Lagranges Theorem for Hopf Monoids in Species

LA GRANGE’S THEOREM F OR HOPF MONOI DS IN SPECIES MAR CELO A GUIAR AND AARON LA UVE Abstract. F ollowing Radford’s pro of of Lagra ng e’s theorem for p o inted Hopf a lge- bras, we prov e Lagr ange’s theor em f or Hopf monoids in the category o f co nnected sp ecies. As a coro llary , we obta in necessary c o nditions for a given subsp e cies k of a Hopf monoid h to be a Hopf submonoid: the quotien t of any one of the gener ating series o f h by the corr e sp onding gener ating ser ies of k must hav e nonnega tive coeffi- cients. Other coro llaries include a necessary condition for a s equence o f nonnega tive int egers to b e the dimension sequence of a Hopf monoid in the form of certain p oly- nomial inequalities, and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter ex press that the binomial transfor m of the seq uence must be nonnegative. Intr oduction Lagrange’s theorem states that for any s ubgroup K of a group H , H ∼ = K × Q as (left) K -sets, where Q = H /K . In particular, if H is finite, then | K | divide s | H | . Pas sing to group algebras ov er a fie ld k , w e ha v e that k H ∼ = k K ⊗ k Q a s (left) k K - mo dules, or that k H is free as a k K - mo dule. Kaplansky [ 10 ] conjectured that the same statemen t holds for Hopf algebras—group algebras b eing principal examples. It turns out that the result does not hold in general, as show n b y Ob erst and Sc hneider [ 21 , Prop osition 10] and [ 18 , Example 3.5.2]. On the other hand, the result do es hold for certain larg e classes of Hopf alg ebras, including the finite dimen sional ones b y a theorem of Nichols and Z o eller [ 20 ], and the p oin ted ones b y a theorem of Ra df o rd [ 22 ]. F urther (and finer) results of this nature were dev elop ed b y Schneid er [ 27 , 28 ]. Additional w ork on the conjecture includes that of Masuok a [ 16 ] and T ak euchi [ 32 ]; more information can b e found in Sommerh¨ auser’s surv ey [ 30 ]. The main result of t his pap er (Theorem 7 ) is a v ersion of Lagrang e’s theorem for Hopf monoids in the catego r y of connected sp ecies: if h is a connected Hopf monoid and k is a Hopf submonoid, the re exists a sp ecie s q suc h that h = k · q . An immediate application is a test for Hopf submonoids (Corolla r y 13 ): if a n y one of the generating series for a species k does not divide in Q ≥ 0 [ [ x ] ] the cor r esp o nding generating series for the Hopf monoid h (in the sense that t he quotien t has a t least one negativ e co efficie n t), then k is not a Hopf submonoid of h . A similar test also holds for connected graded Date : 18 August 201 1. 2010 Mathematics S ubje ct Classific ation. 05A15 ; 05A20; 05E99 ; 16T05 ; 1 6T30; 18D10; 18D35. Key wor ds and phr ases. Ho pf monoids, sp ecies, g raded Ho pf alg ebras, Lagra nge’s theorem, g en- erating series, Poincar´ e-Birkhoff-Witt theor em, H opf k ernel, Lie kernel, primitive element, par tition, comp osition, linear orde r , cy clic order, derang ement . Aguiar suppo r ted in part by NSF gr ant DMS-10019 35. Lauve supp orted in part by NSA grant H982 30-11 -1-01 85. 1 2 MARCELO A GUIAR AN D AARON LAUVE Hopf algebras (Corollary 4 ). The pro o f of Theorem 7 for Hopf monoids in sp ecies parallels Ra df o rd’s pro of fo r Hopf algebras. (Hopf algebras a re Hopf monoids in the category of v ector spaces). The pap er is orga nized a s follows. In Section 1 , w e recall Lagrange’s theorem for Hopf algebras, fo cusing on the case of connected graded Hopf algebras. In Section 2 , w e recall the basics of Hopf monoids in sp ecies and pro v e Lagrange’s t heorem in this setting. Examples and applications are giv en in Section 3 . Among these, w e deriv e certain p olynomial inequalities that a sequence of nonnegativ e in tegers m ust satisfy in order to be