia te an r × r m inor of X as foll o ws: if C h as the en tries i 1 , . . . , i r , then 35 tak e from the ﬁrst r columns of X the ro ws i 1 , . . . , i r . Visually: C =    i 1 . . . i r    − → e C = 1 · · · r ↓ ↓ i 1 → i 2 → . . . i r →            x i 1 , 1 · · · x i 1 , r · · · x i 1 ,n x i 2 , 1 · · · x i 2 , r · · · x i 2 ,n . . . . . . . . . x i r , 1 · · · x i r , r · · · x i r ,n            (Th us if there is a rep eate d num b er in the col umn C , e C = 0, since the same ro w will get c hosen twice. ) Using these monomials e C for eac h column C of the tableau T , we associate a monomial to the en tire tableau, e T = Q C e C . (Th u s, if in an y column of T there is a rep eated num b er, e T = 0. F urthermore, the n umbers m ust all come from { 1 , . . . , n } if they are to sp ecify rows of an n × n matrix. So we restrict our atten tion to n umberin gs of T from { 1 , . . . , n } in whic h the n um b ers in any giv en column are all distinct.) Let V λ b e the v ector sp ace generated b y the set { e T } , where T ranges o ver all such n u mb erings o f shap e λ . Th en GL n ( C ) acts on V λ : for g ∈ GL n ( C ), eac h ro w of gX is a linear com bination o f the rows of X , and since e C is a minor of X , g · e C is a linear c om bination o f minors of X of the same siz e, i.e. g ( e C ) = P D a g C,D e D (this follo ws from standard linear algebra). Then g ( e T ) = g ( e C 1 e C 2 · · · e C k ) = X D a g C 1 ,D e D ! · · · X D a g C k ,D e D ! If w e expand this prod uct out, w e ﬁ nd that e ac h term is in fact e T ′ for s ome T ′ of the appropriate shap e. W e then ha ve the follo wing theorem: Theorem 5.1. 1. V λ is an i rr e ducible r epr e sentat ion of GL n ( C ) . 2. The set { e T | T is a semistanda r d table au of shap e λ } is a b asis for V λ . (R e c al l t hat a semistandar d table au is one whose numb ering is we akly incr e asing acr oss e ach r ow and strictly incr e asing down e ach c olumn.) 3. E very p olynomial irr e ducible r epr esentation of GL n ( C ) of de gr e e d is isomorp hic to V λ for s ome p artition λ of d of h eight a t mo st n . 36 4. E very r ational irr e ducible r epr esentation o f GL n ( C ) (e ach entry of ρ ( g ) is a r ational fu nc tion in the entries of g ∈ GL n ( C ) ) is isom orphic to V λ ⊗ det k for so me p artition λ of height at most n and for so me inte ger k (wher e det i s the determinant r e pr esentation). 5. (We yl’s char acter for mula) Deﬁne the char acter χ λ of V λ by χ λ ( g ) = T r ( ρ ( g )) , wher e ρ : GL n ( C ) → GL ( V λ ) is the r epr esentation map. Then, for g ∈ GL n ( C ) with eigenvalues x 1 , . . . , x n , χ λ ( g ) = S λ ( x 1 , . . . , x n ) :=    x λ i + n − i j       x n − i j    (wher e | y i j | is the determinant of the n × n matrix whose entries ar e y ij = y i j , so, e.g. , the determinant in the denominator is the u sual van der Monde determinant, wh ich is e q ual to Q i 0 . By this lemma, to decide if c γ αβ > 0, it suﬃces to test if P is nonemp ty . The p olytop e P is give n by Ax ≤ b wh ere the en tries of A are 0 or 1– suc h linear programs are called com binatorial. Hence, this can b e done in strongly p olynomial time using T ardos’ algorithm [GLS] for com binatorial linear pr ogramming. This p r o ve s the theorem.  The inte ger programming p roblem is NP-complete, in general. Ho wev er, linear pr ogramming w orks for the sp eciﬁc in teger programming problem here b ecause of the saturation prop ert y [KT]. Problem : Find a gen uinely com binatorial p oly-time algorithm for deciding non-v anishing of c γ αβ . 45 Chapter 7 Littlew o o d-Ric hardson co eﬃcien ts (con t) Scrib e: Paolo Co deno tti Goal: W e con tinue our stud y of Little woo d-Ric h ardson co eﬃcien ts and deﬁne Littlew o o d-Ric hardson coeﬃcien ts for th e orthogona l group O n ( C ). R efer enc es: [FH, F] Recall Let us ﬁ rst r eca ll some deﬁn itio ns a nd results from the last class. Let c γ α,β denote the Littlew o od -Richardson co eﬃcien t for GL n ( C ). Theorem 7.1 (last class) . Non-vanishing of c γ α,β c an b e de cide d i n p oly ( h α i , h β i , h γ i ) time, wher e h i denotes the bit length. The p ositivit y hypotheses w h ic h hold here are: • c γ α,β ∈ # P , and more strongly , • Pos itivity Hypothesis 1 (PH1): Th ere exists a p olytop e P γ α,β of dimension p olynomial in the heights of α, β and γ su c h that c γ α,β = ϕ ( P γ α,β ), wh ere ϕ indicates the n umb er of in teger p oin ts. • Sat uration Hyp othesis (SH): If c k γ k α,kβ 6 = 0 for s ome k ≥ 1, then c γ α,β 6 = 0 [Saturation Theorem]. Pr o of. (of theorem) PH1 + SH + Linear programming. 46 This is the general f orm of algorithms in GCT. The main principle is that linear programming works for in teger programming when PH1 and SH hold. 7.1 The stretc hing function W e deﬁne e c γ α,β ( k ) = c k γ k α,kβ . Theorem 7.2 (Kirillo v, De rke sen W eyman [Der, Ki]) . e c γ α,β ( k ) is a p olyno - mial in k . Here w e pro v e a w eake r result. F or its statemen t, we w ill qu ic kly review the theory of E h rhart quasip olynomials (cf. Stanley [S]). Deﬁnition 7.1. ( Quasip olynomial ) A function f ( k ) is c al le d a qu asipoly- nomial if ther e exist p olynomials f i , 1 ≤ i ≤ ℓ , fo r some ℓ such th at f ( k ) = f i ( k ) if k ≡ i mo d ℓ. We denote such a quasip olynomial f by f = ( f i ) . Her e ℓ is c al le d the p erio d of f ( k ) (we c an assume it is the smal lest such p erio d). The de gr e e of a quasip olynomia l f i s the max of the de gr e es of the f i ’s. No w let P ⊆ R m b e a p olyto p e giv en by Ax ≤ b . Let ϕ ( P ) b e the n um b er of int eger p oin ts ins id e P . W e deﬁne the stretc h ing fu nction f P ( k ) = ϕ ( k P ), where k P is the dilated p olytop e deﬁned by Ax ≤ k b . Theorem 7.3. (Ehrhart) The str etching function f P ( k ) is a quasip olyno- mial. F urthermor e, f P ( k ) is a p olynomial if P is an inte gr al p olytop e (i.e. al l vertic es of P ar e inte gr al). In view of this resu lt, f P ( k ) is called the Ehrhart quasi-p olynomial of P . No w e c γ α,β ( k ) is just the Ehrhart quasip olynomial of P γ α,β , and c γ α,β = ϕ ( P γ α,β ), the num b er of in teger p oints in P γ α,β . Moreo v er P γ α,β is deﬁned by the inequalit y Ax ≤ b , w here A i s constant, and b is a h omogeneous linear form in the co eﬃcients of α , β , and γ . Ho wev er, P γ α,β need n ot b e in tegral. T herefore Theorem (7.2 ) do es not follo w from Ehrhart’s result. Its pro of needs representa tion theory . Deﬁnition 7.2. A quasip olynomial f ( k ) is said to b e p ositiv e if al l the c o eﬃcients of f i ( k ) a r e nonne gative. In p articular, if f ( k ) is a p olynomia l, then it’s p ositive if al l its c o eﬃcients ar e nonne gative. 47 The Ehr hart quasip olynomial of a p olyto p e is p ositiv e only in exceptional cases. In this conte xt: PH 2 (p ositivit y h yp othesis 2) [KTT]: The p olynomial e c γ α,β ( k ) is p ositiv e. There is considerable computer evidence for this. Prop osition 7.1. PH 2 implies SH. Pr o of. Lo ok at: c ( k ) = e c γ α,β ( k ) = X a i k i . If all the co eﬃcien ts a i are nonnegativ e (b y PH2), and c ( k ) 6 = 0, then c (1) 6 = 0. SH has a proof in volvi ng algebraic ge ometry [B]. Th erefore w e susp ect that the stronger PH2 is a deep phenomenon related to algebraic geometry . 7.2 O n ( C ) So far w e h a ve talk ed ab out GL n ( C ). No w we mov e on to the orthogo- nal group O n ( C ). Fix Q , a symmetric bilinear form on C n ; for example, Q ( V , W ) = V T W . Deﬁnition 7.3. The ortho gonal gr oup O n ( C ) ⊆ GL n ( C ) is the gr oup c on- sisting of al l A ∈ GL n ( C ) s.t. Q ( AV , AW ) = Q ( V , W ) for al l V and W ∈ C n . The sub gr oup S O n ( C ) ⊆ S L n ( C ) , wher e S L n ( C ) is the set of matric es w ith determina nt 1 , is deﬁne d similarly. Theorem 7.4 (W eyl) . The gr oup O n ( C ) is r e ductive Pr o of. The pro of is similar to the reductivit y of G L n ( C ), based o n W eyl’s unitary tric k. The next step is to classify all irr educible polynomial represen tations of O n ( C ). Fix a partition λ = ( λ 1 ≥ λ 2 ≥ . . . ) of lengt h at most n . Let | λ | = d = P λ i b e its size. Let V = C n , V ⊗ d = V ⊗ · · · ⊗ V d times, and em b ed the W eyl modu le V λ of GL n ( C ) in V ⊗ d as p er Theorem 5.3. Deﬁne a cont raction map ϕ p,q : V ⊗ d → V ⊗ ( d − 2) for 1 ≤ p ≤ q ≤ d by: ϕ p,q ( v i 1 ⊗ · · · ⊗ v i d ) = Q ( v i p , v i q )( v i 1 ⊗ · · · ⊗ c v i p ⊗ · · · ⊗ c v i q ⊗ · · · ⊗ v i d ) , 48 λ Figure 7.1: The ﬁrst tw o columns of the partition λ are highlighted. where c v i p means omit v i p . It is O n ( C )-equiv arian t, i .e. the follo wing diagram comm utes: V ⊗ d ϕ p,q − − − − → V ⊗ d − 2   y σ ∈ O n ( C )   y σ ∈ O n ( C ) V ⊗ d ϕ p,q − − − − → V ⊗ d − 2 Let V [ d ] = \ pq k er ( ϕ p,q ) . Because the maps are equiv arian t, e ac h kernel is a n O n ( C )-mo d ule, and V [ d ] is an O n ( C )-mo d ule. Let V [ λ ] = V [ d ] T V λ , where V λ ⊆ V ⊗ d is the em b edded W eyl mo dule as ab o v e. Then V [ λ ] is an O n ( C )-mo d ule. Theorem 7.5 (W eyl) . V [ λ ] is an irr e ducible r epr esentation of O n ( C ) . M or e- over, the fol lowing two c onditions hold: 1. If n is o dd, then V [ λ ] is non-zer o if and on ly if th e sum of the lengths of the ﬁrst two c olumns of λ is ≤ n (se e ﬁg u r e 7.1). 2. If n i s o dd, then e ach p olynomial ir r e ducible r epr esentation is isomor- phic to V [ λ ] for some λ . Let V [ λ ] ⊗ V [ µ ] = ⊗ γ d γ λ,µ V [ γ ] 49 b e the decomp osition of V [ λ ] ⊗ V [ µ ] in to irred u cibles. Here d γ λ,µ is called the Littlew o o d-Ric h ard son c o eﬃcien t of t yp e B. The t yp es of v arious connected reductiv e groups are deﬁned as follo ws: • GL n ( C ): t yp e A • O n ( C ), n o dd: type B • S p n ( C ): t yp e C • O n ( C ), n ev en: t yp e D The Littlew o o d-Ric hards on co eﬃcien t can b e deﬁned for any t yp e in a sim- ilar fashion. Theorem 7.6 (Generalized L ittlew o o d-Ric hard s on ru le) . The Li ttlewo o d- R ichar dson c o eﬃcient d γ λ,µ ∈ # P . This also ho lds for any typ e. Pr o of. The most transparen t pr oof of this theorem comes through the theory of qu an tum groups [K]; cf. Chapter 20. As in t yp e A this leads to: Hyp othesis 7.1 (PH1) . Ther e exists a p olytop e P γ λ,µ of dimension p olyno- mial in th e heights of λ, µ and γ such that: 1. d γ λ,µ = ϕ ( P γ λ,µ ) , the numb er of inte ger p oints in P γ λ,µ , and 2. e d γ λ,µ ( k ) = d k γ k λ,kµ is the Eh rhart quasip olynomial of P γ λ,µ . There are sev eral choice s for such polytop es; e.g. the BZ-p olytope [BZ]. Theorem 7.7 (De Lo era, McAlliste r [DM2]) . The str etching function e d γ λ,µ ( k ) is a quasip olynomial of de gr e e a t mo st 2 ; so also for typ es C and D . A v erb atim tran s lation of the saturation prop ert y fails here [Z]): there exist λ, µ and γ suc h that d 2 γ 2 λ, 2 µ 6 = 0 but d γ λ,µ = 0. Th erefore w e change the deﬁnition of saturation: Deﬁnition 7.4. Given a quasip olynomial f ( k ) = ( f i ) , index ( f ) is the smal lest i such that f i ( k ) is not an identic al ly zer o p olynomial. If f ( k ) is identic al ly zer o, index ( f ) = 0 . Deﬁnition 7.5. A quasip olynomial f ( k ) is saturated if f ( index ( f )) 6 = 0 . In p articular, if index ( f ) = 1 , then f ( k ) is s atur ate d if f (1) 6 = 0 . 50 A p ositiv e quasi-p olynomial is clearly saturated. P ositivity Hyp othesis 2 (PH2) [DM2]: The stretc hing quasip oly omial e d γ λ,µ ( k ) is p ositiv e. There is considerable evidence for this. Saturation Hyp othesis (SH) : The stretc hing qu asipolynomial e d γ λ,µ ( k ) is saturated. PH2 implies SH. Theorem 7.8. [GCT5] Assuming SH (or PH 2 ), p ositivity of the Littl ewo o d- R ichar dson c o eﬃcient d γ λ,µ of typ e B c an b e de cide d in pol y ( h λ i , h µ i , h γ i ) time. This is also true for all types. Pr o of. next class. 51 Chapter 8 Deciding non v anishing of Littlew o o d-Ric hardson co eﬃcien ts for O n ( C ) Scrib e: Ha riharan Nara y anan Goal: A p olynomial time algorithm for deciding nonv anish in g of Littlew o o d- Ric h ardson co eﬃcien ts for the orthogonal group assuming SH. R efer enc e: [GCT5] Let d ν λ,µ denote the Little woo d-Ric h ardson coeﬃcien t of type B (i.e. for the orthogonal group O n ( C ), n o dd) as deﬁned in the earlie r lect ure. In this lecture we d escrib e a p olynomial time al gorithm for deciding non v anish ing of d ν λ,µ assuming the f ollo wing p ositivit y h yp othesis PH2. S imilar result also holds for all types, t hough we shall only c oncen trate o n t yp e B in this lecture. Let ˜ d ν λ,µ ( k ) = d k ν k λ,kµ denote the asso ciated stretc hing fu nctio n. It is kno wn to b e a quasi-p olynomial of p erio d at most tw o [DM2]. This means there are p olynomials f 1 ( k ) and f 2 ( k ) suc h that d k ν k λ,kµ =  f 1 ( k ) , if k is o dd; f 2 ( k ) , if k is ev en . P ositivity Hyp othesis (PH2) [DM2 ]: Th e stretc hing qu asi-polynomial ˜ d ν λ,µ ( k ) is p ositiv e. This means the co eﬃcie nts of f 1 and f 2 are all non- negativ e. The main result in this lecture is: 52 Theorem 8.1. [GCT5] If PH2 hol ds, then the pr oblem of de ci ding the p osi- tivity (nonvanishing) of d ν λµ b elongs to P . That is, this pr oblem c an b e solve d in time p olyno mial in the bitlengths of λ, µ and ν . W e need a few lemmas for the p roof. Lemma 8.1. If PH 2 holds, th e fol lowing ar e e quivalent: (1) d ν λµ ≥ 1 . (2) Ther e exists an o dd inte ger k such tha t d k ν k λ k µ ≥ 1 . Pro of: Clearly (1) implies (2). By PH2, there exists a p olynomial f 1 with non-negativ e co eﬃcien ts suc h that ∀ o dd k , f 1 ( k ) = d k ν k λ k µ . Supp ose that f or some o dd k , d k ν k λ k µ ≥ 1 . Th en f 1 ( k ) ≥ 1. Therefore f 1 has at least one non-zero co eﬃcien t. Since all coeﬃcien ts of f 1 are nonnegat iv e, d ν λµ = f 1 (1) > 0. Since d ν λµ is an integ er, (1) follo ws.  Deﬁnition 8.1. L et Z < 2 > b e the subring of Q obtaine d by lo c alizing Z at 2 : Z < 2 > :=  p q | p, q − 1 2 ∈ Z  . This ring c onsists of al l fr actions wh ose denomina tors ar e o dd. Lemma 8.2. L et P ∈ R d b e a c onvex p olytop e sp e ciﬁe d by Ax ≤ B , x i ≥ 0 for al l i , wh er e A and B ar e inte gr al. L et Aﬀ ( P ) denote its aﬃne sp an. The fol lowing ar e e quivalent: (1) P c ontains a p oint in Z d < 2 > . (2) Aﬀ ( P ) c ontains a p oint in Z d < 2 > . Pro of: Since P ⊆ Aﬀ ( P ), (1) implies (2). No w supp ose (2) holds. W e ha v e to sh o w (1). Let z ∈ Z d < 2 > ∩ Aﬀ ( P ). First, consider the case when Aﬀ ( P ) is one dimensional. In this ca se, P is the line segmen t joining t w o p oin ts x and y in Q d . The p oin t z can b e expressed as an aﬃne linear com bination, z = ax + (1 − a ) y for some a ∈ Q . There exists q ∈ Z su c h that q x ∈ Z d < 2 > and q y ∈ Z d < 2 > . Note that { z + λ ( q x − q y ) | λ ∈ Z < 2 > } ⊆ Aﬀ ( P ) ∩ Z d < 2 > . 53 Since Z < 2 > is a dens e subset of Q , the l.h.s. and hence the r.h.s. is a dense subset of Aﬀ ( P ). Cons equen tly , P ∩ Z d < 2 > 6 = ∅ . No w consider the g eneral case. Let u b e any p oint in the int erior of P with rat ional co ord inates, and L th e line through u and z . By restrict ing to L , the lemma r ed u ces to the preceding one dimensional case.  Lemma 8.3. L et P = { x | Ax ≤ B , ( ∀ i ) x i ≥ 0 } ⊆ R d b e a c onvex p olytop e wher e A and B ar e inte gr al. Then, it is p ossible to determine i n p olynomial time whether or not Aﬀ ( P ) ∩ Z d < 2 > = ∅ . Pro of: Using Linear Programming [Kha79, Kar84], a presen tation of the form C x = D can b e obtained for Aﬀ ( P ) in p olynomial time, wh ere C is an in teger matrix and D is a v ector with int eger co ordinates. W e m ay assume that C is s qu are since this can b e ac hiev ed b y padding it with 0’ s if n ecessary , and extending D . The Smith Normal F orm o v er Z of C is a matrix S suc h that C = U S V where U and V are unimo dular and S has the form      s 11 0 . . . 0 0 s 22 . . . 0 . . . . . . . . . 0 0 0 . . . s dd      where for 1 ≤ i ≤ d − 1, s ii divides s i +1 i +1 . It can b e computed in p olynomial time [KB79]. The qu estio n no w reduces to w hether U S V x = D has a solution x ∈ Z d < 2 > . Since V is un imod u lar, its in v erse has integ er entries to o, and y := V x ∈ Z d < 2 > ⇔ x ∈ Z d < 2 > . This is equiv alen t to whether S y = U − 1 D has a solution y ∈ Z d < 2 > . S ince S is diagonal, this can b e answ ered in p olynomial time simply by c hec king eac h co ordinate.  Pro of of T heorem 8.1: By [BZ], there exists a p olyto p e P = P ν λ,µ suc h that the Littlew o o d-Ric h ardson co eﬃcien t d ν λµ is equal to th e n umb er of int eger p oin ts in P . This p olytop e is such that the num b er of in teger p oin ts in the dilated polytop e k P is d k ν k λ k µ . Assu m ing PH2, w e kno w from Lemma 8.1 that P ∩ Z d 6 = ∅ ⇔ ( ∃ o dd k ) , k P ∩ Z d 6 = ∅ . The latter is equiv alent to P ∩ Z d < 2 > 6 = ∅ . 54 The theorem no w follo ws from Lemma 8.2 and Lemma 8.3.  In co m binatorial optimization, LP w orks if the p olytope is inte gral. In our setting, this is not n ece ssarily the case [DM1]: the d enominato rs of the co ord inates of the v ertices of P can b e Ω ( l ), where l is the tota l height o f λ, µ and ν . LP w orks h ere neverthel ess b ecause of PH2; it can b e chec k ed that SH is also suﬃcient. 55 Chapter 9 The pleth ysm problem Scrib e: Joshua A. Gro c how Goal: In this lecture we describ e the general plethysm pr oblem , state anal- ogous p ositivi t y and saturation hyp otheses for it, a nd state the results from GCT 6 whic h imply a p olynomial ti me al gorithm fo r deciding p ositivit y of a plethysm constan t a ssuming these hyp otheses. Reference: [GCT6] Recall Recall th at a fu n ctio n f ( k ) is q uasip olynomial if there are fun ctio ns f i ( k ) for i = 1 , . . . , ℓ such that f ( k ) = f i ( k ) whenev er k ≡ i mo d ℓ . The num b er ℓ is then the p erio d of f . T he index of f is the least i su c h that f i ( k ) is not iden tically zero. If f is identi cally zero, then the index of f is zero b y con v en tion. W e sa y f is p ositive if all the co eﬃcie nts of ea c h f i ( k ) are nonnegativ e. W e sa y f is satur ate d if f (index( f )) 6 = 0. I f f is p ositiv e, then it is saturated. Giv en an y function f ( k ), w e associate to it the rational series F ( t ) = P k ≥ 0 f ( k ) t k . Prop osition 9.1. [S] The fol lowing ar e e quivalent: 1. f ( k ) i s a quasip olynomial of p erio d ℓ . 2. F ( t ) i s a r ational function of the form A ( t ) B ( t ) wher e deg A < deg B and every r o ot of B ( t ) is a n ℓ -th r o ot of u nity. 56 9.1 Littlew o o d-Ric h ardson Problem [GCT 3,5] Let G = GL n ( C ) and c γ α,β the Litt lew o o d-Ric hardson co eﬃcient – i.e. the m ultiplicit y of the W eyl mo dule V γ in V α ⊗ V β . W e saw that the positivit y of c γ α,β can b e decided in pol y ( h α i , h β i , h γ i ) time, where h· i denotes the bit- length. F u rthermore, w e sa w that the stretc hing fun cti on e c γ α,β ( k ) = c k γ k α,kβ is a p olynomial, and the analogous stretc hing function for t yp e B is a quasip olynomial of p erio d at most 2. 9.2 Kronec k er Problem [GCT 4,6] No w w e stud y the analogous problem for the r ep r esen tations of the symmet- ric group (the Sp ec h t mo dules), called the Kr onec k er p r oblem. Let S α b e the S p ec ht mo dule of the symmetric group S m asso cia ted to the partition α . Deﬁne the Kronec k er co eﬃcient κ π λ,µ to b e the multiplici t y of S π in S λ ⊗ S µ (considered as an S m -mo dule via the diagonal action). In other wo rds, write S λ ⊗ S µ = L π κ π λ,µ S π . W e ha v e κ π λ,µ = ( χ λ χ µ , χ π ), wh ere χ λ denotes the c h aract er of S λ . By the F rob enius charact er formula, this can b e computed in PSP A CE. More strongly , a nalogous to the Littlew o o d- Ric h ardson pr oblem: Conjecture 9.1. [GCT4, GCT6] The Kr one cker c o eﬃcie nt κ π λ,µ ∈ # P . In other wor ds, ther e is a p ositive # P -formula for κ π λ,µ . This is a fun damen tal problem in represen tation theory . More concretel y , it can b e ph rased as asking for a set of com binatorial ob jects I and a c har- acteristic fu nctio n χ : { I } → { 0 , 1 } su ch th at χ ∈ FP and κ π λ,µ = P I χ ( I ). Con tin uing our an alogy: Conjecture 9.2. [GCT6] The pr oblem of de ciding p ositivity of κ π λ,µ b elongs to P . Theorem 9.1. [G CT6] The str etching function e κ π λ,µ ( k ) = κ k π k λ,kµ is a quasip oly- nomial. Note that κ k π k λ,kµ is a Kronec k er co eﬃcien t fo r S k m . There is also a dual deﬁn itio n of the Kronec k er co eﬃcien ts. Namely , consider th e em b edding H = GL n ( C ) × GL n ( C ) ֒ → G = GL ( C n ⊗ C n ) , where ( g , h )( v ⊗ w ) = ( g v ⊗ hw ). Then 57 Prop osition 9 .2. [FH] The Kr one c k er c o eﬃcient κ π λ,µ is the multiplicity of the tensor pr o duct of Weyl mo dules V λ ( GL n ( C )) ⊗ V µ ( GL n ( C )) (this is an irr e ducible H -mo dule) in the W e yl mo dule V π ( G ) c onsider e d as an H - mo dule via the emb e dding ab ove. 9.3 Pleth ysm Problem [GCT 6,7] Next we consider the more general plethysm problem. Let H = GL n ( C ), V = V µ ( H ) the W eyl mod ule of H corresp onding t o a partition µ , and ρ : H → G = GL ( V ) the corresp onding representa tion map. Th en the W eyl mo dule V λ ( G ) of G for a giv en partition λ can b e considered an H -mo dule vi a the map ρ . By complete reducibilit y , we ma y decomp ose this H -represen tation as V λ ( G ) = M π a π λ,µ V π ( H ) . The coeﬃcien ts a π λ,µ are kno w n as plethsym c onstants (this deﬁn itio n can easily b e generalized to an y redu cti ve group H ). T h e Kronec k er co eﬃcien t is a sp ecial case of the plethsym constant [Ki]. Theorem 9.2 (GCT 6) . The plethysm c onstant a π λ,µ ∈ PSP A CE. This is based on a parallel algorithm to compu te the pleth ysm constan t using W eyl’s charac ter formula. Conti nuing in our pr evious tren d : Conjecture 9.3. [GCT6] a π λ,µ ∈ # P and the pr oblem of de ciding p ositivity of a π λ,µ b elongs to P . F or the stretc hing function, we need to b e a bit careful. Deﬁne e a π λ,µ = a k π k λ,µ . Here the subscript µ is not stretc hed, since that w ould c hange G , while stretching λ and π only alters the representa tions of G . As in the b eginning of the lec ture, we can a sso ciate a f unction A π λ,µ ( t ) = P k ≥ 0 e a π λ,µ ( k ) t k to the pleth ysm constan t. Kir illo v conjectured that A π λ,µ ( t ) is rational. In view of Prop osition 9 .1, this follo ws from the follo wing stronger result: Theorem 9.3 (GCT 6) . The str etching function e a π λ,µ ( k ) is a qua sip olyno- mial. This is the mai n result of GCT 6, whic h in some sense allo ws GCT to go forwa rd. Without it, there wo uld b e little hop e for pro ving that the 58 p ositivit y of pleth ysm constan ts ca n b e decided in p olynomial time. Its pro of is essen tially algebro-geometric . The basic idea is to show that the stretc hin g function is the Hilb ert function of some algebraic v ariet y w ith n ice (i.e. “rational ”) singularities. Similar results are shown for the stretc hin g functions in the algebro-ge ometric problems arising in GCT. The main complexi t y-theoretic result in [GCT6] sho ws that, under the follo wing p ositivi t y and saturation hypotheses (for whic h there is muc h ex- p erimen tal evidence), the p ositivit y of th e plethysm constan ts can indeed b e decided in p olynomial time (cf. Conjecture 9.3). The ﬁrst p ositivit y hyp othesis is su gge sted by Theorem 9.3: sin ce the stretc hin g fun ctio n is a quasipolynomial, w e ma y susp ect that it is ca ptured b y some p olytop e: P ositivity Hypot hesis 1 (PH1). There exists a p olytop e P = P π λ,µ suc h that: 1. a π λ,µ = ϕ ( P ), where ϕ denotes the n umber of in teger p oints inside the p olytope, 2. The stretc hing quasip olynomial (cf. T hm. 9.3) e a π λ,µ ( k ) is equal to the Ehrhart quasip olynomial f P ( k ) of P , 3. The dimension of P is p olynomial in h λ i , h µ i , and h π i , 4. the m emb ership in P π λ,µ can b e decided in p oly( h λ i , h µ i , h π i ) time, and there is a p olynomial time separation oracle [GLS] for P . Here (4) do es not imply that the p olytop e P has only p olynomially man y constrain ts. In fact, i n the p lethysm problem there may b e a su p er- p olynomial num b er of constrain ts. P ositivity Hyp othesis 2 (PH2). Th e stretc hing quasip olynomia l e a π λ,µ ( k ) is p ositiv e. This imp lie s: Saturation Hyp othesis (SH). Th e stretc hing quasip olynomial is sat - urated. Theorem 9.3 is essen tial to state these h yp otheses, since p ositivit y and saturation a re prop erties that only apply to qu asipolynomials. Eviden ce for PH1, PH2, and S H can b e foun d in GCT 6. Theorem 9.4. [GCT6] Assuming PH 1 and SH (or PH2), p ositivity of the plethysm c onstant a π λ,µ c an b e d e cide d in p oly ( h λ i , h µ i , h π i ) time. 59 This follo ws from the p olynomial time algorithm for saturated intege r programming d escribed in the next class. As with Th eorem 9.3 , this also holds for more general problems in algebraic geometry . 60 Chapter 10 Saturated and p osi tiv e in teger programming Scrib e: Sour a v Chak rab ort y Goal : A p olynomial time algorithm for saturated in teger pr ogrammi ng and its application to the pleth ysm problem. R efer enc e: [GCT6] Notation : In this class w e denote by h a i the bit-length of the a . 10.1 Saturated, p ositiv e in teger programming Let Ax ≤ b b e a set of in equalit ies. The num b er of constrain ts can b e exp onen tial. Let P ⊂ R n b e the polytop e d eﬁned by these inequalitie s. The bit length of P is deﬁned to b e h P i = n + ψ , where ψ is the maxim um bit-length of a constrain t in the set of inequalitie s. W e assume that P is giv en b y a separating oracle. T his means m emb ership in P can be decided in p oly( h P i ) t ime, and if x 6∈ P then a separating h yp erplane is g iv en as a pro of as in [GLS]. Let f P ( k ) b e the Ehrhart qu asi- p olynomial of P . Quasi-p olynomialit y means there exist p olynomials f i ( k ), 1 ≤ i ≤ l , l the p erio d, so that f P ( k ) = f i ( k ) if k = i mo dulo l . Then Index( f P ) = m in { i | f i ( k )not iden tically 0 as a p olynomial } The in teger programming problem is called p ositive if f P ( k ) i s p ositiv e whenev er P is non-empt y , and satur ate d if f P ( k ) is saturated wh enev er P is non-empt y . 61 Theorem 10.1 (GCT6) . 1. Index ( f P ) c an b e c ompute d in tim e p olyno- mial in the bit length h P i of P assuming that the sep ar ation or acle works in p oly- h P i -time. 2. Satur ate d and henc e p ositive i nte g er pr o gr amming pr oblem c an b e solve d in p oly- h P i -time. The second statemen t follo w from the ﬁrst. Pr o of. Let Af f ( P ) denote the aﬃne span of P . B y [G LS] w e can compute the speciﬁcations C x = d , C and d integ ral, of Af f ( P ) in p oly( h P i ) time. Without loss of generalit y , b y padding, w e can assume that C is s qu are. By [KB79] w e ﬁnd the Smith-normal f orm of C in p olynomial time. Let it b e ¯ C . So, ¯ C = AC B where A and B are unimo dular, and ¯ C is a diagonal m atrix, where the diagonal entries c 1 , c 2 , . . . are such that with c i | c i +1 . Clearly C x = d iﬀ ¯ C z = ¯ d wh ere z = B − 1 x and ¯ d = Ad . So all equations h ere are of form ¯ c i z i = d i (10.1) Without loss of generalit y we can assume th at c i and d i are relativ ely prime. Let ˜ c = l cm ( c i ). Claim 10.1. I ndex ( f P ) = ˜ c . F rom this claim the theorem clearly follo ws. Pr o of of the claim. Let f P ( t ) = P k ≥ 0 f P ( k ) t k b e the Ehr hart Series of P . No w k P will not ha v e an int eger p oint u nless ˜ c divides k b ecause of (10.1) . Hence f P ( t ) = f ¯ P ( t ˜ c ) where ¯ P is the stretc hed p olytop e ˜ cP and f ¯ P ( s ) is the Eh rhart series of ¯ P . F rom this it follo ws that I ndex ( f P ) = ˜ cI ndex ( f ¯ P ) No w w e sho w that I ndex ( f ¯ P ) = 1. The equations of ¯ P are of the form z i = ˜ c c i d i 62 where eac h ˜ c c i is an inte ger. Therefore without loss of generalit y we can ignore these equations and assume the ¯ P is f u ll dimensional. Then it suﬃces to sho w that ¯ P con tains a rational p oint wh ose denomi- nators are all 1 mo dulo ℓ ( ¯ P ), the p erio d of the quasi-p olynomial f ¯ P ( s ). This follo ws from a simple densit y argument that w e sa w earlier (c f. the pro of of Lemma 8.2). F rom this the claim follo ws . 10.2 Application to the p leth ysm problem No w we can pro ve the result stated in the last class: Theorem 10.2. Assuming PH1 a nd SH, p ositivity of the plethysm c onstant a π λ,µ c an b e de cide d in time p olynomial in h λ i , h µ i and h π i . Pr o of. Let P = P π λ,µ b e the polytop e as in PH1 such that a π λ,µ is the num b er of integ er points in P . The goal is to decide if P conta ins a n inte ger p oin t. This inte ger programming problem is saturated b ecause of S H. Hence the result f oll o ws from Theorem 10.1. 63 Chapter 11 Basic algebraic geometry Scrib e: Paolo Co deno tti Goal: So far we h a v e fo cussed on purely r epresen tation-theoretic asp ect s of GCT. No w we ha ve to b ring in algebraic geometry . In this le cture we review the basic deﬁnitions and results in algebraic ge ometry that will b e needed for th is purp ose. The pro ofs w ill b e omitted o r only sk etc h ed. F or details, see the b o oks b y Mumford [Mm] and F u lton [F]. 11.1 Algebraic geometry deﬁnitio ns Let V = C n , and v 1 , . . . , v n the co ordinates of V . Deﬁnition 11.1. • Y is an aﬃne algebraic set in V if Y is the set of simultane ous zer os of a set of p olynomials in v i ’s. • An algebr aic set that c annot b e written as the union of two pr op er algebr aic sets Y 1 and Y 2 is c al le d irreducible . • An irr e ducible aﬃne algebr aic set i s c al le d an aﬃne v ariet y . • The ide al of an a ﬃne algebr aic set Y is I ( Y ) , the set of al l p olynomials that vanish o n Y . F or example, Y = ( v 1 − v 2 2 + v 3 , v 2 3 − v 2 + 4 v 1 ) is an irreducible aﬃn e algebraic set (and therefore an aﬃne v ariety) . Theorem 11.1 (Hilb ert) . I ( Y ) is ﬁnitely ge ner ate d, i.e. ther e exist p oly- nomials g 1 , . . . , g k that gener ate I ( Y ) . This me ans every f ∈ I ( Y ) c an b e written as f = P f i g i for s ome p olynomials f i . 64 Let C [ V ], the co ordinate rin g of V , b e the ring of p olynomials o ve r the v ariables v 1 , . . . , v n . Th e co ordinate ring of Y is deﬁned to be C [ Y ] = C [ V ] /I ( Y ). It is the set of p olynomial functions ov er Y . Deﬁnition 11.2. • P ( V ) is the pro jectiv e space ass o ciate d with V , i. e . the set of lines thr ough th e origin in V . • V is c al le d the cone of P ( V ) . • C [ V ] is c al le d the homogeneous co ordinate ring o f P ( V ) . • Y ⊆ P ( V ) is a pro jectiv e algebraic set if it is the set of simultane ous zer os of a set of homo g ene ous form s (p olynom ials) in the v ariables v 1 , . . . , v n . It is ne c essary that the p olynomials b e homo gene ous b e c ause a p oint in P ( V ) is a line in V . • A pr oje ctive algebr aic set Y is irredu cible if it c an not b e expr esse d as the union of two pr op er algebr aic sets in P ( V ) . • An irr e ducible pr oje ctive al gebr aic set is c al le d a pro jectiv e v ariet y . Let Y ⊆ P ( V ) b e a pro jectiv e v ariet y , and deﬁne I ( Y ), the ideal o f Y to b e the set of all homogeneous forms that v anish on Y . Hilb ert’s result implies that I ( Y ) is ﬁ nitely generated. Deﬁnition 11.3. The cone C ( Y ) ⊆ V of a pr oje c tive variety Y ⊆ P ( V ) is deﬁne d to b e the set of al l p oints on the lines in Y . Deﬁnition 11.4. We deﬁne the h omogeneous coordin ate ring of Y as R ( Y ) = C [ V ] /I ( Y ) , the set of homo gene ous p olynomial forms on the c one of Y . Deﬁnition 11.5. A Zariski o p en subset of Y is the c omplement of a pr o- je ctive algebr aic subset of Y . It is c al le d a quasi-pro jectiv e v ariet y . Let G = GL n ( C ), and V a ﬁnite dimensional represen tation of G . Th en C [ V ] is a G -mo dule, with the action of σ ∈ G d eﬁned b y: ( σ · f )( v ) = f ( σ − 1 v ) , v ∈ V . Deﬁnition 11.6 . L et Y ⊆ P ( V ) b e a pr oje ctive variety with ide al I ( Y ) . We say that Y is a G -v ariet y if I ( Y ) is a G -mo dule, i.e., I ( Y ) is a G -submo dule of C [ V ] . 65 If Y is a pro jectiv e v ariet y , then R ( Y ) = C [ V ] /I ( Y ) is also a G -mo dule. Therefore Y is G -in v arian t, i.e. y ∈ Y ⇒ σ y ∈ Y , ∀ σ ∈ G. The alge braic geometry of G -v arieties is calle d geometric in v arian t theory (GIT). 11.2 Orbit closures W e no w deﬁne sp ecial classes of G -v arieties called orbit closur es . Let v ∈ P ( V ) b e a p oint, and Gv the orbit of v : Gv = { g v | g ∈ G } . Let the stabilizer of v b e H = G v = { g ∈ G | g v = v } . The orbit Gv is isomorphic to the space G/H of cosets, called the ho- mogeneous space. This is a very sp ecial kind of algebraic v ariet y . Deﬁnition 11.7. The orbit closure of v is deﬁne d by: ∆ V [ v ] = Gv ⊆ P ( V ) . Her e Gv is the closur e of the orbit Gv in the c omplex top olo gy on P ( V ) (se e ﬁgur e 1 1.1). A b asic fact of algebraic geometry: Theorem 11.2. The orbit closur e ∆ V [ v ] is a pr oje ctiv e G -variety It is also called an almost h omogeneous space. Let I V [ v ] b e the ideal of ∆ V [ v ], and R V [ v ] the homogeneous co ordinate ring of ∆ V [ v ]. The algebraic geometry of general orbit closures is hop eless, since the closures can b e h orrendous (see ﬁgure 11.1). F ortunately we shall only b e in terested in ve ry sp ecial kin ds of orb it closures w ith goo d a lgebraic geometry . W e now deﬁne the simplest kin d of orbit closure, whic h is obtained wh en the orb it itself is closed. Let V λ b e an irreducible W eyl mo dule of GL n ( C ), where λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0) is a partition. Let v λ b e the highest w eigh t p oin t in P ( V λ ), i.e., the p oin t corresp onding to th e highest weig ht 66 v Gv ∆ V [ v ] Limit p oints of Gv Figure 11.1: The limit p oints of Gv in ∆ V [ v ] c a n b e horrendous. v ector in V λ . This means bv λ = v λ for all b ∈ B , where B ⊆ GL n ( C ) is the Borel subgroup of lo wer triangular matrices. Reca ll that the highest w eigh t v ector is u n ique. Consider the orbit Gv λ of v λ . Basic fact: Prop osition 11.1. The orbit G v λ is alr e ady close d in P ( V ) . It ca n b e sho wn that the stabilizer P λ = G v λ is a group of b lo c k lo w er triangular matrices, w here the blo c k lengths only dep end on λ (see ﬁgur e 11.2). S uc h subgroups of GL n ( C ) are call ed p ar ab olic sub gr oups , and will b e denoted by P . Clearly Gv λ ∼ = G/P λ = G/P . 11.3 Grassmanians The simplest examples of G/P are Gr assmanians . Deﬁnition 11.8. L et G = Gl n ( C ) , and V = C n . The Gr assmanian Gr n d is the sp ac e of d -dimensional subsp ac es (c ontaining the origin) of V . Examples: 1. Gr 2 1 is the set of lines in C 2 (see ﬁgure 11.3). 2. More generally , P ( V ) = Gr n 1 . Prop osition 11.2. The Gr assmanian Gr n d is a p r oje ctive variety (just like P ( V ) = Gr n 1 ). 67 * A 1 A 2 A 3 A 4 A 5 m 1 m 2 m 3 m 4 m 5 Figure 11.2: The parab olic s ubgroup of blo ck lo wer triang ular matrices. The sizes m i only depend on λ . Figure 11.3: Gr 2 1 is the set of lines in C 2 . 68 It is easy to see that Gr n d is closed (since the limit of a sequence of d - dimensional subsp ace s of V is a d -dimensional s u bspace). Hence this follo ws from: Prop osition 11.3. L et λ = (1 , . . . , 1) b e the p artition o f d , whose al l p arts ar e 1 . Then Gr n d ∼ = Gv λ ⊆ P ( V λ ) . Pr o of. F or th e giv en λ , V λ can b e iden tiﬁed with the d th w edge pro duct Λ d ( V ) = span { ( v i 1 ∧ · · · ∧ v i d ) | i 1 , . . . , i d are distinct } ⊆ V ⊗ · · · ⊗ V (d times) , where ( v i 1 ∧ · · · ∧ v i d ) = 1 d ! X σ ∈ S d sg n ( σ )( v σ ( i 1 ) ⊗ · · · ⊗ v σ ( i d ) ) . Let Z b e a v ariable d × n matrix. Th en C [ Z ] is a G -module: giv en f ∈ C [ Z ] and σ ∈ GL n ( C ), we deﬁne the action of σ b y ( σ · f )( Z ) = f ( Z σ ) . No w Λ d ( V ), as a G -mo dule, is iso morphic to th e span in C [ Z ] of all d × d minors of Z . Let A ∈ Gr n d b e a d -dimensional su bspace of V . T ak e any b asis { v 1 , . . . , v d } of A . The p oin t v 1 ∧ · · · ∧ v d ∈ Λ d ( V ) dep ends only on the subspace A , and not on the basis, since the c hange of basis do es not c hange the wedge prod - uct. Let Z A b e the d × n complex matrix whose rows are the basis v ectors v 1 , . . . , v d of A . The Plucker map asso ciates with A the tuple of all d × d minors A j 1 ,...,j d of Z A , wh ere A j 1 ,...,j d denotes the minor of Z A formed b y the columns j 1 , . . . , j d . This dep ends only on A , and not on the choic e of basis f or A . The pr oposition follo ws f rom: Claim 1 1.1. The Plucker map is a G -e quivariant map fr om Gr n d to Gv λ ⊆ P ( V λ ) and G r n d ≈ Gv λ ⊆ P ( V λ ) . Pr o of. Exercise. Hin t: tak e the usual basis, and n ote that the highest w eigh t p oin t v λ corresp onds to v 1 ∧ · · · ∧ v d . 69 Chapter 12 The class v arieties Scrib e: Ha riharan Nara y anan Goal: Asso ciate class v arieties with the complexit y classes # P and N C and reduce the N C 6 = P # P conjecture o v er C to a conjecture that the class v ariet y for # P cannot b e em b edded in the class v ariet y for N C . r efer enc e: [GCT1] The N C 6 = P # P conjecture o v er C sa ys that the p ermanen t of an n × n complex matrix X cannot b e exp r essed as a determinant of an m × m com- plex matrix Y , m = p oly( n ), whose en tries a re (p ossibly nonhomogeneous) linear forms in the en tries of X . This obvio usly implies the N C 6 = P # P conjecture o v er Z , s in ce m ultiv ariate p olynomials o v er C n are determined b y the v alues that they tak e ov er the subset Z n . The conjecture o v er Z is implied by the usual N C 6 = P # P conjecture o v er a ﬁn ite ﬁeld F p , p 6 = 2, and hen ce, has to b e pr o ve d ﬁrst anyw ay . F or th is reason, w e concen trate on the N C 6 = P # P conjecture o v er C in this lecture. T h e goal is to redu ce this conjecture to a statemen t in geome tric in v arian t theory . 12.1 Class V ariet ies in GCT T o w ards that end, w e associate with the complexi t y classes # P and N C certain pro j ect iv e algebraic v arieties, which we ca ll class varieties . F or this, w e n eed a few deﬁnitions. Let G = GL ℓ ( C ), V a ﬁnite d imensional represent ation of G . Let P ( V ) b e the associated pro jectiv e space, w h ic h inherits the group action. Give n a p oin t v ∈ P ( V ), let ∆ V [ v ] = Gv ⊆ P ( V ) b e its orbit closure. Here Gv is the 70 closure of the orbit Gv in the c omplex t op ology on P ( v ). It is a pro jectiv e G -v ariet y; i.e ., a pro jectiv e v ariet y with the action of G . All cla ss v arieties in GCT are orbit closures (o r their slig ht ge neraliza- tions), w h ere v ∈ P ( V ) co rresp onds to a complete fun ctio n fo r th e class in question. The c hoice of the co mplete fun ctio n is crucial, since it determines the algebraic geometry of ∆ V [ v ]. W e no w asso ciate a class v ariet y with N C . Let g = det ( Y ), Y an m × m v ariable matrix. This is a complete fu nctio n for N C . Let V = sy m m ( Y ) b e the space of h omog eneous forms in th e e ntrie s of Y of degree m . It is a G -mod ule, G = GL m 2 ( C ), with the action of σ ∈ G giv en by: σ : f ( Y ) 7− → f ( σ − 1 Y ) . Here σ − 1 Y is deﬁned thinking of Y as an m 2 -v ector. Let ∆ V [ g ] = ∆ V [ g , m ] = Gg , where w e think of g as an elemen t of P ( V ). T his is the class v ariet y associated with N C . If g is a diﬀeren t function instead of det ( Y ), the algebraic geometry of ∆ V [ g ] wo uld ha v e b een unmanageable. The main point is that the a lgebraic geometry o f ∆ V [ g ] is nice, b ecause of the v ery sp ecial nature of the d eterminan t function. W e next asso ciate a class v ariet y with # P . Let h = per m ( X ), X an n × n v ariable matrix. Let W = sy m n ( X ). It is s imila rly an H -modu le, H = GL k ( C ), k = n 2 . Think of h as an element of P ( W ), and let ∆ W [ h ] = H h b e its orb it closure. It is called the class v ariet y asso ciated w ith # P . No w assume that m > n , and think of X as a submatrix of Y , sa y the lo wer prin cipal subm atrix. Fix a v ariable en try y of Y outside of X . Deﬁne the map φ : W → V wh ic h tak es w ( x ) ∈ W to y m − n w ( x ) ∈ V . This induces a map from P ( V ) to P ( W ) whic h w e call φ as w ell. Let φ ( h ) = f ∈ P ( V ) and ∆ V [ f , m, n ] = Gf its orbit closure. It is called the extended class v ariet y asso cia ted with # P . Prop osition 12.1 (GCT 1) . 1. If h ( X ) ∈ W c an b e c ompute d by a cir- cuit (over C ) of depth ≤ log c ( n ) , c a c onstant, then f = φ ( h ) ∈ ∆ V [ g , m ] , for m = O (2 log c n ) . 2. Conversely if f ∈ ∆ V [ g , m ] for m = 2 log c n , then h ( X ) c an b e appr ox- imate d inﬁnitesima l ly closely by a cir cuit of depth log 2 c m . That is, ∀ ǫ > 0 , th er e e xists a function ˜ h ( X ) that c an b e c ompute d by a cir cuit of depth ≤ l og 2 c m such th at k ˜ h − h k < ǫ in the usual norm on P ( V ) . If the p ermanen t h ( X ) can b e approxima ted inﬁnitesimally closely b y small depth circuits, then ev ery function in # P can b e appro ximated in- ﬁnitesimally closely b y small depth circuits. This is n ot exp ected. Hence: 71 Conjecture 12.1 (GCT 1) . L et h ( X ) = per m ( X ) , X an n × n varia ble matrix. Then f = φ ( h ) 6∈ ∆ V [ g ; m ] if m = 2 poly log ( n ) and n is suﬃciently lar ge. This is equiv alen t to: Conjecture 12.2 (GCT 1) . The G -variety ∆ V [ f ; m, n ] c annot b e emb e dde d as a G -subvariety of ∆ V [ g , m ] , symb olic al ly ∆ V [ f ; m, n ] 6 ֒ → ∆ V [ g , m ] , if m = 2 poly log ( n ) and n → ∞ . This is the state ment in geometric inv arian t theory (GIT) that w e sought . 72 Chapter 13 Obstructions Scrib e: Paolo Co deno tti Goal: Deﬁne an obstru ction to the emb edding of the # P -class v ariet y in the N C -class-v ariet y and describ e wh y it should exist. R efer enc es: [GCT 1, GCT2] Recall Let us ﬁ rst recall some deﬁn itio ns and r esults from the last class. Let Y b e a generic m × m v ariable matrix, and X an n × n minor of Y (see ﬁgure 13.1). Let g = d et( Y ), h = p erm ( X ), f = φ ( h ) = y m − n p erm( X ), and V = Sym m [ Y ] the set of homogeneous forms of d egree m in the entries of Y . Then V is a G -mo dule fo r G = GL ( Y ) = GL l ( C ), l = m 2 , with the act ion Y X n n m m Figure 13 .1: Here Y is a generic m by m matrix, a nd X is an n b y n minor. 73 of σ ∈ G giv en by σ : f ( Y ) → f ( σ − 1 Y ) , where Y is thought of as an l -v ector, and P ( V ) a G -v ariet y . Let ∆ V [ f ; m, n ] = Gf ⊆ P ( V ) , and ∆ V [ g ; m ] = Gg ⊆ P ( V ) b e the class v arieties asso ciated with # P and N C . 13.1 Obstructions Conjecture 13.1. [GCT1] Ther e do es not exist an emb e dding ∆ V [ f ; m, n ] ֒ → ∆ V [ g ; m ] with m = 2 p olylo g ( n ) , n → ∞ . This implies V alian t’s conjecture that the p ermanent cannot b e com- puted by circuits of p olylog dep th . Now w e discuss ho w to go a b out pr oving the conjecture. Supp ose to the cont rary , ∆[ f ; m, n ] ֒ → ∆ V [ g ; m ] . (13.1) W e den ote ∆ V [ f ; m, n ] by ∆ V [ f ], and ∆ V [ g ; m ] by ∆ V [ g ]. Let R V [ g ] b e the homogeneous coordin ate rin g of ∆ V [ g ]. T h e embedd ing (1 3.1) implies existence of a su r jecti on: R V [ f ] և R V [ g ] (13.2) This is a basic fac t from algebraic geometry . T he r easo n is that R V [ g ] is the set of homoge neous p olynomial functions on t he cone C of ∆ V [ g ], and an y suc h fu nction τ can b e restricted to ∆ V [ f ] (see ﬁgur e 13 .2). Con versely , an y p olynomial function on ∆ V [ f ] can b e extended to a homogeneous p olynomial function on the cone C . Let R V [ f ] d and R V [ g ] d b e the degree d comp onen ts of R V [ f ] and R V [ g ]. These are G -modules since ∆ V [ f ] and ∆ V [ g ] are G -v arieties. The surjection (13.2) is degree pr eserving. So there is a s u rjectio n R V [ f ] d և R V [ g ] d (13.3) for every d . Since G is r eductiv e, b oth R V [ f ] d and R V [ g ] d are direct sums of irreducible G -mo dules. Hence the su rjectio n (13.3) implies that R V [ f ] d can b e em b edded as a G submo dule of R V [ g ] d . 74 C τ ∆ V [ f ] Figure 13.2: C denotes the cone of ∆ V [ g ]. Deﬁnition 13.1. We say that a Weyl-mo dule S = V λ ( G ) is an obstruction for the emb e dding (13.1) (o r, e quivalently, for the p air ( f , g ) ) if V λ ( G ) o c curs in R V [ f ; m, n ] d , but not in R V [ g ; m ] d , for some d . Her e o c curs me ans the multiplicity of V λ ( G ) in th e de c omp osition of R V [ f ; m, n ] d is nonzer o. If a n obstruction exists for giv en m, n , then t he em b edding (13.1) does not exist. Conjecture 13.2 (GCT2) . An obstruction e xists for t he p air ( f , g ) for al l lar ge enough n if m = 2 p olylo g ( n ) . This implies Conjecture 13.1. In essence, this turn s a nonexistence prob- lem (of p olylog depth circuit for the p ermanent ) into an existence pr oblem (of an obstruction). If we replace the d ete rminant here by an y other complete function in N C , an obstru cti on n eed not exist. Because, as we sh all see in the n ext lecture, the existence of an obstruction c rucially dep ends on the exceptional nature of the class v ariet y constructed from the determinan t. Th e main goals of GCT in this con text are: 1. understand the exceptional nature of the class v arieties for N C and # P , and 2. use it to prov e the existence of obstru ctio ns. 13.1.1 Why are t he class v arieties exceptional? W e no w elaborate on the e xceptional nature of th e class v arieties. Its signif- icance for the exi stence of obstru cti ons will b e discussed in the n ext lecture. Let V b e a G -mo dule, G = GL n ( C ). Let P ( V ) b e a pro jectiv e v ariet y o ver V . Let v ∈ P ( V ), and rec all ∆ V [ v ] = Gv . Let H = G v b e the sta bilizer of v , that is, G v = { σ ∈ G | σ v = v } . 75 Deﬁnition 13.2. We say that v is c h aract erized b y i ts sta bilizer H = G v if v is the only p oint in P ( V ) such that hv = v , ∀ h ∈ H . If v is charact erized by its stabilizer, then ∆ V [ v ] is completely determined b y the group triple H ֒ → G ֒ → K = GL ( V ). Deﬁnition 13.3. The orbit closur e ∆ V [ v ] , when v is char acterize d by its stabilizer, i s c al le d a group-theoretic v ariet y . Prop osition 13.1. [GCT1] 1. The d eterminant g = det( Y ) ∈ P ( V ) is char acterize d by its st abilizer. Ther efor e ∆ V [ g ] is gr oup the or etic. 2. The p ermanent h = p erm ( X ) ∈ P ( W ) , wher e W = Sym n ( X ) , is also char acterize d by its stabilizer. Ther efor e ∆ W [ h ] is also gr oup the or etic. 3. Final ly, f = φ ( h ) ∈ P ( V ) is also char acterize d by its stabilizer. Henc e ∆ V [ f ] is also gr oup the or etic. Pr o of. (1) It is a fact in classical represent ation theory that the stabilizer of det( Y ) in G = GL ( Y ) = GL m 2 ( C ) is t he subgroup G det that consists o f linear transform ati ons of the form Y → AY ∗ B , where Y ∗ = Y or Y t , for any A, B ∈ GL m ( C ). It is clear that linear trans f ormatio n of this form stabilize the d ete rminant since: 1. det( AY B ) = det ( A ) det ( B ) det ( Y ) = c det( Y ), where c = det( A ) det( B ). Note that th e constan t c do esn’t matter b ecause we get the same p oin t in the p ro jectiv e spac e. 2. det( Y ∗ ) = d et( Y ). It is a basic fact in classical in v arian t th eory that d et( Y ) is the o nly p oin t in P ( V ) stabiliz ed b y G det . F u rthermore, the sta bilizer G det is r eductiv e, since its connected part is ( G det ) ◦ ≈ GL m × GL m with the natural embedd ing ( G det ) ◦ = GL m × GL m ֒ → GL ( C m ⊗ C m ) = GL m 2 ( C ) = G. (2) T h e stabilizer of p erm( x ) is the subgroup G p erm ⊆ GL ( X ) = GL n 2 ( C ) generated by linear transformations of the form X → λX ∗ µ , wh ere X ∗ = X or X t , and λ a nd µ are diagonal (which c hange the p ermanent b y a con- stan t factor) or p ermutat ion matrices (which do n ot c hange the p ermanent) . Finally , the d iscrete comp onent of G p erm is isomorph ic to S 2 ⋊ S n × S n , where ⋊ denotes semidirect pro duct. The con tinuous part is ( C ∗ ) n × ( C ∗ ) n . So G p erm is r eductiv e. 76 (3) S imila r. The main signiﬁcance of this pr oposition is the follo win g. Beca use ∆ V [ g ] , ∆ V [ f ], and ∆ W [ h ] are group theoretic, the algebraic geomet ric prob- lems concerning these v arieties can b e “reduced” to problems in the theory of qu an tum groups. So the p lan is: 1. Use the theory of qu antum groups to und erstand the structure of the group triple asso ciated with the algebraic v ariet y . 2. T ranslate this und erstanding to the stru cture of the algebraic v ariet y . 3. Use this to sho w the existence of obstru ctio ns. 77 Chapter 14 Group theoretic v arieties Scrib e: Joshua A. Gro c how Goal: In this lecture w e conti nue our d iscussion of group-theoretic v arieties. W e describ e wh y obstructions should exist, and wh y the exceptional group- theoretic nature of th e class v arieties is crucial f or this existence. Recall Let G = G L n ( C ), V a G -mod u le, a nd P ( V ) the asso ciated pro jectiv e space. Let v ∈ P ( V ) b e a p oin t charact erized by its stabilizer H = G v ⊂ G . In other w ords, v is the only p oint in P ( V ) stabilized b y H . Then ∆ V [ v ] = Gv is called a gr oup-the or etic variety b ecause it is completely d etermined by the group triple H ֒ → G ֒ → GL ( V ) . The simp lest example of a group -theo retic v ariet y is a v ariet y of the form G/P that w e describ ed in the earlier lecture. Let V = V λ ( G ) b e a W eyl mo dule of G and v λ ∈ P ( V ) the h ighest w eigh t p oin t (recall: the unique p oin t stabil ized b y the Borel subgroup B ⊂ G of lo we r triangular matrice s). Then the stabilizer of v λ consists of blo c k-up p er triangular matrices, w here 78 the b loc k sizes are determined b y λ : P λ := G v λ =             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗             The orbit ∆ V [ v λ ] = Gv λ ∼ = G/P λ is a group-theoretic v ariet y determined en tirely b y the triple P λ = G v λ ֒ → G ֒ → K = G L ( V ) . The g roup-theoretic v arieties of main inte rest in GCT are the class v a- rieties asso ciated with the v arious complexit y c lasses. 14.1 Represen tation theoret ic data The main principle guiding GCT is th at the algebraic geometry of a group- theoretic v ariet y ough t to b e completely determined by the representat ion theory of the corresp onding group triple. Th is is a natur al extension of w ork already pursued in mathemati cs b y De ligne and Milne on T annakien catego ries [DeM], sho wing that an algebraic group is completely d ete rmined b y its represen tation theo ry . So the go al is to associate to a group-theoretic v ariet y some repr esen tation-theoretic data that w ill analogo usly capture the information in the v ariet y completel y . W e shall no w illustrate this for the class v ariet y for N C . First a few deﬁnitions. Let v ∈ P ( V ) b e the p oint as ab ov e charact erized b y its stabilizer G v . This means the line C v ⊆ V corresp ondin g t o v is a one-dimensional repre- sen tation of G v . Th us ( C v ) ⊗ d is a one-dimensional degree d represent ation, i.e. the represen tation ρ : G → GL ( C v ) ∼ = C ∗ is p olynomial of degree d in the en tries of the matrix of an elemen t in G . Reca ll that C [ V ] is the co ord inate ring o f V , and C [ V ] d is its degree d homog eneous comp onen t, so ( C v ) ⊗ d ⊆ C [ V ] d . T o eac h v ∈ P ( V ) th at is c haracterized b y it s stabilizer, w e a sso ciate a represen tation-theoreti c data, whic h is the set of G -mo dules Π v = [ d Π v ( d ) , 79 where Π v ( d ) is the set of all irredu cible G -sub modu les S of C [ V ] d whose duals S ∗ do not con tain a G v -submo dule isomorphic to ( C v ) ⊗ d ∗ (the dual of ( C v ) ⊗ d ). The follo wing prop osit ion elucidates the imp ortance of this data: Prop osition 14.1. [GCT2] Π v ⊆ I V [ v ] (wher e I V [ v ] is the ide al of the pr oje ctive variety ∆ V [ v ] ⊆ P ( V ) ). Pr o of. Fix S ∈ Π v ( d ). Su p p ose, f or the sak e of con tradiction, that S * I V [ v ]. Since S ⊆ C [ V ], S consists of “functions” on the v ariet y P ( V ) (ac- tually homoge neous p olynomials on V ). The co ordinate ring of ∆ V [ v ] is C [ V ] /I V [ v ], and since S * I V [ v ], S must not v anish identi cally on ∆ V [ v ]. Since the orbit Gv is dense in ∆ V [ v ], S must n ot v anish ident ically o n this single orbit Gv . S ince S is a G -mo dule, if S were to v anish identic ally on the line C v , then it wo uld v anish o n the e ntire orbit Gv , so S do es not v anish iden tically on C v . No w S consists of functions of degree d . Restrict them to the line C v . The dual of this restriction giv es an inj ect ion of ( C v ) ⊗ d ∗ as a G v -submo dule of S ∗ , contradic ting the deﬁnition of Π v ( d ). 14.2 The second fu n damen tal theorem W e no w ask essen tially the rev erse questio n: when do es th e represen tation theoretic data Π v generate the id eal I V [ v ]? F or if Π v generates I V [ v ], then Π v completely captures the coordinate ring C [ V ] /I V [ v ], and h ence the v ariet y ∆ V [ v ]. Theorem 14.1 (Second f u ndamen tal theorem of inv arian t theory for G/P ) . The G -mo dules in Π v λ (2) ge ner ate th e ide al I V [ v λ ] of the orbit Gv λ ∼ = G/P λ , when V = V λ ( G ) . This theorem justiﬁes the main principle for G/P , so w e c an hop e that similar results hold for the class v arieties in GCT (though n ot alw a ys exactly in the same form). No w , let ∆ V [ g ] b e the class v ariet y for NC (in other words, tak e g = det( Y ) for a matrix Y of indeterminates). Based on the main pr inciple, w e hav e the follo wing conjecture, whic h essen tially general izes the second fundamenta l theorem of in v arian t theory for G/P to the class v ariet y for NC : Conjecture 14.1 (GCT 2) . ∆ V [ g ] = X (Π g ) wher e X (Π g ) is the zer o-set of al l forms in the G -mo dules c ontaine d in Π g . 80 Theorem 14.2 (GCT 2) . A we aker ve rsion of the ab ove c onje ctur e holds. Sp e ciﬁc al ly, assuming that the Kr one cker c o eﬃcients satisfy a c ertain sep- ar ation pr op erty, ther e exists a G -invariant (Zariski) op en neighb ourho o d U ⊆ P ( V ) of the orbit Gg such that X (Π g ) ∩ U = ∆ V [ g ] ∩ U . There is a notion of algebro-geomet ric co mplexit y called Luna-V ust c om- plexity w hic h qu an tiﬁes the gap b et we en G/P and class v arieties. The Luna- V ust complexit y of G/P is 0. T he Lu na-V ust complexi t y of the NC class v ariet y is Ω(dim( Y )). This is analogo us to the diﬀerence b etw een circuits of constan t depth and circuits of sup erp olynomial depth. This is wh y the previous c onjecture and theorem turn out to b e far harder than the co rre- sp onding facts for G/P . 14.3 Wh y should obstruc tio ns exist? The follo w in g prop osition explains wh y obstructions sh ould exist to separate NC fr om P # P . Prop osition 14.2 (GCT 2) . L et g = det( Y ) , h = p erm ( X ) , f = φ ( h ) , n = dim( X ) , m = dim( Y ) . If Conje ctur e 14.1 holds and the p ermanent c an- not b e appr oximate d arbitr arily closely by cir cuits of p oly-lo garithmic depth (har dness assumption), then an obstruction for the p air ( f , g ) exists fo r al l lar ge enough n , when m = 2 log c n for some c onstant c . Henc e , under these c onditions, NC 6 = P # P over C . This p rop osit ion ma y seem a b it circular at ﬁr st, since it relies on a hard- ness assumption. But w e do not plan to pro v e the existence of obstructions b y pro ving the assumptions of this prop osition. Rather, this pr oposition should b e take n as evidence that obstructions exist (since we exp ect the hardness assum ption therein to hold, giv en that th e p ermanent is # P - complete), and we will dev elop other metho ds to pr o v e their existence. Pr o of. The hardness assum ption implies that f / ∈ ∆ V [ g ] if m = 2 log c n [GCT 1]. Conjecture 14.1 sa ys that X (Π g ) = ∆ V [ g ]. So there exists an irred u cible G -mod ule S ∈ Π g suc h that S do es not v anish on f . S o S o ccurs in R V [ f ] as a G -submo dule. On the other h and, since S ∈ Π g , S ⊆ I V [ g ] by Prop osition 14.1. S o S do es not o ccur in R V [ g ] = C [ V ] /I V [ g ]. Thus S is not a G -submod ule of R V [ g ], bu t it is a G -submo dule of R V [ f ], i.e., S is an obstruction. 81 Chapter 15 The ﬂip Scrib e: Ha riharan Nara y anan Goal: Describ e the basic principle of GCT, called the ﬂ ip, in the con text of the N C vs. P # P problem ov er C . r efer enc es: [GCTﬂ ip1 , GCT1, GCT2, GCT6] Recall As in the pr evious lectures, let g = det ( Y ) ∈ P ( V ), Y an m × m v ari- able matrix, G = GL m 2 ( C ), and ∆ V ( g ) = ∆ V [ g ; m ] = Gg ⊆ P ( V ) the class v ariet y for NC. Let h = per m ( X ), X an n × n v ariable matrix, f = φ ( h ) = y m − n h ∈ P ( V ), and ∆ V ( f ) = ∆ V [ f ; m, n ] = Gf ⊆ P ( V ) th e class v ariet y for P # P . Let R V [ f ; m, n ] denote the homog eneous co ordinate ring of ∆ V [ f ; m, n ], R V [ g ; m ] the homogeneo us co ordinate ring of ∆ V [ g ; m ], and R V [ f ; m, n ] d and R V [ g ; m ] d their degree d -comp onen ts. A W eyl mo dule S = V λ ( G ) of G is an obstruction of degree d for the pair ( f , g ) if V λ o ccurs in R V [ f ; m, n ] d but n ot R V [ g ; m ] d . Conjecture 15.1. [GCT2] An obstruction (of d e gr e e p olynomial in m ) ex- ists if m = 2 p olylo g ( n ) as n → ∞ . This imp lie s N C 6 = P # P o ver C . 82 15.1 The ﬂip In this lecture w e describ e an approac h to prov e the existence of suc h ob- structions. It is based on the follo wing complexit y theoretic p ositivit y hy- p othesis: PHﬂip [GCTﬂ ip 1]: 1. Giv en n , m and d , whether an obstruction of degree d for m and n exists can b e decided in poly ( n, m, h d i ) time, and if it exists, the lab el λ of such an obstruction can b e constructed in poly ( n, m , h d i ) time. Here h d i denotes the bitlength of d . 2. (a) Whether V λ o ccurs in R V [ f ; m, n ] d can b e decided in pol y ( n, m, h d i , h λ i ) time. (b) Whet her V λ o ccurs in R V [ g ; m ] d can b e decided in pol y ( n, m, h d i , h λ i ) time. This su gge sts the follo wing appr oac h for pro ving Conjecture 15.1: 1. Find p olynomial time algorithms sought in PHﬂip-2 for t he basic d e- cision pr oblems (a) and (b) therein. 2. Using these ﬁnd a p olynomial time algorithm sought in PHﬂip-1 for deciding if an obstru cti on exists. 3. T ransform (the tec hniques underlying) this “easy” ( p olynomial time) algorithm for deciding if an obstruction exists for giv en n and m into an “easy” (i.e., feasible) pro of of existence of an obstruction for ev ery n → ∞ when d is large enough and m = 2 p olylog ( n ) . The ﬁrst step here is the crux of the matter. Th e main results of [GCT6] sa y that the p olynomial time algorithms for the basic decision problems as sough t in PHﬂip-2 indeed exist assuming natural analogues of PH1 and SH (PH2) that we h a ve seen earlier in the con text of the pleth ysm problem. T o state them, we need some d eﬁ n itio ns. Let S λ d [ f ] = S λ d [ f ; m, n ] b e the m ultiplicit y of V λ = V λ ( G ) in R V [ f ; m, n ]. The stretc hing fu nction ˜ S d [ f ] = ˜ S λ d [ f ; m, n ] is deﬁ n ed b y ˜ S λ d [ f ]( k ) := S k λ k d [ f ] . The stretc hing fu nction for g , ˜ S λ d [ g ] = ˜ S λ d [ g ; m ], i s deﬁned analogously . The main mathematical r esult of [GCT6] is: 83 Theorem 15.1. [GCT6] The str etching functions ˜ S λ d [ g ] and ˜ S λ d [ f ] ar e quasip oly- nomials assuming that the singularities of ∆ V [ f ; m, n ] and ∆ V [ g ; m ] ar e r a- tional. Here rational means “nice”; we s h all not w orr y ab out the e xact deﬁni- tion. The main complexit y-theoretic result is: Theorem 15.2. [GCT6] A ssuming the fol lowing math ematic al p ositivity hyp othesis P H 1 and the sa tur ation hyp othesis S H (or the str onger p ositivity hyp othesis P H 2 ), PHﬂip-2 holds. PH1: There exists a p olytop e P = P λ d [ f ] such that 1. The Ehrhart quasi-p olynomial of P , f P ( k ), is ˜ S λ d [ f ]( k ). 2. dim ( P ) = poly ( n, m, h d i ). 3. Mem b ership in P can b e ans w ered in p olynomial time. 4. There is a p olynomial time separation oracle [GLS] for P . Similarly , there exists a p olytop e Q = Q λ d [ g ] suc h that 1. The Ehrhart quasi-p olynomial of Q , f Q ( k ), is ˜ S λ d [ g ]( k ). 2. dim ( Q ) = pol y ( m, h d i ). 3. Mem b ership in Q can b e ans w ered in p olynomial time. 4. There is a p olynomial time separation oracle f or Q . PH2: The quasi-p olynomials ˜ S λ d [ g ] and ˜ S λ d [ f ] are p ositiv e. This imp lie s: SH: T he quasi-p olynomials ˜ S λ d [ g ] and ˜ S λ d [ f ] are saturated. PH1 and SH imply that the decision problems in PHﬂip-2 ca n b e trans- formed in to saturated p ositiv e in teger p r ogramming pr ob lems. Hence Th e- orem 15.2 foll o ws fr om the p olynomial time a lgorithm for saturated linea r programming that w e describ ed in an earlier class. The decision problems in PHﬂip -2 are “h yp ed” up v ersions of the pleth ysm problem discussed earlier. The article [GCT6] pro vides evidence for P H 1 84 and P H 2 for the pleth ysm problem. Th is constitutes the main evidence for PH1 and PH2 for the class v arieties in view of their group-theo retic nature; cf. [GCTﬂip1]. The follo wing pr oblem is imp ortan t in the con text of PHﬂip-2: Problem 15.1. Understand the G -mo dule structur e of the homo gene ous c o or dinate rings R V [ f ] d and R V [ g ] d . This is an instance of the follo wing abstract: Problem 15.2. L et X b e a pr oje ctive gr oup-the or etic G -variety. L et R = L ∞ d =0 R d b e its homo gene ous c o or dinate ring. Understand the G -mo dule structur e of R d . The simplest group-theoretic v ariet y is G/P . F or it, a solution to this abstract p roblem is giv en b y the follo wing results: 1. The Borel-W e il theorem. 2. The Second F u ndamen tal theorem of inv ariant theory [SFT]. These will b e co vered in t he next cla ss for the simplest case of G/P , the Grassmanian. 85 Chapter 16 The Grassmani an Scrib e: Ha riharan Nara y anan Goal: The Borel-W e il and the second fundamenta l theorem of in v arian t theory for the Grassmanian. R efer enc e: [F] Recall Let V = V λ ( G ) b e a W eyl mo dule of G = GL n ( C ) and v λ ∈ P ( V ) the p oin t corresp onding to its highest wei ght v ector. The orbit ∆ V [ v λ ] := Gv λ , whic h is already closed, is of the form G/P , wh ere P is the parab olic stabilizer of v λ . When λ is a single column, it is called the Gr assmannian . An alternativ e description of the Grassmanian is as follo ws. Assume that λ is a single column of length d . Let Z b e a d × n ma trix of v ariables z ij . Then V = V λ ( G ) can b e id entiﬁed with the sp an of d × d minors of Z with the action of σ ∈ G giv en by: σ : f ( z ) 7→ f ( z σ ) . Let Gr n d b e the space of all d-dimensional s ubspaces of C n . Let W b e a d -dimensional sub s pace of C n . L et B = B ( W ) be a basis of W . Constru ct the d × n matrix z B , wh ose rows are v ectors in B . Consider th e Pl ¨ uc ker map from Gr n d to P ( V ) which maps any W ∈ Gr n d to the tuple of d × d minors of Z B . Here the c h oic e of B = B ( W ) do es not m att er, since an y c h oice giv es the same p oin t in P ( V ). Then the image of Gr n d is precisely th e Grassmanian Gv λ ⊆ P ( V ). 86 16.1 The second fu n damen tal theorem No w we ask: Question 16.1. What is t he ide al of Gr n d ≈ Gv λ ⊆ P ( V ) ? The homogeneous co ordinate ring of P ( V ) is C [ V ]. W e w an t an ex- plicit set of generators of this ideal in C [ V ]. This is given b y the second fundamenta l theorem of in v ariant theory , whic h w e describ e next. The co ordinates of P ( V ) are in one-to-one corresp ondence with the d × d minors of the m atrix Z . Let eac h minor of Z b e ind exed by its columns. Thus for 1 ≤ i 1 < · · · < i d ≤ n , Z i 1 ,...,i d is a co ordinate of P ( V ) corresp ond in g to the minor of Z formed by the columns i 1 , i 2 , . . . . Let Λ( n, d ) b e the set of ordered d -tuples of { 1 , . . . , n } . T he tuple [ i 1 , . . . , i d ] in this set will b e identi ﬁed with the coordin ate Z i 1 ,...,i d of P ( V ). There is a bijection b et ween the elemen ts of Λ( n, d ) and of Λ ( n, n − d ) obtained by asso ciati ng complemen tary set s: Λ( n, d ) ∋ λ ! λ ∗ ∈ Λ( n, n − d ) . W e deﬁne sg n ( λ, λ ∗ ) to b e the sign of the p erm utation that take s [1 , . . . , n ] to [ λ 1 , . . . , λ d , λ ∗ 1 , . . . , λ ∗ n − d ]. Giv en s ∈ { 1 . . . . , d } , α ∈ Λ( n, s − 1), β ∈ Λ( n, d + 1), and γ ∈ Λ( n, d − s ), w e n o w deﬁne the V an der W a erden Syzygy [[ α, β , γ ]], whic h is an elemen t of the degree t w o comp onen t C [ V ] 2 of C [ V ], as follo ws: [[ α, β , γ ]] = P τ ∈ Λ( d +1 ,s ) sg n ( τ , τ ∗ )[ α 1 , . . . , α s − 1 , β τ ∗ 1 , . . . , β τ ∗ d +1 − s ][ β τ 1 , . . . , β τ s , γ 1 , . . . , γ d − s ] . It is easy to sho w that this syzygy v anishes on the Grassmanian Gr n d : b ecause it is an alternating ( d + 1)-m ultilinear-form, and hence has to v anish on an y d - dimensional sp ace W ∈ G r n d . Thus it b elongs to the ideal of t he Grassmanian. Moreo v er: Theorem 16.1 (Second fu n damen tal theorem) . The ide al of the Gr assma- nian Gr n d is gener ate d by the V an-der-Waer den syzygies. An alternativ e form ulation of this result is as follo ws. Let P λ ⊆ G b e the stabilizer of v λ . Let Π v λ (2) b e the s et of irr educible G -submo dules of C [ V ] 2 whose duals do not con tain a P λ -submo dule isomorphic to C v ⊗ 2 ∗ λ (the dual of C v ⊗ 2 λ ). Here C v λ denotes the line in P ( V ) corresp onding to v λ , wh ic h is a one-dimensional rep r esen tation of P λ since it stabilizes v λ ∈ P ( V ). It can 87 b e sh o wn that the span of the G -mo dules in Π v λ (2) is equal to the span of the V an-d er-W ae rden syzyg ies. Hence, Th eorem 16.1 is equiv alen t to: Theorem 16.2 (Second F un damen tal Th eorem(SFT)) . The G -mo dules in Π v λ (2) gener ate the ide al of Gr n d . This form ulation o f SFT for the Grassmanian l o oks very similar t o the generalize d conjectural SFT for t he N C -class v ariet y describ ed in the ear- lier c lass. T h is in d icat es that t he class v arieties i n GCT a re “qualita tiv ely similar” to G/P . 16.2 The Borel-W eil theorem W e no w describ e the G -module structure of the homogeneo us co ordinate ring R of the Grassmannian Gv λ ⊆ P ( V ), wh ere λ is a single column of heigh t d . The goal is to giv e an e xplicit basis for R . Let R s b e the degree s component of R . Corresp onding to an y n u mb ering T of the shap e sλ , whic h is a d × s rectangle, w hose columns h a v e strictly increasing elemen ts top to b ottom, we ha ve a monomial m T = Q c Z c ∈ C [ V ] s , were Z c is the co ord inate of P ( V ) ind exed by the d -tuple c , and c r anges o v er th e s columns of T . W e s ay that m T is (semi)-standard if the ro ws o f T are nond ecrea sing, when r ead left to r igh t. It is called n onstandard otherwise. Lemma 16.1 (Straigh tening Lemma) . Each non-standar d m T c an b e str aight- ene d to a normal form, as a line ar c ombination of sta ndar d monomials, by using V an der Waer den Syzygies as str aightening r elations (r ewriting rules). F or any n u m b ering T as ab o v e, exp r ess m T in a normal form as p er the lemma: m T = X (Semi)-Standard T ableau S α ( S, T ) , m S where α ( S, T ) ∈ C . Theorem 16.3 (Borel-W eil Th eorem for Grassmannians) . Standar d mono- mials { m T } form a b asis of R s , wher e T r anges over al l semi-standar d table aux of r e ctangular shap e sλ . Henc e, R s ∼ = V ∗ sλ , the dual of the Weyl mo dule V sλ . This gi v es the G -module structure of R c ompletely . It follo ws that t he problem of deciding if V β ( G ) o ccurs in R s can b e so lv ed in p olynomial time: this is so if and only i f ( sλ ) ∗ = β , where ( sλ ) ∗ denotes the d ual partition, whose description is left as an exercise. 88 The second fun damen tal theorem as well as the Borel-W eil theorem easily follo w fr om the straigh tening lemma and linear indep endence of the standard monomials (as functions on the Grassmanian). 89 Chapter 17 Quan tum group: basic deﬁnitions Scrib e: Paolo Co deno tti Goal: Th e basic p lan to implemen t the ﬂip in [GCT6] is to pro ve PH1 and SH vi a the theory of qu antum groups. W e in tro duce the basic concepts i n this theory in this and the next tw o lectures, and brieﬂy sh o w their r elev ance in the con text of P H1 in the ﬁnal lecture. R efer enc e: [KS] 17.1 Hopf Algebras Let G b e a group, and K [ G ] the ring of functions on G with v alues i n the ﬁeld K , w h ic h will b e C in our applications. The group G is deﬁned by the follo wing o p erations: • multiplica tion: G × G → G , • id entit y e : e → G , • inv erse: G → G . In order for G to b e a group, the f oll o wing p r operties ha v e to hold: • eg = g e = g , • g 1 ( g 2 g 3 ) = ( g 1 g 2 ) g 3 , • g − 1 g = g g − 1 = e . 90 W e no w w an t to translate these prop erties to prop erties of K [ G ]. This should b e p ossible since K [ G ] con tains all the information that G has. In other w ords, we wa nt to translate t he notion of a group in terms of K [ G ]. This translate is called a Ho pf algebra. Th us if G is a group, K [ G ] is a Hopf algebra. Let us ﬁ rst deﬁne the du al op erati ons. • Multiplication is a map: · : G × G → G. So co-m ultiplication ∆ will b e a map as follo w s: K [ G × G ] = K [ G ] ⊗ K [ G ] ← K [ G ] . W e w an t ∆ to b e the pullbac k of m ultiplication. So for a giv en f ∈ K [ G ] we deﬁne ∆( f ) ∈ K [ G ] ⊗ K [ G ] b y: ∆( f )( g 1 , g 2 ) = f ( g 1 g 2 ) . Pictoriall y: G × G · − − − − → G ∆( f )   y   y f k k • T he unit is a map: e → G. Therefore we w an t the co-unit ǫ to b e a map: K ǫ ← − K [ G ] , deﬁned by: for f ∈ K [ G ], ǫ ( f ) = f ( e ). • Inv ers e is a map: ( ) − 1 : G → G. W e wan t the dual antipo de S to b e the map: K [ G ] ← K [ G ] deﬁned by: for f ∈ K [ G ], S ( f )( g ) = f ( g − 1 ). The follo wing are the abstract axioms satisﬁed by ∆ , ǫ and S . 91 1. ∆ and ǫ are algebra homomorphisms . ∆ : K [ G ] → K [ G ] ⊗ K [ G ] ǫ : K [ G ] → K. 2. co-a sso ciativit y: Asso ciati vit y is deﬁn ed so that the follo wing diagram comm utes: G × G × G G × G × G ·   y id   y   y id   y · G × G G × G ·   y   y · G G Similarly , we deﬁne co-asso ciat ivit y so that the follo wing d ual diagram comm utes: K [ G ] ⊗ K [ G ] ⊗ K [ G ] K [ G ] ⊗ K [ G ] ⊗ K [ G ] ∆ x   id x   x   id x   ∆ K [ G ] ⊗ K [ G ] K [ G ] ⊗ K [ G ] ∆ x   x   ∆ K [ G ] K [ G ] Therefore co-asso cia tivit y sa ys: (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ . 3. The prop ert y ge = g is deﬁned so that the follo wing diagram com- m utes: e × G G e   y   y id   y G × G id   y ·   y G G 92 W e deﬁn e the c o o f this prop ert y so that the follo wing diagram com- m utes: K × K [ G ] K [ G ] ǫ x   x   id x   K [ G ] × K [ G ] id x   ∆ x   K [ G ] K [ G ] That is, id = ( ǫ ⊗ id) ◦ ∆ . Similarly , g e = g translates to: id = (id ⊗ ǫ ) ◦ ∆ . Therefore we get id = ( ǫ ⊗ id) ◦ ∆ = (id ⊗ ǫ ) ◦ ∆ . 4. The last pr operty is g g − 1 = e = g − 1 g . The ﬁrst equalit y is equiv alen t to requiring that the follo win g diag ram comm ute: G G diag   y   y G × G   y () − 1   y   y id e G × G   y   y ·   y G G Where diag : G → G × G is the diag onal em b edding. The co of diag is m : K [ G ] ← K [ G ] ⊗ K [ G ] deﬁn ed by m ( f 1 , f 2 )( g ) = f 1 ( g ) · f 2 ( g ). So the co of this p rop ert y w ill hold when the follo wing diagram comm utes: 93 K [ G ] K [ G ] m x   x   K [ G ] ⊗ k [ G ] ν x   S x   x   id K K [ G ] ⊗ K [ G ] x   x   ∆ ǫ x   K [ G ] K [ G ] Where ν is the em b edding of K into K [ G ]. Therefore the last pr op erty w e wan t to b e satisﬁed is: m ◦ ( S ⊗ id) ◦ ∆ = ν ◦ ǫ. F or e = g − 1 g , we similarly get: m ◦ (id ⊗ S ) ◦ ∆ = ν ◦ ǫ. Deﬁnition 17.1 (Hopf algebra) . A K - algebr a A i s c al le d a Hopf alge bra if ther e exist homomorp hisms ∆ : A ⊗ A → A , S : A → A , ǫ : A → K , and ν : A → K tha t sa tisfy (1) − (4) ab ove, with A in plac e of K [ G ] . W e h a ve s ho wn that if G is a group, the ring K [ G ] of functions on G is a (co mmuta tiv e) Hopf algebra, whic h is non-co-comm u tat iv e if G is non-comm utativ e. Th u s for ev ery usual group, w e get a commutat iv e Ho pf algebra. Ho we ve r, in general, Hopf algebras may b e non-comm utativ e. Deﬁnition 17.2. A quan tum group is a (non-c ommutative and non-c o- c ommutative) Hopf algebr a. A nont rivial example of a quant um group w ill b e constru cte d in the next lecture. Next we wa nt to look a t what happ ens to group theoretic n otio ns su c h as represen tations, actions, and homomorphisms, in the con text of Hopf algebras. These will corresp ond to co-represen tations, co-ac tions, and co- homomorphisms. Let us lo ok closely a t the n oti on of co-represen tation. A r epresen tation is a map · : G × V → V , such that • ( h 1 h 2 ) · v = h 1 · ( h 2 · v ), and 94 • e · v = v . Therefore a (righ t) co-represen tation of A w ill b e a linear mapping ϕ : V → V ⊗ A , where V is a K -v ector space, and ϕ satisﬁes the follo win g: • T he follo wing diagram comm utes: V ⊗ A ⊗ A id ⊗ ∆ ← − − − − V ⊗ A ϕ ⊗ id x   x   ϕ V ⊗ A ← − − − − ϕ V That is, the follo wing equalit y holds: ( ϕ ⊗ id) ◦ ϕ = (id ⊗ ∆) ◦ ϕ. • T he follo wing diagram comm utes: V ⊗ K id ← − − − − V ⊗ K id ⊗ ǫ x      V ⊗ A ← − − − − ϕ V That is, the follo wing equalit y holds: (id ⊗ ǫ ) ◦ ϕ = id In fact all usual group theo retic n oti ons can b e “Hopﬁﬁed” in th is sen s e [exercise]. Let us lo ok now at an example. Let G = GL n ( C ) = GL ( C n ) = GL ( V ) , where V = C n . Let M n b e the matrix space of n × n C -matrices, and O ( M n ) the co ordinate r ing of M n , O ( M n ) = C [ U ] = C [ { u i j } ] , where U is a n n × n v ariable m atrix with ent ries u i j . Let C [ G ] = O ( G ) b e the coordin ate ring of G obtained by adjoining det ( U ) − 1 to O ( M n ). T hat is, C [ G ] = O ( G ) = C [ U ][det( U ) − 1 ], whic h is the C algebra generated b y u i j ’s and det( U ) − 1 . 95 Prop osition 17.1. C [G] is a Hopf algebr a, with ∆ , ǫ , and S as fol lows. • R e c al l that the axiom s of a Hopf algebr a r e quir e that ∆ : C [ G ] → C [ G ] ⊗ C [ G ] , ∆( f )( g 1 , g 2 ) = f ( g 1 g 2 ) . Ther efor e we deﬁne ∆( u i j ) = X k u i k ⊗ u k j , wher e U denotes the generic matrix in M n as ab ove. • A gain, it is r e quir e d that ǫ ( f ) = f ( e ) . Ther efor e we deﬁne ǫ ( u i j ) = δ ij , wher e δ ij is the Kr one cker delta fu nc tion. • Final ly, the antip o de is r e quir e d to satisfy S ( f )( g ) = f ( g − 1 ) . L et e U b e th e c ofactor matrix of U , U − 1 = 1 det( U ) e U , and e u i j the entr ies of e U . Then we deﬁne S by: S ( u i j ) = 1 det( U ) e u i j = ( U − 1 ) i j . 96 Chapter 18 Standard quan tum group Scrib e: Paolo Co deno tti Goal: In this lecture w e construct the standard (Drinfeld-Jim b o) quant um group, whic h is a q -deformation of the general linear group GL n ( C ) with remark able p rop erties. R efer enc e: [KS] Let G = GL ( V ) = GL ( C n ), and V = C n . In the earlier lecture, we constructed the co mmutat iv e and n on co-comm utativ e Hopf algebra C [ G ]. In this lec ture w e quan tize C [ G ] to get a non-comm utativ e and non-co- comm utativ e Hopf algebra C q [ G ], and then deﬁne the standard quan tum group G q = G L q ( V ) = GL q ( n ) as th e virtual ob ject wh ose co ordinate ring is C q [ G ]. W e s tart b y deﬁn in g GL q (2) and S L q (2), for n = 2. Then w e will generalize this construction to arb itrary n . Let O ( M 2 ) b e the co ordinate ring of M 2 , the s et of 2 × 2 complex mat rices, C [ V ] the co ordinate ring of V generated b y the coordinates x 1 and x 2 of V whic h satisfy x 1 x 2 = x 2 x 1 . Let U =  a b c d  b e the g eneric (v ariable) matrix in M 2 . It acts on V = C 2 from the left and from the right . Let x =  x 1 x 2  . The left action is deﬁned by x → x ′ := U x. 97 Let x ′ =  x ′ 1 x ′ 2  . Similarly , the right action is deﬁned by x T → ( x ′′ ) T := x T U. Let x ′′ =  x ′′ 1 x ′′ 2  . The action of M 2 on V satisﬁes x ′ 1 x ′ 2 = x ′ 2 x ′ 1 , and x ′′ 1 x ′′ 2 = x ′′ 2 x ′′ 1 . No w instead of V , we tak e it s q -d eformati on V q , a quantum space, whose co ord inates x 1 and x 2 satisfy x 1 x 2 = q x 2 x 1 , (18.1) where q ∈ C is a parameter. In tuitiv ely , in quantum physic s if x 1 and x 2 are p osition and m omentum, then q = e i ~ when ~ is Planck’ s constan t. Let C q [ V ] b e the r in g generate d by x 1 and x 2 with the r ela tion (18 . 1) . T hat is, C q [ V ] = C [ x 1 , x 2 ] / < x 1 x 2 − q x 2 x 1 > . It is the c o ordinate ring of the quantum s pace V q . No w w e w ant to quan tize M (2) to get M q (2), the space of quant um 2 × 2 matrices, and GL (2) to GL q (2), the space of quantum 2 × 2 n ons ingular matrices. Intuitiv ely , M q (2) is the space of linear transformations of the quan tum sp ace V q whic h p reserv e the equation (18.1) under the le ft and right actions, and s imil arly , GL q (2) is the space of non-singular linear transformation that preserv e the equation (18.1) u n der the left and right actio ns. W e n o w formalize this in tuition. Let U =  a b c d  b e a quan tum matrix wh ose coordinates d o not comm ute. The left and right actions of U m us t preserv e 18.1. [Left act ion:] Let the left a ction b e ϕ L : x → U x , and U x = x ′ . Then w e m ust ha v e:  a b c d   x 1 x 2  =  ax 1 + bx 2 cx 1 + dx 2  =  x ′ 1 x ′ 2  . 98 [Righ t action:] Let the right action b e ϕ R : x T → x T U , and let x ′′ = ( x T U ) T = U T x . Then w e m ust hav e:  x 1 x 2   a b c d  =  ax 1 + cx 2 bx 1 + dx 2  =  x ′′ 1 x ′′ 2  . The pr eserv ation of x 1 x 2 = q x 2 x 1 under left m ultiplication means x ′ 1 x ′ 2 = q x ′ 2 x ′ 1 . That is, ( ax 1 + bx 2 )( cx 1 + dx 2 ) = q ( cx 1 + dx 2 )( ax 1 + bx 2 ) . (18.2) The left hand side of (18.2) is acx 2 1 + bcx 2 x 1 + adx 1 x 2 + bdx 2 2 = acx 2 1 + ( bc + adq ) x 2 x 1 + bdx 2 2 . Similarly , the right hand side of (18.2) is q ( cax 2 1 + ( da + cbq ) x 2 x 1 + bdx 2 2 ) . Therefore equation (18.2) imp lie s: ac = q ca bd = q db bc + adq = da + q cb. That is, ac = q ca bd = q db ad − da − q cb + q − 1 bc = 0 . Similarly , since x ′′ 1 x ′′ 2 = q x ′′ 2 x ′′ 1 , we get: ab = q ba cd = q dc ad − da − q bc + q − 1 cb = 0 . The last equations from eac h of these sets imply bc = cb . So we deﬁne O ( M q (2)), the co ordinate ring of the sp ace of 2 × 2 quantum matrices M q (2), to b e the C -algebra w ith generators a , b , c , and d , satisfying the r elations: ab = qba, ac = q ca, bd = q db, cd = q dc, bc = cb, ad − da = ( q − q − 1 ) bc. 99 Let U =  a b c d  =  u 1 1 u 1 2 u 2 1 u 2 2  . Deﬁne the quan tum determinan t of U to b e D q = det( U ) = ad − q bc = da − q − 1 bc. Deﬁne C q [ G ] = O ( GL q (2)), the co ord inate ring of the virtual quant um group GL q (2) of in v ertible 2 × 2 qu an tum matrices, to b e O ( GL q (2)) = O ( M q (2))[ D − 1 q ] , where the square brac k ets indicate adjoining. Prop osition 18.1. The c o or dinate ring O ( GL q (2)) is a Hopf algebr a, w ith ∆( u i j ) = X k u i k ⊗ u k j , S ( u i j ) = 1 D q e u i j = ( U − 1 ) i j , ǫ ( u i j ) = δ ij , wher e e U = [ ˜ u i j ] is th e c ofacto r matrix e U =  d − q − 1 b − q c a  . (deﬁne d so that U e U = D q I ) and U − 1 = ˜ U /D q is the invers e of U . This is a n on-co mmutat iv e a nd non-co-comm u tat iv e Hopf algebra. No w we go to the general n . Let V q b e the n -dimensional quan tum space, the q -deformation of V , with co ordinates x i ’s w hic h satisfy x i x j = q x j x i ∀ i < j. (18.3) Let C q [ V ] b e the co ordinate ring of V q deﬁned b y C q [ V ] = C [ x 1 , . . . , x n ] / < x i x j − q x j x i > . 100 Let M q ( n ) b e the space of quan tum n × n matrices, that is the set of linear transformations on V q whic h preserve (18.3) under the left as well as the righ t action. The left action is giv en by:   x 1 ... x n   = x → U x = x ′ , where U is the n × n generic qu an tum m atrix. Similarly , the righ t action is giv en b y: x T → x T U = ( x ′′ ) T . Preserv ation of (18.3) und er the left and r igh t actions means: x ′ i y ′ j = q x ′ j x ′ i , for i < j x ′′ i y ′′ j = q x ′′ j x ′′ i , for i < j. After straigh tforw ard calculations, these yield the follo wing relations on the entries u ij = u i j of U : u j k u ik = q − 1 u ik u j k ( i < j ) u k j u k i = q − 1 u k i u k j ( i < j ) u j k u iℓ = u iℓ u j k ( i < j, k < ℓ ) u j l u ik = u ik u j ℓ − ( q − q − 1 ) u j k u iℓ ( i < j, k < ℓ ) . (18.4) The quantum determinan t is deﬁn ed as D q = X σ ∈ S n ( − q ) ℓ ( σ ) u i σ (1) j 1 . . . u i σ ( n ) j n , where ℓ ( σ ) denotes the length of the p ermutatio n σ , that is, the num b er of in v ersions in σ . This d ete rminant form ula is the s ame as the usual formula substituting ( − q ) for ( − 1). W e d eﬁne the coord inate r ing of the space M q ( n ) of quantum n × n matrices by O ( M q ( n )) = C [ U ] / < (18 . 4) >, and and the co ordinate ring of the virtual qu an tum group GL q ( n ) by C q [ G ] = O ( GL q ( n )) = O ( M q ( n ))[ D − 1 q ] . W e d eﬁne the qu an tum minors and , u sing these, the quantum co-factor matrix e U and th e quan tum inv erse matrix U − 1 = e U /D q in a str aightfo rward fashion (these constructions are left as exercises). 101 Theorem 18.1. The algebr a O ( GL q ( n )) is a Hopf algebr a, with ∆( u i j ) = X k u i k ⊗ u k j ǫ ( u i j ) = δ ij S ( u i j ) = 1 D q e u i j = ( U − 1 ) i j S ( D − 1 q ) = D q . W e also denote the quant um group GL q ( n ) by G q , GL q ( C n ) or GL q ( V ). It has to b e emp hasize d that this is only a virtual ob ject. On ly its co ordinate ring C q [ G ] is real. Henceforth, whenever w e sa y representat ion or action of G q , we actually mean corepresen tation or coaction of C q [ G ], and so forth. 102 Chapter 19 Quan tum uni tary group Scrib e: Joshua A. Gro c how Goal: Deﬁne the qu an tum u nitary subgroup of the standard qu antum group. R efer enc e: [KS] Recall Let V = C n , G = GL n ( C ) = GL ( V ) = GL ( C n ), and O ( G ) t he coord in ate ring of G . The quantum group G q = GL q ( V ) is the virtual ob ject whose co ord inate ring is O ( G q ) = C [ U ] / h relations i , where U is the ge neric n × n matrix of indeterminates, and the relat ions are the quadratic relat ions on the co ordinates u j i deﬁned in the last class so as to preserv e the non-comm uting relatio ns among the co ordinates of the quan tum v ector space V q on which G q acts. This co ordinate rin g is a Hopf algebra. 19.1 A q -analogue of the unitary group In this lecture w e deﬁne a q -analo gue of th e unitary subgroup U = U n ( C ) = U ( V ) ⊆ GL n ( C ) = GL ( V ) = G . T h is is a q -deformation U q = U q ( V ) ⊆ G q of U ( V ). Since G q is only a virtual ob ject, U q will also b e virtual. T o deﬁne U q , w e m us t determine ho w to capture t he notion of unitarit y in t he setting of Hopf algebras. As w e sh all see, it is captured b y the notion of a Hopf ∗ -alge bra. 103 Deﬁnition 19.1. A ∗ -v ector space is a ve ctor sp ac e V with an involution ∗ : V → V satisfying ( αv + β w ) ∗ = αv ∗ + β w ∗ ( v ∗ ) ∗ = v for a l l v , w ∈ V , and α, β ∈ C . W e think of ∗ as a generaliza tion of complex conjugation; and in fact ev ery complex v ector space is a ∗ -ve ctor space, where ∗ is e xactly c omplex conjugation. Deﬁnition 19.2. A Hopf ∗ -algebra is a Hopf algebr a ( A, ∆ , ǫ, S ) with an involution ∗ : A → A such tha t ( A, ∗ ) is a ∗ -ve ctor sp ac e, and: 1. ( ab ) ∗ = b ∗ a ∗ , 1 ∗ = 1 2. ∆ ( a ∗ ) = ∆( a ) ∗ (wher e ∗ acts diagonal ly on the tensor p r o duct A ⊗ A : ( v ⊗ w ) ∗ = ( v ∗ ⊗ w ∗ ) ) 3. ǫ ( a ∗ ) = ǫ ( a ) There is no e xplicit condition here on ho w ∗ in teracts with the an tip o de S . Let O ( G ) = C [ G ] b e the co ordinate ring of G as deﬁned earlier. Prop osition 19.1. Then O ( G ) is a Hop f ∗ -al gebr a. Pr o of. W e think of the elemen ts in O ( G ) as C -v alued functions on G and deﬁne ∗ : O ( G ) → O ( G ) so that i t sa tisﬁes the three co nditions f or a Hopf ∗ -alge bra, and (4) F or all f ∈ O ( G ) and g ∈ U ⊆ G , f ∗ ( g ) = f ( g ) Let u j i b e the c o ordinate functions whic h, together w ith D − 1 , D = det( U ), generate O ( G ). Be cause of the ﬁrst condition on a Hopf ∗ -algebra (relating the in vo lution ∗ to multi plication), sp ecifying ( u j i ) ∗ and D ∗ suﬃces to deﬁn e ∗ completely . W e deﬁne ( u j i ) ∗ = S ( u i j ) = ( U − 1 ) i j and D ∗ = D − 1 . W e can c h ec k that this satiﬁes ( 1)-(4). Here w e will only c h ec k (4), and l ea v e the remaining veriﬁcat ion as an exercise. Let g b e a n elemen t of th e unitary group U . Then ( u j i ) ∗ ( g ) = S ( u i j )( g ) = ( g − 1 ) i j = ( g ) j i , where the last equalit y follo ws from the fact that g is unitary (i.e. g − 1 = g † , where † denotes conjugate transp ose). 104 Th us, we ha ve deﬁned a map f 7→ f ∗ purely algebraically in such a w a y that the restrict ion o f f ∗ to the un itary group U is the same a s ta king the complex conjugate f on U . Prop osition 19.2. The c o or dinate ring C q [ G ] = O ( G q ) of the quantum gr oup G q = GL q ( V ) i s al so a Hopf ∗ -algebr a. Pr o of. The pro of is syntac tically iden tical to the pro of for O ( G ), except that the co ordinate function u j i no w liv es in O ( G q ) and the determinan t D b ecomes the q -determinant D q . The deﬁnition of ∗ is: ( u j i ) ∗ = S ( u i j ) and D ∗ q = D − 1 q , essen tially the same as in the classical case. In tuitiv ely , the “quan tum su bgroup” U q of G q is the virtual ob ject suc h that the restriction to U q of the in v olution ∗ ju s t deﬁned co incides with the complex conjugate. 19.2 Prop erties of U q W e w ould l ik e t he nice p rop erties of the c lassical unitary group to trans fer o ver to the quantum u nitary group , and this is indeed th e ca se. Some of the nice pr operties of U are: 1. It is compact, so we can in tegrate o ve r U . 2. w e can do harmonic analysis on U (viz. the P eter-W eyl Theorem, whic h is an analogue for U of the F ourier analysis on the circle U 1 ). 3. Ev ery ﬁnite d imens ional r epresen tation of U has a G -in v arian t Hermi- tian form, and thus a unitary b asis – w e sa y that ev ery ﬁnite dimen- sional representat ion of U is unitarizable . 4. Ev ery ﬁnite d imensional represen tation X of U is completely red u cible; this follo ws from (3) , since an y subrepresen tation W ⊆ X has a p er- p endicular su b represen tation W ⊥ under the G -in v ariant Herm itian form. Compactness is in some sens e the key h ere. The qu estion is ho w to deﬁn e it in the quan tum set ting. F ollo wing W oronowicz , we deﬁn e compactness to mean that ev ery ﬁnite dimensional representa tion of U q is unitarizable. Let us see what this means formally . Let A be a Hopf ∗ - algebra, and W a corepresen tation of A . Let ρ : W → W ⊗ A be the corepresen tation map. Let { b i } be a basis of W . Then, under 105 ρ , b i 7→ P j b j ⊗ m j i for some m j i ∈ A . W e can th us deﬁne the matrix of the (c o)r epr esentation M = ( m j i ) in the basis { b i } . W e deﬁne M ∗ suc h that ( M ∗ ) j i = ( M i j ) ∗ . Thus, in the classical case (i.e. when q = 1), M ∗ = M † . W e sa y that the corepresen tation W is unitarizable if it h as a basis B = { b i } suc h that the corresp onding matrix M B of corepresen tation satisﬁes the unitarit y condition: M B M ∗ B = I . In this case, we sa y B is a unitary basis of the corepresen tation W . Deﬁnition 19.3. A Hopf ∗ -algebr a A is compact if every ﬁnite dimensional c or epr esentation of A is unitarizable. Theorem 19.1 (W orono wicz) . The c o or dinte ring C q [ G ] = O ( G q ) i s a c omp act Hopf ∗ -algebr a. This implies that every ﬁnite dimensional r epr e- sentation of G q , by which we me an a ﬁnite dimensional c o or e pr esentation of C q [ G ] , is c ompl etely r e ducible. W orono w icz go es fu r ther to show that w e can q -integ rate on U q , and that w e ca n do quan tized harmonic anal ysis on U q ; i.e., a quantum analogue of the Pete r-W eyl t heorem holds. No w that w e kno w the ﬁ nite dimensional represen tations of G q are com- pletely r educible, w e can ask wh at the irreducible represen tations are. 19.3 Irreducible Represen tations of G q W e pro ceed by analogy with th e W eyl mo dules V λ ( G ) for G . Recall that ev ery p olynomial irreducible repr esen tation of G = GL n ( C ) is of th is form. Theorem 19.2. 1. F or al l p artitions λ of length at most n , ther e exists a q -Weyl mo dule V q ,λ ( G q ) which is an irr e ducible r epr esentation of G q such that lim q → 1 V q ,λ ( G q ) = V λ ( G ) . 2. The q -Weyl mo dules give al l p olynomial irr e ducible r epr esentation s of G q . 19.4 Gelfand-Tsetlin basis T o understand the q -W eyl mo dules b etter, w e wish to get an explicit b asis for eac h mo d ule V q ,λ . W e b egin by deﬁning a v ery u s efu l basis – the Gel’fand- T estlin basis – in the classical ca se f or V λ ( G ), and then describ e the q - analogue of this b asis. 106 By Pieri’s rule [FH] V λ ( GL n ( C )) = M λ ′ V λ ′ ( GL n − 1 ( C )) where the sum is tak en o ver all λ ′ obtained from λ b y removi ng an y num b er of b o xes (in a legal w a y) suc h that no tw o remov ed b oxes come from the same column. Th is is an orthogonal decomp osition (relativ e to th e GL n ( C )- in v arian t He rmitian form on V λ ) and it is a lso multiplic it y-free, i.e., eac h V λ ′ app ears only once. Fix a G -in v arian t Hermitian form on V λ . Th en the Gel’fand-Tsetlin basis for V λ ( GL n ( C )), denoted GT n λ , is the unique orthonormal basis for V λ suc h that GT n λ = [ λ ′ GT n − 1 λ ′ , where the disjoin t union is o ve r the λ ′ as in P ieri’s ru le, and GT n − 1 λ ′ is d eﬁned recursiv ely , the case n = 1 b eing trivial. The dimens ion of V λ is the n umb er of semistandard tableau of shap e λ . With a ny tableau T of this shap e, one can also exp licitly asso cia te a basis elemen t G T ( T ) ∈ GT n λ ; we shall not wo rry ab out ho w. W e can deﬁne the Gel’fand-Tsetlin basis GT n q ,λ for V q ,λ ( G q ( C n )) a nalo- gously . W e h a v e the q -analogue of Pieri’s r ule: V q ,λ ( G q ( C n )) = M λ ′ V q ,λ ′ ( G q ( C n − 1 )) where the d eco mp osition is orthogonal and m ultiplicit y-free, and the s u m ranges o v er the same λ ′ as ab o v e. S o w e can d eﬁ n e GT n q ,λ to b e the un iqu e unitary basis of V q ,λ suc h that GT n q ,λ = [ λ ′ GT n − 1 q ,λ ′ . With any semistandard ta bleau T , one can also explicitly asso ciate a basis elemen t G T q ( T ) ∈ GT n q ,λ ′ ; d eta ils omitted. 107 Chapter 20 T o w ards p ositivit y h yp otheses via q uan tum groups Scrib e: Joshua A. Gro c how Goal: In this ﬁnal b risk lecture, w e indicate the role of quantum groups in the conte xt of the p ositivit y h yp othesis PH1. Sp eciﬁcally , we sk etc h ho w the Littlew o o d-Ric h ard son ru le – the gist of PH1 in the Littlew o od -Richardson problem – follo ws from the theory of standard quantum groups. W e then brieﬂy mention analogous (nonstandard) qu an tum groups for the Kronec k er and plethysm pr oblems deﬁn ed in [GCT4, GCT7], and the theorems and conjectures for them that would imply PH1 for these problems. R efer enc es: [KS, K, Lu2, GCT4, GCT6, GCT7, GCT8] Let V = C n , G = GL n ( C ) = GL ( V ), V λ = V λ ( G ) a W eyl mo dule of G , G q = GL q ( V ) the standard quantum group, V q the q -deformation of V on w hic h GL q ( V ) ac ts, V q ,λ = V q ,λ ( G q ) the q -deformation of V λ ( G ), and GT q ,λ = GT n q ,λ the Gel’fand-Tsetlin basis for V q ,λ . 20.1 Littlew o o d-Ric hardson rule via standard quan- tum groups W e now ske tc h ho w the Littlew o o d-Ric h ardson ru le falls out of the standard quan tum group mac hinery , sp eciﬁcally the prop erties of the Gelfand-Tsetlin basis. 108 20.1.1 An em b edding of the W eyl mo dule F or this, we ha v e to embed the q -W eyl mo dule V q ,λ in V ⊗ d q , where d = | λ | = P λ i is the siz e o f λ . W e ﬁrst describ e ho w t o e mbed the W eyl mod ule V λ of G in V ⊗ d in a standard wa y that can b e qu an tized. If d = 1, then V λ ( G ) = V = V ⊗ 1 . Otherwise, obtain a Y oung diagram µ from λ by remo ving its top-righ tmost b o x that c an b e r emo v ed to get a v alid Y oung diagram, e.g.: x λ µ In the follo wing, the b ox m u st b e remo ve d from the second ro w , since remo ving from the ﬁrst ro w would result in an illegal Y oung diagram: x λ µ By induction o n d , w e hav e a standard embed d ing V µ ( G ) ֒ → V ⊗ d − 1 . Th is giv es us an em b edding V µ ( G ) ⊗ V ֒ → V ⊗ d . By Pieri’s rule [FH] V µ ( G ) ⊗ V = M β V β ( G ) , where the sum is o v er all β obtained from µ by adding one b o x in a legal wa y . In p articular, V λ ( G ) ⊂ V µ ( G ) ⊗ V . By restricting th e ab o ve em b edding, we get a standard emb edding V λ ( G ) ֒ → V ⊗ d . No w Pieri’s rule also holds in a quantiz ed setting: V q ,µ ⊗ V q = M β V q ,β ( G ) , where β is as ab o ve . Hence, the standard em b eddin g V λ ֒ → V ⊗ d ab o ve can b e quan tized in a straigh tforwa rd fashion to get a standard emb ed d ing V q ,λ ֒ → V ⊗ d q . W e shall denote it by ρ . Here th e tensor p rod uct is mean t to b e ov er Q ( q ). Actually , Q ( q ) d oesn’t quite work. W e ha v e to a llo w square ro ots of elemen ts of Q ( q ), but we won’t wo rry ab out this. F or a semistandard tableau b of shap e λ , we denote the image of a Gelfand-Tsetlin basis elemen t GT q ,λ ( b ) ∈ GT q ,λ under ρ b y GT ρ q ,λ ( b ) = ρ ( GT q ,λ ( b )) ∈ V ⊗ d q . 109 20.1.2 Crystal op erators and crystal bases Theorem 20.1 (Crystallization) . [DJM] The Gelfand -Tsetlin b asis ele- ments crystal lize at q = 0 . This me ans: lim q → 0 GT ρ q ,λ ( b ) = v i 1 ( b ) ⊗ · · · ⊗ v i d ( b ) , (20.1) for s ome int e ger functions i 1 ( b ) , . . . , i d ( b ) , and lim q →∞ GT ρ q ,λ ( b ) = v j 1 ( b ) ⊗ · · · ⊗ v j d ( b ) , (20.2) for s ome int e ger functions j 1 ( b ) , . . . , j d ( b ) . The phenomenon that these limits co nsists of mon omials, i.e., s imp le tensors is kno wn as crystal lization . It is r elated to the ph ysical phenomenon of crystallizat ion, hence the name. T h e maps b 7→ i ( b ) = ( i 1 ( b ) , . . . , i d ( b )) and b 7→ j ( b ) = ( j 1 ( b ) , . . . , j d ( b )) are compu table in pol y ( h b i ) time (where h b i is the b it-length of b ). No w w e wa nt to deﬁne a sp ecial crystal b asis of V q ,λ based on this p he- nomenon of crystalliza tion. T o wards that end, consider the f oll o wing family of n × n matrices: E i =           0 0 · · · 0 . . . . . . . . . 0 1 · · · 0 0 · · · 0 . . . . . . 0           , where the only nonzero en try is a 1 in the i -th ro w and ( i + 1)-st column. Let F i = E T i . Corresp onding to E i and F i , Kashiw ara asso ciat es certa in op erators ˆ E i and ˆ F i on V q ,λ ( G q ). W e shall not w orry about their actual construction here (for the readers f amiliar w ith Lie algebras: these are closely related to the u sual op erators in th e Lie algebra of G asso ciated with E i and F i ). If we let ˆ E i act on GT ρ q ,λ ( b ), we get some linear com bination ˆ E i ( GT ρ q ,λ ( b )) = X b ′ a b b ′ ( q ) GT ρ q ,λ ( b ′ ) , where a b b ′ ( q ) ∈ Q ( q ) (actual ly an algebraic extensio n of Q ( q ) as men tioned ab o ve) . Essen tially because of crystal lization (Theorem 20.1 ), it turns out 110 that lim q → 0 a b b ′ ( q ) is alw a ys either 0 or 1, and for a give n b , this limit is 1 for at most one b ′ , if any . A similar result holds for ˆ F i ( GT ρ q ,λ ( b )). This all o ws us to deﬁne the crystal o p er ators (due to Kashiw ara): e e i · b =  b ′ if lim q → 0 a b b ′ ( q ) = 1 , 0 if n o suc h b ′ exists, and similarly for e f i . Although these op erators are deﬁn ed according to a particular em b edding V q ,λ ֒ → V ⊗ d q and a basis, they can b e deﬁned in trinsi- cally , i.e., without reference to the em b edding or the Gel’fand-Tsetlin basis. No w , let W b e a ﬁn ite -dimensional representa tion of G q , and R the subring of functions in Q ( q ) regular at q = 0 (i.e. without a p ole at q = 0). A lattic e within W is an R -submo dule of W suc h that Q ( q ) ⊗ R L = W . (In tuition b ehind this d eﬁ n itio n: R ⊂ Q ( q ) is analog ous to Z ⊂ Q . A latt ice in R n is a Z -submo dule L of R n suc h that R ⊗ Z L = R n .) Deﬁnition 20.1. An (upp er) crystal basis of a r epr esentation W of G q is a p air ( L, B ) such that • L is a lattic e in W pr eserve d by the Kashiwar a op er ators ˆ E i and ˆ F i , i.e. ˆ E i ( L ) ⊆ L and ˆ F i ( L ) ⊆ L . • B is a b asis of L/q L pr eserve d by the crystal op er ators e e i and e f i , i.e., e e i ( B ) ⊆ B ∪ { 0 } an d e f i ( B ) ⊆ B ∪ { 0 } . • The crystal op er ators e e i and e f i ar e inverse to e ach other wher ever p ossible, i.e., for al l b, b ′ ∈ B , if e e i ( b ) = b ′ 6 = 0 then e f i ( b ′ ) = b , and similarly, if e f i ( b ) = b ′ 6 = 0 then e e i ( b ′ ) = b . It can be sho wn that if W = V q ,λ ( G q ), then there exists a unique b ∈ B suc h that e e i ( b ) = 0 for all i ; this corresp onds to the h ighest w eigh t v ector of V q ,λ (the w eigh t vec tors in V q ,λ are anal ogous to the wei gh t v ectors in V λ ; w e d o not giv e their exact deﬁn itio n here) . By the w ork of Ka shiwara and Date et al [K, DJM] ab o ve , the Gel’fand-Tsetlin basis (after appropriate rescaling) is in fact a crystal basis: just let L = L GT = the R -mo dule generated by GT q ,λ , and B GT = GT q ,λ ( b ) , where GT q ,λ ( b ) is the image un der the pro jection L 7→ L/q L of the set of basis vect ors in GT q ,λ ( b ). 111 Theorem 20.2 (Kashiw ara) . 1. E v ery ﬁnite-dimensional G q -mo dule has a unique c rystal b asis (up to isomorph ism). 2. L et ( L λ , B λ ) b e the unique crystal b asis c orr esp onding to V q ,λ . Then ( L α , B α ) ⊗ ( L β , B β ) = ( L α ⊗ L β , B α ⊗ B β ) is the unique crystal b asis of V q ,α ⊗ V q ,β , wher e B α ⊗ B β denotes { b a ⊗ b b | b a ∈ B α , b b ∈ B β } . It can b e sho wn that every b ∈ B λ has a w eigh t; i.e., it is the image of a w eigh t v ector in L λ under the pro j ect ion L λ → L λ /q L λ . No w let us see ho w the Litt lew o o d-Ric hardson rule falls out of the prop- erties of the crystal b ases. Recall that the sp ecializatio n of V q ,α at q = 1 is the W eyl m o du le V α of G = GL n ( C ), and V α ⊗ V β = M γ c γ α,β V γ (20.3) where c γ α,β are the Littlew o o d-Ric hard son co eﬃcien ts. The Littlew o o d- Ric h ardson ru le no w follo ws from the follo wing fact: c γ α,β = # { b ⊗ b ′ ∈ B α ⊗ B β |∀ i, e e i ( b ⊗ b ′ ) = 0 and b ⊗ b ′ has wei ght γ } . In tuitiv ely , b ⊗ b ′ here corresp ond to the highest w eigh t v ectors of the G - submo dules of V α ⊗ V β isomorphic to V γ . 20.2 Explicit decomp osition of the tensor pro d uct The decomp ositi on (20.3) is only an abstract decomposition of V α ⊗ V β as a G -mod ule. Next we consider the explicit d eco mp osition p roblem. The goal is to ﬁ nd an explicit basis B = B α ⊗ β of V α ⊗ V β that is compatible with this abstract d eco mp osition. Sp eciﬁcall y , we wa nt to construct an explicit basis B of V α ⊗ V β in terms of suitable explicit bases of V α and V β suc h that B has a ﬁltration B = B 0 ⊇ B 1 ⊇ · · · ⊇ ∅ where eac h hB i i / hB i +1 i is an irreducible r epresen tation of G and hB i i denotes the linear span of B i . F urth ermore, eac h el ement b ∈ B should hav e a suﬃcien tly explicit represent ation in terms of the basis B α ⊗ B β of V α ⊗ V β . The explicit d eco mp osition p roblem for th e q -analogue V q ,α ⊗ V q ,β is similar. F or example, we h a ve already constructed explicit Gelfand-Tsetlin bases of W eyl mo du les. But it is n ot kno wn how to construct an explicit b asis B 112 with ﬁltration as ab o v e in terms of the Gelfand-Tsetlin bases of V α and V β (except wh en the Y oung diagram of either α or β is a single ro w). Kashiw ara and L usztig [K, Lu2] construct certain c anonic al b ases B q ,α and B q ,β of V q ,α and V q ,β , and Lusztig f u rthermore constructs a c anonic al b asis B q = B q ,α ⊗ β of V q ,α ⊗ V q ,β suc h that: 1. B q has a ﬁltration as ab o ve, 2. Eac h b ∈ B q has an expansion of the form b = X b α ∈B q,α ,b β ∈B q,β a b α ,b β b b α ⊗ b β , where eac h a b α ,b β b is a p olynomial in q and q − 1 with nonnegativ e i nte - gral co eﬃcien ts, 3. Crysta l lization : F or eac h b , as q → 0, exactl y one co eﬃcien t a b α ,b β b → 1, and the remaining all v anish. The pr o of of nonnega tivit y of the co eﬃcien ts of a b α ,b β b is b ased on t he Rie- mann h yp othesis (theorem) o v er ﬁn ite ﬁelds [Dl2], and explicit formulae for these co eﬃcien ts are kno wn in terms of p erv erse shea v es [BBD] (whic h are certain typ es of algebro-ge ometric ob jects). This then provi des a sa tisfactory solution to the explicit decomp osition problem, which is far harder and deep er than the abstract decomp osition pro vided by the Little woo d-Ric h ardson rule. By sp ecializing at q = 1 , we also get a s olutio n to the explicit decomp osition pr oblem for V α ⊗ V β . This (i.e. via quan tum groups ) is t he only kno wn solution to the explicit decom- p osition problem ev en at q = 1. This ma y giv e some idea of the pow er of the qu an tum group m achinery . 20.3 T o w ards nonstandard quan tum groups for the Kronec k er and pleth ysm problems No w the goal is to constru ct quant um groups w hic h can b e used to de- riv e PH1 and explicit decomp osition for the K ronec ke r and pleth ysm prob- lems just as th e st andard quan tum group ca n b e used for the same in the Littlew o o d-Ric h ard son problem. In the Kronec k er p roblem, we let H = GL ( C n ) and G = GL ( C n ⊗ C n ). The Kronec ker co eﬃcient κ γ α,β is the multiplici t y of V α ( H ) ⊗ V β ( H ) in V γ ( G ): V γ ( G ) = M α,β κ γ α,β V α ( H ) ⊗ V β ( H ) . 113 The g oal is to get a p ositiv e # P -form u la for κ γ α,β ; this i s the gist of PH1 for the Kronec k er problem. In the pleth ysm problem, w e let H = GL ( C n ) and G = GL ( V µ ( H )). The plethysm constan t a π λ,µ is the m ultiplicit y of V π ( H ) in V λ ( G ): V λ ( G ) = M π a π λ,µ V π ( H ) . Again, the goal is to get a p ositiv e # P -form ula for the pleth ys m constant; this is the gist of P H1 for the p leth ysm problem. T o apply the quantum group approac h, w e n eed a q -analogue of the em b edding H ֒ → G . Unfortun ate ly , th ere is no suc h q -analog ue in the theory of standard quant um groups. Because there is n o non trivial quan tum group homomorphism fr om the stand ard quantum group H q = GL q ( C n ) and to the standard quan tum group G q . Theorem 20.3. (1) [GCT4]: L et H and G b e as i n the Kr one cke r pr oblem. Then ther e exists a quantum gr oup ˆ G q such tha t the homomorp hism H → G c an b e quantize d in the form H q ֒ → ˆ G q . F urthermo r e, ˆ G q has a unitary quantum sub gr oup ˆ U q which c orr esp onds to the maximal unitary sub gr oup U ⊆ G , and a q -analo gue of the Peter-We yl the or em holds for ˆ G q . The latter implies that every ﬁnite dimensional r epr esentation of ˆ G q is c ompletely de c omp osible into irr e ducibles. (2) [GCT7] Ther e is an analo g ous (p ossibly singular) quantum gr oup ˆ G q when H and G ar e as in the pl ethysm pr oblem. This al so holds for gener al c onne cte d r e ductive (classic al) H . Since the Kronec k er p roblem is a sp ecial case of the (generalized) p lethysm problem, the quantum group in GCT 4 is a sp ecial case of the quan tum group in GCT 7. T he quantum group in the plethysm problem can b e singular, i.e., its determinan t can v anish and hence the ant ip o de n eed not exist. W e still call it a quan tum group b ecause its prop erties are v ery similar to those of the standard qu an tum group; e.g. q -analog ue of the P eter-W eyl theorem, whic h allo ws q -h armonic analysis on these group s. W e call the qu an tum group ˆ G q nonstand ar d , b ecause though it is qualitat iv ely similar to the sta ndard (Drinfeld-Jim b o) quan tum group G q , it is al so, as exp ected, fund amen tally diﬀeren t. The article [GCT8] giv es a conjecturally correct algorithm to construct a canonical basis of an irr educible p olynomial repr esen tation of ˆ G q whic h gen- eralizes the canonical basis f or a p olynomial representa tion of the standard quan tum group as p er Kashiwa ra and Lusztig. It also giv es a conjecturally 114 correct algorithm to constru ct a canonical basis of a certain q -deformation of the symmetric group algebra C [ S r ] whic h generalizes the Kazhdan-Lu sztig basis [KL] of the Hec ke algebra (a standard q -deformation of C [ S r ]). It is sho wn in [GCT7, GCT8] that PH1 for the Kronec k er and p lethysm pr oblems follo ws assuming th at these canonical b ases in the nonstandard setting h a ve prop erties akin to the o nes in the standard setting. F or a discussion on SH, see [GCT6]. 115 P art I I In v arian t theory with a view to w ards GCT By Milind Sohoni 116 Chapter 21 Finite Groups R efer enc es: [FH, N] 21.1 Generaliti es Let V b e a v ector s pace ov er C , and let GL ( V ) denote the group of all isomorphisms on V . F or a ﬁxed basis of V , GL ( V ) is isomorphic to the group GL n ( C ), the group of all n × n inv ertible matrices. Let G b e a group and ρ : G → GL ( V ) b e a represen tation. W e also denote this b y the tuple ( ρ, V ) or sa y that V is a G -mo dule. Let Z ⊆ V b e a sub space suc h that ρ ( g )( Z ) ⊆ Z f or all g ∈ G . Then, w e say that Z is an in v arian t subspace . W e sa y that ( ρ, V ) is irreducible if there is no prop er sub space W ⊂ V suc h that ρ ( g )( W ) ⊆ W for all g ∈ G . W e sa y that ( ρ, V ) is indecomposable is there is no expression V = W 1 ⊕ W 2 suc h that ρ ( g )( W i ) ⊆ W i , for all g ∈ G . F or a p oint v ∈ V , the orbit O ( v ), and the stabilizer S tab ( v ) are deﬁned as: O ( v ) = { v ′ ∈ V |∃ g ∈ G with ρ ( g )( v ) = v ′ } S tab ( v ) = { g ∈ G | ρ ( g )( v ) = v } One may also d eﬁne v ∼ v ′ if there is a g ∈ G suc h that ρ ( g )( v ) = v ′ . I t is then easy to sho w th at [ v ] ∼ = O ( v ). Let V ∗ b e the d ual-space of V . Th e represen tation ( ρ, V ) induces the dual rep r esen tation ( ρ ∗ , V ∗ ) deﬁn ed as ρ ∗ ( v ∗ )( v ) = v ∗ ( ρ ( g − 1 )( v )). It will b e con v enien t f or ρ ∗ to act on the right, i.e., (( v ∗ )( ρ ∗ ))( v ) = v ∗ ( ρ ( g − 1 )( v )). When ρ is ﬁ xed, w e abbrieviate ρ ( g )( v ) as j u st g · v . Along w ith this, there are the standard constructions of the tensor T d ( V ), the symmetric p o wer S y m d ( V ) and the alternat ing po wer ∧ d ( V ) repr esen tations. 117 Of special signiﬁcance is S y m d ( V ∗ ), the space of homogeneous polyno- mial functio ns o n V of degree d . Let dim ( V ) = n and le t X 1 , . . . , X n b e a basis of V ∗ . W e deﬁne as follo ws: R = C [ X 1 , . . . , X n ] = ⊕ ∞ d =0 R d = ⊕ ∞ d =0 S y m d ( V ∗ ) Th us R is the ring of all p olynomial functions on V and is isomorphic to the algebra (o ve r C ) of n in determinate s. Since G acts on the domain V , G also acts on all f unctions f : V → C as follo ws: ( f · g )( v ) = f ( g − 1 · v ) This acti on of G on all fun ctio ns extends the action of G on p olynomial functions ab o ve. Indeed, for an y g ∈ G , the map t g : R → R giv en by f → f · g is an algebra isomorphism. This is called the translation map . F or an f ∈ R , w e sa y t hat f is an in v ariant if f · g = f for all g ∈ G . The follo wing are equiv alen t: • f ∈ R is an inv arian t. • S tab ( f ) = G . • f ( g · v ) = f ( v ) for all g ∈ G and v ∈ V . • F or all v , v ′ suc h that v ′ ∈ O r bit ( v ), we ha ve f ( v ) = f ( v ′ ). If W 1 and W 2 are t w o mo dules of G and φ : W 1 → W 2 is a linear ma p suc h that g · φ ( w 1 ) = φ ( g · w 1 ) for all g ∈ G and w 1 ∈ W 1 then we sa y that φ is G -equiv arian t o r that φ is a morphism of G -mo dules . 21.2 The ﬁnite group action Let G b e a ﬁ nite group and ( µ, W ) b e a represent ation. Recall that a complex inner product on W is a m ap h : W × W → C suc h that: • h ( αw + β w ′ , w ′′ ) = αh ( w, w ′′ ) + β h ( w ′ , w ′′ ) for all α, β ∈ C and all w, w ′ , w ′′ ∈ W . • h ( w ′′ , αw + β w ′ ) = αh ( w ′′ , w ) + β h ( w ′′ , w ′ ) for all α, β ∈ C and all w, w ′ , w ′′ ∈ W . • h ( w , w ) > 0 for all w 6 = 0. 118 Also recall that if Z ⊆ W is a sub space, then Z ⊥ is d eﬁned as: Z ⊥ = { w ∈ W | h ( w, z ) = 0 ∀ z ∈ Z } Also recall that W = Z ⊕ Z ⊥ . W e sa y that an inner pro duct h is G -in v ariant if h ( g · w , g · w ′ ) = h ( w , w ′ ) for all w , w ′ ∈ W and g ∈ G . Prop osition 21.1. L et W b e as ab ove, and Z b e an i nvariant subsp ac e of W . Then Z ⊥ is als o an invariant subsp ac e . Thus every r e ducible r epr esen- tation of G is also d e c omp osable. Pro of : Let x ∈ Z ⊥ , z ∈ Z and let us examine ( g · x, z ). Applying g − 1 to b oth sides, we see that: h ( g · x, z ) = h ( g − 1 · g · x, g − 1 · z ) = h ( x, g − 1 · z ) = 0 Th us, G p r eserv es Z ⊥ as claimed.  Let h be a complex inner p rod uct on W . W e d eﬁne the inner pro duct h G as follo ws: h G ( w, w ′ ) = 1 | G | X g ′ ∈ G h ( g ′ · w , g ′ · w ′ ) Lemma 21.1. h G is a G -invarian t inner pr o duct. Pro of : First w e see that h G ( w, w ) = 1 | G | X g ′ ∈ G h ( w ′ , w ′ ) where w ′ = g ′ · w . Th us h G ( w, w ) > 0 unless w = 0. Secondly , by the linearit y of the action of G , w e see that h G is indeed an inner p r odu ct. Finally , w e see that: h G ( g · w, g · w ′ ) = 1 | G | X g ′ ∈ G h ( g ′ · g · w , g ′ · g · w ′ ) Since as g ′ ranges o v er G , so will g ′ · g for an y ﬁxed g , we hav e that h G is G -in v arian t.  Theorem 1. • L et G b e a ﬁnite gr oup and ( ρ, V ) b e an inde c omp osable r epr esentation, th en it is also irr e ducible. 119 • E very r epr esentation ( ρ, V ) may b e de c omp ose d into irr e ducib le r epr e- sentations V i . Thus V = ⊕ i V i , wher e ( ρ i , V i ) is an irr e ducible r epr e- sentation. Pro of : Sup p ose th at Z ⊆ V is an in v ariant su bspace, then V = Z ⊕ Z ⊥ is a non-trivial decomp osition of V con tradicting the hyp othesis. Th e second part is prov ed by applying the ﬁ rst, recursiv ely .  W e hav e seen the op eration of a v eraging o v er the group in going from th e inner pr odu ct h to the G -in v arian t inner pro duct h G . A similar approac h ma y b e u s ed for constructing in v arian t p olynomials functions. So let p ( X ) ∈ R = C [ X 1 , . . . , X n ] b e a p olynomia l function. W e deﬁne the function p G : V → C as: p G ( v ) = 1 | G | X g ∈ G p ( g · v ) The transition from p to p G is called the Reynold’s ope ra t or . Prop osition 21.2. L et p ∈ R b e of de gr e e atm ost d , then p G is also a p olynomia l function of de gr e e atmos t d . Next, p G is an invaria nt. Let R G denote the set of all in v arian t p olynomial fun cti ons on th e space V . It is easy to s ee that R G ⊆ R is actually a subring of R . Let Z ⊆ V b e an arbitrary s u bset of V . W e sa y that Z is G -closed if g · z ∈ Z for all g ∈ G and z ∈ Z . Th us Z is a union of orbits of p oints in V . Lemma 21.2. L et p ∈ R G b e an inv ariant and let Z = V ( p ) b e the variety of p . Then Z is G -close d. W e ha v e already s een that O ( v ), the orbit of v arises from the equiv alence class ∼ on V . Since the group is ﬁn ite , | O ( v ) | ≤ | G | for any v . Let O 1 and O 2 b e d isj oint orbits. It is essen tial to determine if elemen ts of R G can separate O 1 and O 2 . Lemma 21.3. L et O 1 and O 2 b e as ab ove, and I 1 and I 2 b e th eir ide als in R . Then ther e ar e p 1 ∈ I 1 and p 2 ∈ I 2 so that p 1 + p 2 = 1 . Pro of : This follo w s from the Hilbert Nullstellensat z. Since the p oin t sets are ﬁnite, t here is an explici t construction based o n Lagrange in terp olatio n.  Let G b e a ﬁnite group and ( ρ, V ) b e a represen tation as ab o ve . W e ha v e see that this induces an actio n on C [ X 1 , . . . , X n ]. Also note that this action is homogeneous : for a g ∈ G a nd p ∈ R d , w e ha ve that p · g ∈ R d 120 as well. T hus R G , the ring of inv ariants, is a homo gene ous subring of R . In other words: R G = ⊕ ∞ d =0 R G d where R G d are in v arian ts which are homogeneo us of degree d . Th e exi stence of the ab o ve d eco mp osition imp lies that ev ery in v arian t is a sum of homo- geneous inv arian ts. No w, since R G d ⊆ R d as a v ector sp ace o ver C . Thus dim C ( R G d ) ≤ dim C ( R d ) ≤  n + d − 1 n − 1  W e deﬁ n e the hilb ert function h ( R G ) of R G (or for that matter, of an y homogeneous ring) as: h ( R G ) = ∞ X d =0 dim C ( R G d ) z d W e will see now that h ( R G ) is actually a rational fun cti on w hic h is easily computed. W e need a lemma. Let ( ρ, W ) b e a representa tion of the ﬁnite group G . Let W G = { w ∈ W | g · w = w } b e the set of all ve ctor inv ariants in W , and this is a su bspace of W . . Lemma 21.4. L et ( ρ, W ) b e as ab ove. We have: dim C ( W G ) = 1 | G | X g ∈ G tr ace ( ρ ( g )) Pro of : Deﬁne P = 1 | G | P g ∈ G ρ ( g ), as the a v erage of the representa tion matrices. W e see that ρ ( g ) · P = P · ρ ( g ) and that P 2 = P . T h u s P is diagonaliza ble and the eige nv alues of P are in the set { 1 , 0 } . Let W 1 and W 0 b e the correspond ing eigen-spaces. It is clear that W G ⊆ W 1 and that W 1 is ﬁxed b y eac h g ∈ G . W e now argue that ev ery w ∈ W 1 is actually an in v arian t. F or that, let w g = g · w . W e then h av e that P w = w implies that w = 1 | G | X g ∈ G w g Note that a c hange-of-basis does not aﬀect the h yp othesis nor the assertion. W e ma y thus assume that eac h ρ ( g ) is unitary , w e ha v e th at w g = w for all g ∈ G . No w, the claim follo w s by computing t race ( P ).  W e are n o w ready to state Molien’s Theorem : 121 Theorem 2. L et ( ρ, W ) b e as ab ove. We ha ve: h ( R G ) = 1 | G | X g ∈ G 1 det ( I − z ρ ( g )) Pro of : Let dim C ( W ) = n and let { X 1 , . . . , X n } b e a basis of W ∗ . Since R G = P d R G d and eac h R G d ⊆ C [ X 1 , . . . , X n ] d . Note th at eac h C [ X 1 , . . . , X n ] d is also a r epresen tation ρ d of G . F ur thermore, it is easy to see that if { λ 1 , . . . , λ n } are the eig en v alues of ρ ( g ), then the eig en-v alues of the matrix ρ d ( g ) are p recisel y (including multiplicit y) { Y i λ d i i | X i d i = d } Th us tr ace ( ρ d ( g )) = X d : | d | = d Y i λ d i i W e then h av e: h ( R G ) = P d z d dim C ( R G d ) = P d z d [ 1 | G | P g tr ace ( ρ d ( g ))] = 1 | G | P g 1 (1 − λ 1 ( g ) z ) ... (1 − λ n ( g ) z ) = 1 | G | P g 1 det ( I − z ρ ( g )) This p ro ve s the theorem.  21.3 The Symmetric Group S n will den ote the sym metric group of all bijectio ns on the set [ n ]. The standard r ep r esen tation of S n is ob viously on V = C n with σ · ( v 1 , . . . , v n ) = ( v σ (1) , . . . , v σ ( n ) ) Th us, r eg arding V as column v ectors, and S n as the group of n × n - p erm utation matrices, w e s ee that the action of p erm utation P on v ector v is give n by the matrix multi plication P · v . Let X 1 , . . . , X n b e a basis of V ∗ . S n acts on R = C [ X 1 , . . . , X n ] b y X i · σ = X σ ( i ) . The orbit of a ny point v = ( v 1 , . . . , v n ) is th e collection of all p ermuta tion of the e ntrie s of the vecto r v a nd th us the s ize of the orb it is b oun ded b y n !. 122 The in v arian ts for this act ion are giv en by the elemen tary symmetric p olynomials e k ( X ), for k = 1 , . . . , n , wh ere e k ( X ) = X i 1 : Γ( G ) × X ( G ) → Z which is a u nimo dular p airing o n la ttic es. Exercise 25.1. L et G = ( C ∗ ) 3 and λ and χ b e as fol lows: λ ( t ) = ( t 3 , t − 1 , t 2 ) χ ( t 1 , t 2 , t 3 ) = t − 1 1 t 2 t 2 3 Then, λ ∼ = [3 , − 1 , 2] and χ ∼ = [ − 1 , 1 , 2] . We evaluat e the p airing: < λ, χ > = 3 · − 1 + ( − 1) · 1 + 2 · 2 = 0 W e no w tu r n to the s p ecial case o f D ⊆ S L n , the ma ximal toru s w hic h is isomorphic to ( C ∗ ) n − 1 . By th e ab ov e theorem, Γ( D ) , X ( D ) ∼ = Z n − 1 . Ho wev er, it will m ore con v enien t to iden tify this space as a s u bset of Z n . So let: Y n = { [ m 1 , . . . , m n ] ∈ Z n | m 1 + . . . + m n = 0 } It is e asy to see that Y n ∼ = Z n − 1 . I n fact, w e will set up a sp ecial b iject ion θ : Y n → Z n − 1 deﬁned as: θ ([ m 1 , m 2 , . . . , m n ]) = [ m 1 , m 1 + m 2 , . . . , m 1 + . . . + m n − 1 ] 144 The inv erse θ − 1 is also easily computed: θ − 1 [ a 1 , . . . , a n − 1 ] = [ a 1 , a 2 − a 1 , a 3 − a 2 , . . . , a n − 1 − a n − 2 , − a n − 1 ] This θ corresp onds to the Z -basis of Y n consisting of the ve ctors e 1 − e 2 , . . . , e n − 1 − e n where e i is th e standard basis of Z n . This is also equiv alen t to the em b edding θ ∗ : ( C ∗ ) n − 1 → D as follo ws: ( t 1 , . . . , t n − 1 ) →        t 1 0 . . . 0 0 t − 1 1 t 2 0 . . . 0 . . . 0 . . . 0 t − 1 n − 2 t n − 1 0 0 . . . 0 t − 1 n − 1        A useful computation is to c onsider the inclusion D ⊆ D ∗ , wh ere D ∗ ⊆ GL n is subgroup of al l d iag onal matrices. Clearly Γ( D ) ⊆ Γ( D ∗ ), ho w ev er there is a surjection X ( D ∗ ) → X ( D ). It will be useful to w ork out this su rjectio n explicitly via θ and θ ∗ . If [ m 1 , . . . .m n ] ∈ Z n ∼ = X ( D ∗ ), then it map s to [ m 1 − m 2 , . . . , m n − 1 − m n ] ∈ Z n − 1 ∼ = X (( C ∗ ) n − 1 ) via θ ∗ . If we push this bac k into Y n via θ − 1 , we get: [ m 1 , . . . , m n ] → [ m 1 − m 2 , 2 m 2 − m 1 − m 3 , , . . . , 2 m n − 1 − m n − 2 − m n , m n − m n − 1 ] W e are no w ready to deﬁne the w eigh t spaces of an S L n -mo dule W . So let W b e suc h a mo dule. By restricting this mo dule to D ⊆ G , via Prop osition 25.2, we see that W is a direct sum W = C χ 1 ⊕ . . . ⊕ C χ N , where N = dim C ( W ). Collecti ng identica l c haracters, we see that: W = ⊕ χ ∈ X ( D ) C m χ χ Th us W is a su m of m χ copies of the mo dule C χ . Clearly m χ = 0 for all b ut a ﬁn ite n um b er, and is called the m ultiplicit y of χ . F or a give n mo dule W , computing m χ is an in tricate combinato rial exercise and is giv en by the celebrated W eyl Character F ormula . Exercise 25.2. L et us lo ok at S L 3 and the weight-sp ac es f or some mo dules of S L 3 . A l l mo dules that we discuss wil l also b e GL 3 -mo dules and thus D ∗ mo dules. The formula for c onverting D ∗ -mo dules to D -mo dules wil l b e useful. This m ap is Z 3 → Y 3 and is given b y: [ m 1 , m 2 , m 3 ] → [ m 1 − m 2 , 2 m 2 − m 1 − m 3 , m 3 − m 2 ] 145 The simplest S L n mo dule is C 3 with the b asis { X 1 , X 2 , X 3 } with D ∗ weights [1 , 0 , 0] , [0 , 1 , 0] and [0 , 0 , 1] . This c onverte d to D -weights give us { [1 , − 1 , 0] , [ − 1 , 2 , − 1] , [0 , − 1 , 1] } , with C [1 , − 1 , 0] ∼ = C · X 1 and so on. The next mo dule is S y m 2 ( C 3 ) with the b asis X 2 i and X i X j . The six D ∗ and D -weights with the weight-sp ac es ar e given b elow: D ∗ -wie ghts D -weights weight-sp ac e [2 , 0 , 0] [2 , − 2 , 0] X 2 1 [0 , 2 , 0] [ − 2 , 4 , − 2] X 2 2 [0 , 0 , 2] [0 , − 2 , 2] X 2 3 [0 , 1 , 1] [ − 1 , 1 , 0] X 2 X 3 [1 , 0 , 1] [1 , − 2 , 1] X 1 X 3 [1 , 1 , 0] [0 , 1 , − 1] X 1 X 2 The ﬁnal example is the sp ac e of 3 × 3 -matr ic es M acte d up on by c onju- gation. We se e at onc e tha t M = M 0 ⊕ C · I wher e M 0 is the 8 -dim ensional sp ac e of tr ac e-zer o matric e s, and C · I is 1 -dimensional sp ac e of multiples of the idenity matrix. Weight ve ctors ar e E ij , with 1 ≤ i, j ≤ 3 . The D ∗ weights ar e [1 , − 1 , 0] , [1 , 0 , − 1] , [0 , 1 , − 1] , [ − 1 , 0 , 1] , [0 , − 1 , 1] , [ − 1 , 1 , 0] and [0 , 0 , 0] . The multiplicity o f [0 , 0 , 0] in M is 3 and in M 0 is 2 . Note th at E ii 6∈ M 0 . The D -weights ar e [2 , − 3 , 1] , [1 , 0 , − 1] , [ − 1 , 3 , − 2] and its ne gatives, and ob- viously [0 , 0 , 0] . The normalizer N ( D ) giv es us an action of N ( D ) on the weig ht spaces. If w is a we igh t-v ector of w eigh t χ , t ∈ D and g ∈ N ( D ), then g · w is al so a weig ht ve ctor. Afterall t · ( g · w ) = g · t ′ · w where t ′ = g − 1 tg . T h u s t · ( g · w ) = χ ( t ′ )( g · w ) whence g · w must also b e a we igh t-v ector with some w eigh t χ ′ . This χ ′ is easily compu ted via the action of D ∗ . Here the a ction of N ( D ∗ ) is clear: if χ = [ m 1 , . . . , m n ], then χ ′ = [ m σ (1) , . . . , m σ ( n ) ] for some p ermutati on σ ∈ S n determined b y the component of N ( D ∗ ) c ont aining g . Thus the map χ to χ ′ for D -w eigh ts in the case of S L 3 is as follo ws : [ m 1 − m 2 , 2 m 2 − m 1 − m 3 , m 3 − m 2 ] → [ m σ (1) − m σ (2) , 2 m σ (2) − m σ (1) − m σ (3) , m σ (3) − m σ (2) ] Caution : Note that though Y 3 ⊆ Z 3 is an S 3 -in v arian t subset, the a ction of S 3 on χ ∈ Y 3 is diﬀeren t . Note that, e.g., in the last example ab o ve, [2 , − 3 , 1] is a weig ht bu t not the ‘p erm uted’ vect or [ − 3 , 2 , 1]. This is b ecause of our p eculiar embeddin g of Z n − 1 → Y n . 146 Chapter 26 The Null-cone and the Destabilizing ﬂag R efer enc e: [Ke, N] The fu n damen tal result of Hilb ert states: Theorem 11. L et W b e an S L n -mo dule, and let w ∈ W b e an element of the nul l-c one. Then ther e is a 1 -p ar ameter sub gr oup λ : C ∗ → S L n such tha t lim t → 0 λ ( t ) · w = 0 W In other words, if the zero-v ector 0 W lies in the orb it-c losure of w , then there is a 1-parameter su bgroup ta king it there, in the limit. W e will not pro v e this statemen t here. Our ob j ect iv e for this c hapter is to in terpret the geometric con ten t of th e theorem. W e w ill sh o w that there is a standar d form for an elemen t of the n ull-cone. F or w ell-kno w n represen tatio ns, this standard form is easily identiﬁed b y geometric concepts. 26.1 Characters and the half-space criterion T o b egin, let D b e the ﬁ xed maximal torus. F or an y w ∈ W , we m a y express: w = w 1 + w 2 + . . . + w r where w i ∈ W χ i , the wei ght -space for c haracter χ i . Note the th e ab o ve ex- pression is un ique if we insist that eac h w i b e non-zero. The set of charac ters { χ 1 , . . . , χ r } will b e called the supp ort of w and denote d a s sup p ( w ). Let 147 λ : C ∗ → S L n b e su ch that I m ( λ ) ⊆ D . In this case, the action o f t ∈ C ∗ via λ is easily describ ed: t · w = t ( λ,χ 1 ) w 1 + . . . + t ( λ,χ r ) w r Th us, if lim t → 0 t · w exists (and is 0 W ), then for all χ ∈ sup p ( w ), we hav e ( λ, χ ) ≥ 0 (and furth er ( λ, χ ) > 0). Note that ( λ, χ ) is implement ed as a linear functional on Y n . Th us, if lim t → 0 t · w exists (and is ) W ) then there is a hyperplane in Y n suc h that the supp ort of w is on one side o f the h yp erplane ( strictly on one side of the hyperplane). The normal to this hyperp lane is giv en b y the conv ersion of λ in to Y n notation. On the other hand if the su pp ort s upp ( w ) enjoys the geometric/com binatorial prop ert y , then by th e app r o ximabilit y of r ea ls by rationals , w e see that there is a λ suc h that lim t → 0 t · w exists (and is zero). Th us for 1-parameter sub groups of D , Hilb ert’s theorem translates int o a com b inato rial statemen t on the lattice sub set supp ( w ) ⊂ Y n . W e call this the ( strict ) half-space prop ert y . In th e general case, we know that giv en an y λ : C ∗ → S L n , there is a maximal torus T cont aining I m ( λ ). By the conjugacy result on maximal tori, w e kn o w th at T = AD A − 1 for s ome A ∈ S L n . Thus, we ma y sa y that w is in the null- cone iﬀ there is a tran s late A · w suc h that supp ( A · w ) sat isﬁes th e strict half-space pr operty . Exercise 26.1. L et us c onsider S L 3 acting of the sp ac e of forms of de gr e e 2 . F or the standar d torus D , the weight-sp ac es ar e C · X 2 i and C · X i X j . Consider th e form f = ( X 1 + X 2 + X 3 ) 2 . We se e tha t supp ( f ) is set of al l char acters of S y m 2 ( C 3 ) and do es not satisfy the c ombinato rial pr op erty. However, under a b asis c ha nge A : X 1 → X 1 + X 2 + X 3 X 2 → X 2 X 3 → X 3 we se e that A · f = X 2 1 . Thus A · f do es satisfy the strict ha lf-sp ac e pr op erty. Inde e d c onsider the λ λ ( t ) =   t 0 0 0 t − 1 0 0 0 1   We se e that lim t → 0 t · ( A · f ) = t 2 X 2 1 = 0 148 Thus we se e tha t every form in the nul l-c one ha s a standar d form with a very limite d sets of p ossible supp orts. L et us lo ok at the mo dule M of 3 × 3 -matric es under c onjugation. L et us ﬁx a λ : λ ( t ) =   t n 1 0 0 0 t n 2 0 0 0 t n 3   such that n 1 + n 2 + n 3 = 0 . We may assume that n 1 ≥ n 2 ≥ n 3 . L o oking at the action of λ ( t ) on a gener al matrix X , we se e that: t · X = ( t n i − n j x ij ) Thus if lim t → 0 t · X is to b e 0 then x ij = 0 for al l i > j . In other wor ds, X is strictly upp er-triangular. Considering the gener al 1 -p ar ameter gr oup tel ls us that X is in the nul l-c one i ﬀ ther e is an A such that AX A − 1 is strictly upp er-triangular. In other wor ds, X is nilp oten t . The 1 -p ar ameter sub gr oup identiﬁes this tr ansformation and thus t he ﬂ ag of nilp otency. 26.2 The destabilizing ﬂ ag In this section w e do a more reﬁned analysis of elemen ts of the n ull-cone. The basic motiv ation is to iden tify a unique set of 1- parameter sub groups wh ic h driv e a null-p oin t to zero. T he simplest example is give n by X 2 1 ∈ S y m 2 ( C 3 ). Let λ , λ ′ and λ ′′ b e as b elo w: λ ( t ) =   t 0 0 0 t − 1 0 0 0 1   λ ′ ( t ) =   t 0 0 0 1 0 0 0 t − 1   λ ′′ ( t ) =   t 0 0 0 0 − 1 0 t − 1 0   W e see that all the thr ee λ , λ ′ and λ ′′ driv e X 2 1 to zero. Th e question is w hether these are related, and to classify such 1-parameter su bgroups. Alternately , one ma y view this to a more r eﬁned classiﬁcatio n of p oints in the n ull-cone, su c h as the stratiﬁcation of the nilp oten t matrices by th eir Jordan canonical form. There are t w o asp ects to this analysis. Firstly , to identify a me tric by whic h to choose the ’b est’ 1-parameter subgroup d riving a null- p oin t to zero. Next, to sh o w that there is a unique equiv alence class of s u c h ’b est’ subgroups. 149 T o w ards the ﬁrst ob jectiv e, let λ : C ∗ → S L n b e a 1-parameter subgroup . Without loss of generalit y , w e ma y assume that I m ( λ ) ⊆ D . If w is a null- p oin t then we ha ve: t · w = t n 1 w 1 + . . . + t n k w k where n i > 0 for all i . Clearly , a measure of how fast λ drive s w to zero is m ( λ ) = min { n 1 , . . . , n k } . V erify that this really does n ot d epen d o n t he c h oice of the m aximal torus at all, and thus is well- deﬁned. Next, w e see that for a λ as ab ov e, w e consider λ 2 : C ∗ → S L n suc h that λ 2 ( t ) = λ ( t 2 ). It is ea sy to s ee that m ( λ 2 ) = 2 · m ( λ ). Clearly , λ and λ 2 are in trinsically id en tical and we would lik e to hav e a measure i nv ariant und er suc h scaling. This comes ab out by associating a length to eac h λ . Let λ b e as ab o ve and let I m ( λ ) ⊆ D . Then, there are integ ers a 1 , . . . , a n suc h that λ ( t ) =      t a 1 0 0 0 0 t a 2 0 0 . . . 0 0 0 t a n      W e deﬁne k λ k as k λ k = q a 2 1 + . . . + a 2 n W e m ust sho w that this d o es not dep end on the c hoice of the maximal torus D . L et T ( S L n ) denote the collectio n of all m aximal tori of S L n as abstract subgroups. F or eve ry A ∈ S L n , we ma y deﬁne the map φ A : T → T deﬁned by T → AT A − 1 . T he stabilizer o f a torus T for this a ction of S L n is clearly N ( T ), the normalizer of T . Also recall that N ( T ) /T = W is the (discrete) weyl group. Let I m ( λ ) ⊆ D ∩ D ′ for some tw o maximal tori D and D ′ . Since there is an A suc h that AD ′ A − 1 = D , it is clear that k λ k = k AλA − 1 k . Thus, w e are left to chec k if k λ ′ k = k λ k wh en (i) I m ( λ ) , I m ( λ ′ ) ⊆ D , and (ii) λ ′ = AλA − 1 for some A ∈ S L n . Th is thro ws the questio n to in v ariance of k λ k und er N ( D ), or in other w ords, symmetry under the weyl group. Since W ∼ = S n , the sym m etric group, and since p a 2 1 + . . . + a 2 n is a symmetric functio n on a 1 , . . . , a n , w e ha v e that k λ k is w ell deﬁned. W e no w d eﬁne the eﬃci e ncy of λ on a null-p oin t w to b e e ( λ ) = m ( λ ) k λ k W e immediately s ee that e ( λ ) = e ( λ 2 ). 150 Lemma 26.1. L et W b e a r e pr esentation of S L n and let w ∈ W b e a nul l- p oint. L et N ( w, D ) b e the c ol le ction of al l λ : C ∗ → D such that lim t → 0 t · w = 0 W . If N ( w, D ) is non-empty then ther e is a unique λ ′ ∈ N ( w, D ) which maximizes the eﬃciency, i.e., e ( λ ′ ) > e ( λ ) for al l λ ∈ N ( w, D ) and λ 6 = ( λ ′ ) k for any k ∈ Z . This 1 -p ar ameter sub gr oup wil l b e denote d by λ ( w, D ) . Pro of : Supp ose that N ( w, D ) is non-empty . Then in the w eigh t-space expansion of w for the maximal toru s D , we see that sup p ( w ) s tai sﬁes the half-space pr operty for some λ ∈ Y n . Note th at the λ ∈ N ( w, D ) are parametrized by lattic e p oints λ ∈ Y n suc h that ( λ, χ ) > 0 for all χ ∈ supp ( w ). Let C one ( w ) b e th e conical combinatio n (o ver R ) of all χ ∈ supp ( w ) and C one ( w ) ◦ its p olar . Thus, in other words, N ( w , D ) is precisely the collection of lattice p oin ts in the cone C one ( w ) ◦ . Next, w e see that e ( λ ) is a con ve x function of C on e ( w ) ◦ whic h is c onstan t o ve r ra ys R + · λ for all λ ∈ C one ( w ) ◦ . By a r outine anal ysis, the maximum of suc h a function must b e a unique ra y with r ati onal ent ries. This p ro ve s the lamma.  This co ve rs one imp ortan t part in our task of identifying the ’b est’ 1- parameter sub group driving a null-point to zero. The next p a rt is to r elate D to other maximal tori. Let λ : C ∗ → S L n and let P ( λ ) b e deﬁned as follo ws : P ( λ ) = { A ∈ S L n | lim t → 0 λ ( t ) Aλ ( t − 1 ) = I ∈ S L n } Ha vin g ﬁxed a maximal torus D con taining I M ( λ ), w e easily identify P ( λ ) as a parab olic su bgroup, i.e., blo c k up p er-triangula r. Indeed, let λ ( t ) =      t a 1 0 0 0 0 t a 2 0 0 . . . 0 0 0 t a n      with a 1 ≥ a 2 ≥ . . . ≥ a n (ob viously with a 1 + . . . + a n = 0). Then P ( λ ) = { ( x ij | x ij = 0 for all i, j suc h that a i < a j } The unip oten t radical U ( λ ) is a normal sub group of P ( λ ) d eﬁned as: U ( λ ) = ( x ij ) wh ere =  x ij = 0 if a i < a j x ij = δ ij if a i = a j 151 Lemma 26.2. L et λ ∈ N ( w, D ) and let g ∈ P ( λ ) , then (i) g λg − 1 ⊆ P ( λ ) and P ( g λg − 1 ) = P ( λ ) , (ii) g λg − 1 ∈ N ( w , g Dg − 1 ) , and (iii) e ( λ ) = e ( gλg − 1 ) . This actually f oll o ws from the construction of the explicit S L n -mo dules and is left to the r eader. W e now come to the unique ob ject that w e will deﬁne for eac h w ∈ W in the n u ll-cone. This is t he p arabolic su bgroup P ( λ ) for an y ’b est’ λ . W e hav e already seen ab o v e that if λ ′ is a P ( λ )-conjugate of a b est λ then λ ′ is ’equally b est’ and P ( λ ) = P ( λ ′ ). W e now relate t wo general equally b est λ and λ ′ . F or this we need a preliminary deﬁn itio n and a lemma: Deﬁnition 26.1. L et V b e a ve ctor sp ac e over C . A ﬂag F of V is a se quenc e ( V 0 , . . . , V r ) of neste d subsp ac es 0 = V 0 ⊂ V 1 ⊂ . . . ⊂ V r = V . Lemma 26.3. L et dim C ( V ) = r and let F = ( V 0 , . . . , V r ) and F ′ = ( V ′ 0 , . . . , V ′ r ) b e two (c omplete) ﬂags for V . Then ther e is a b asis b 1 , . . . , b r of V and a p ermutation σ ∈ S r such that V i = { b 1 , . . . , b i } and V ′ i = { b σ (1) , . . . , b σ ( i ) } for a l l i . This is pro ve d by induction on r . Corollary 26.1. L et λ and λ ′ b e two 1 -p ar ameter sub gr oups and P ( λ ) an d P ( λ ′ ) b e their c orr esp onding p ar ab olic sub gr oups. Then ther e is a maximal torus T of S L n such that T ⊆ P ( λ ) ∩ P ( λ ′ ) . Pro of : It is clear that th ere is a corresp ondence b et we en parab olic sub- groups of S L n and ﬂags. W e reﬁne the ﬂags asso ciated to the parab olic subgroups P ( λ ) and P ( λ ′ ) to complete ﬂags and apply the ab o ve lemma.  W e are n o w prepared to prov e Kempf ’d theorem: Theorem 12. L e t W b e a r epr esentation of S L n and w ∈ W a n ul l-p oint. Then th er e is a 1 -p ar ameter sub gr oup λ ∈ Γ( S L n ) such that (i) f or al l λ ′ ∈ Γ( S L n ) , we have e ( λ ) ≥ e ( λ ′ ) , an d (ii) for al l λ ′ such th at e ( λ ) = e ( λ ′ ) w e have P ( λ ) = P ( λ ′ ) and t hat th er e is a g ∈ P ( λ ) such th at λ ′ = g λg − 1 . Pro of : Let N ( w ) b e all elemen ts of Γ( S L n ) whic h driv e w to zero. Let Ξ ( W ) b e the (ﬁnite) collection of D -c haracters app earing in the represen tation W . F or ev ery λ ( w , T ) such that I m ( λ ) ⊆ D , we may consider a n A ∈ S L n suc h that AλA − 1 ∈ N ( A · w , D ) and e ( λ ) = e ( AλA − 1 ). S ince the ’b est’ elemen t of N ( A · w, D ) is determined b y supp ( A · w ) ⊆ Ξ , we see that there are only ﬁnitely man y p ossibilities for e ( A · w , AλA − 1 ) and therefore for e ( λ ) for the ’b est’ λ d riving w to zero. 152 Th us the length k of any sequence λ ( w , T 1 ) , . . . , λ ( w , T k ) suc h that e ( λ ( w , T 1 )) < . . . < e ( λ ( w , T k )) must b e b ounded by the num b er 2 Ξ . This pro v es (i). Next, let λ 1 = λ ( w , T 1 ) and λ 2 = λ ( w , T 2 ) b e tw o ’b est’ elemen ts of N ( w , T 1 ) and N ( w, T 2 ) resp ectiv ely . By corollary 26.1, we hav e a torus , sa y D , and P ( λ ( w , T i ))-conjugate s λ i suc h that (i) e ( λ i ) = e ( λ ( w, T i )) a nd (ii) I m ( λ i ) ⊆ D . By le mma 26.1, we h av e λ 1 = λ 2 and th us P ( λ 1 ) = P ( λ 2 ). On the other h and, P ( λ ( w, T i )) = P ( λ i ) and this p ro ves (ii).  Th us 12 asso ciates a unique parab olic s u bgroup P ( w ) to ev ery p oint in the null-cone. This subgroup is cal led the destabilizing ﬂag of w . Clearly , if w is in the null-c one then so is A · w , where A ∈ S L n . F urthermore, it is clear that P ( A · w ) = AP ( w ) A − 1 . Corollary 26.2. L et w ∈ W b e in the nul l- c one and let G w ⊆ S L n stabilize w . Then G w ⊆ P ( w ) . Pro of : Let g ∈ G w . Since g · w = w , w e see that g P ( w ) g − 1 = P ( w ), and that g normalizes P ( w ). S ince the normalizer of an y p arabolic sub group is itself, we see th at g ∈ P ( w ).  153 Chapter 27 Stabilit y R efer enc e: [Ke, GCT1] Recall that z ∈ W is stable iﬀ i ts orbit O ( z ) is cl osed in W . In the last c h apter, w e tac kled the p oint s in the n u ll-c one, i.e., p oin ts in the s et [0 W ] ≈ , or in other w ords, p oin ts wh ic h close ont o the stable p oint 0 W . A similar analysis may b e done for arbitrary s table p oints. F ollo wing ke mpf, let S ⊆ W b e a clo sed S L n -in v arian t subset. Let z ∈ W b e arbitrary . If the orbit-c losure ∆( z ) in tersects S , then we asso ciate a unique parab olic subgroup P z ,S ⊆ S L n as a witness to this fact. The construction of this p arabolic subgroup is in several steps. As the ﬁrs t step, we construct a represen tation X of S L n and a closed S L n -in v arian t embedd ing φ : W → X suc h th at φ − 1 (0 X ) = S , s cheme- theoretica lly . Th is may b e done as follo ws: since S is a cl osed sub-v ariet y of W , there is an ideal I s = ( f 1 , . . . , f k ) of deﬁnition for S . W e may further assume that the v ector space { f 1 , . . . , f k } is itself an S L n -mo dule, sa y X . W e assume that X is k -dimensional. W e no w constru ct the map φ : W → X as follo ws: φ ( w ) = ( f 1 ( x ) , . . . , f k ( x )) Note that φ ( S ) = 0 X and that I S = ( f 1 , . . . , f k ) ensure that the requiremen ts on our φ d o hold. Next, there is an adaptation of (Hilb ert’s) Theorem 11 whic h we do not pro v e: Theorem 13. L et W b e an S L n -mo dule and let y ∈ W b e a stable p oint. L et z ∈ [ y ] ≈ b e an element which closes onto y . Then ther e is a 1 -p ar ameter sub gr oup λ : C ∗ → S L n such that lim t → 0 λ ( t ) · w ∈ O ( y ) 154 Thus the limit e xists and lies in the close d orbit of y . No w supp ose that ∆( z ) ∩ S is n on-empt y . Then there must b e stable y ∈ ∆( z ). W e a pply the theorem to O ( y ) and o btain t he λ as abov e. This sho ws that there is in deed a 1 -parameter su bgroup driving z in to S . Next, it is easy to see that lim t → 0 [ λ ( t ) · φ ( z )] = 0 X Th us φ ( z ) actually lies in the null-c one of X . W e ma y no w b e tempted to apply the te c hniques of the previous c hapter to come up with the ’b est’ λ and its p arabolic, now called P ( z , S ). Th is is almost the tec h n ique to b e adopted , except that this ’b est’ λ driv es φ ( z ) in to 0 X but lim t → 0 [ λ ( t ) · z ] (whic h is sup p osed to b e in S ) ma y not exist! Th is is b ecause w e are usin g the unprov ed (and untrue) con v erse of the assertion that 1-parameter sub group s whic h d riv e z in to S dr iv e φ ( z ) into 0 X . This ab o v e argument is rectiﬁed b y limiting the domain of allo w ed 1- parameter subgroups to (i) C one ( supp ( φ ( z )) ◦ as b efore, and (ii) those λ suc h that lim t → 0 [ λ ( t ) · z ] exists. This second condition is also a ’con v ex’ condition and then the ’b est’ λ do es exist. This completes the construction of P ( z , S ). As b efore, if G z ⊆ S L n stabilizes z then it normalize s P ( z , S ) th u s must b e con tained in it: Prop osition 27.1. If G z stabilizes z then G z ⊆ P ( z , S ) . Let us no w consider the p ermanen t and the det erminan t . Let M b e th e n 2 -dimensional sp ace of all n × n -matrices. Since det and per m are h omog eneous n -forms on M , w e consider the S L ( M )-mo dule W = S y m n ( M ∗ ). W e r ecall no w certain stabiliz ing groups of the det and the per m . W e wil l need the deﬁnitio n o f a c ertain group L ′ . This is deﬁned as the group generated by the p ermutatio n and diag onal matrices in GL n . In other wo rds, L ′ is the norm alizer of the complete sta ndard to rus D ∗ ⊆ GL n . L is deﬁned as that subgroup of L ′ whic h is cont ained in S L n . Prop osition 27.2. (A) Consider the gr oup K = S L n × S L n . W e deﬁne the action µ K of typic al element ( A, B ) ∈ K on X ∈ M as given by: X → AX B − 1 Then (i) M is an irr e ducible r epr esentation of K and I m ( K ) ⊆ S L ( M ) , and (ii) K stabilizes the determinant. 155 (B) Consider the gr oup H = L × L . W e deﬁne the action µ H of typic al element ( A, B ) ∈ H on X ∈ M as given by: X → AX B − 1 Then (i) M is an irr e ducible r epr esentation of H and I m ( H ) ⊆ S L ( M ) , and (ii) H stabilizes the p ermanent. W e are n o w ready to show: Theorem 14. Th e p oints det and per m in the S L ( M -mo dule W = S y m n ( M ∗ ) ar e sta ble. Pro of : Lets lo ok at det , the per m b eing similar. If det we re n ot stable, then there w ould b e a closed S L ( M )-in v arian t sub set S ⊂ W such that det 6∈ S but closes on to S : just tak e S to b e the unique closed orbit in [ det ] ≈ . Whence there is a parab olic P ( det, S ) whic h, by Prop osition 27. 1, w ould con tain K . This w ould mean that there is a K -in v arian t ﬂag in M corresp onding to P ( det, S ). This con tradicts the irreducibilit y of M as a K -mo dule.  156 Bibliograph y [BBD] A. Beilinson, J. Bernstein, P . Deli gne, F aisceaux p erv ers, Ast ´ erisqu e 100, (1982), So c. Math. F rance. [B] P . Belk ale, Geometric pro ofs of Horn and saturation conject ures, math.A G/0208107. [BZ] A. Be renstein, A. Zelevinsky , T ensor pro duct multiplic ities and con v ex p olytopes in p artit ion space, J. Geom. Ph ys. 5(3): 453 -472, 1988. [DJM] M. Date, M. Jimb o, T . 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