the dimension seque nce of a connected Hopf monoid in sp ecies. In the case of a set-theoretic Hopf monoid structure, we obtain additiona l necessary conditions in the form of linear inequalities whic h expres s that the binomial transform of the en umerating sequence m ust b e no nnegativ e. In Section 4 w e provide information on the gro wth a nd supp ort of the dimension sequence of a connected Hopf monoid. The la tter m ust b e an additiv e submonoid of the natura l n umbers. W e conclude in Section 5 with information on t he sp ecies q en tering in Lagrange’s theorem. In the dual setting, q is the Hopf k ernel of a morphism, and for co comm utative Hopf monoids it can b e describ ed in terms of Lie k ernels and primitiv e elemen ts via the Poincar ´ e-Birkhoff-Witt theorem. All v ector spaces are o v er a fixed field k o f c hara cteristic 0, except in Section 4 , where the c haracteristic is arbitrary . 1. Lagrange’s theorem for Hopf algebras W e begin b y recalling a c ouple of v ersions of this theorem. Theorem 1. L et H b e a fi n ite dimensional Hopf alge br a ove r a field k . If K ⊆ H is any Hopf sub algebr a, then H is a fr e e left (an d right) K -mo dule. This is the Nichols -Zo eller theorem [ 20 ]; see also [ 18 , Theorem 3.1.5]. W e will not mak e direct use of this result, but ins tead of the related results dis cussed b elo w. A Hopf algebra H is p ointe d if all its simple subcoa lgebras are 1-dimensional. Equiv- alen tly , the group-lik e elemen ts of H linearly span the coradical of H . Giv en a subspace K of H , set K + := K ∩ ker( ǫ ) , where ǫ : H → k is the counit of H . Let K + H denote the righ t H -ideal generated b y K + . Theorem 2. L et H b e a p ointe d Hopf algebr a. I f K ⊆ H is any Hopf sub a lgebr a, then H is a f r e e left (and right) K -mo dule. Mor e over, H ∼ = K ⊗ ( H /K + H ) as left K -mo dules. The first statemen t is due to Radford [ 22 , Section 4] and the second (stronger) state- men t to Schneid er [ 27 , Remark 4.14], [ 28 , Corollary 4 .3]. V ario us generalizations can b e found in these references as w ell as in Masuok a [ 16 ] and T ak euc hi [ 32 ]; see also Sommerha ¨ user [ 30 ]. W e are in terested in the particular v ariant giv en in Theorem 3 b elo w. LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 3 A Hopf algebra H is gr ade d if there is giv en a decomposition H = M n ≥ 0 H n in to linear subspaces that is preserv ed b y all op eratio ns. It is c on ne cte d gr ade d if in addition H 0 is linearly spanned b y the unit elemen t. Theorem 3. L et H b e a c onne cte d gr ade d Hopf algebr a. If K ⊆ H is a gr ade d Hopf sub algebr a, then H is a fr e e left (and right) K -mo dule. Mor e ov e r, H ∼ = K ⊗ ( H /K + H ) as left K -mo dules and as gr ade d ve ctor sp ac es. Pr o of. Since H is connected g raded, its coradical is H 0 = k , so H is p oin ted and Theorem 2 applies. Radford’s pro of sho ws that there exists a graded v ector space Q suc h that H ∼ = K ⊗ Q as left K -mo dules and as graded v ector spaces. (The argumen t w e giv e in the parallel setting of Theorem 7 mak es this clear.) Note that K + = L n ≥ 1 K n , hence K + H a nd H /K + H inhe rit the grading of H . T o complete the pro of, it suffice s to sho w that Q ∼ = H /K + H as graded vec tor spaces. Let ϕ : K ⊗ Q → H b e an isomorphism of left K -mo dules and of gr aded v ector spaces. W e claim that ϕ ( K + ⊗ Q ) = K + H . In fact, since ϕ is a morphism of left K -mo dules , ϕ ( K + ⊗ Q ) = K + ϕ (1 ⊗ Q ) ⊆ K + H . Con v ersely , if k ∈ K + and h ∈ H , writing h = P i ϕ ( k i ⊗ q i ) with k i ∈ K and q i ∈ Q , w e obtain k h = X i ϕ ( k k i ⊗ q i ) ∈ ϕ ( K + ⊗ Q ) , since K + is an ideal of K . No w, s ince K = K 0 ⊕ K + , w e ha v e K ⊗ Q = ( K 0 ⊗ Q ) ⊕ ( K + ⊗ Q ) and therefore H /K + H = ϕ ( K ⊗ Q ) /ϕ ( K + ⊗ Q ) ∼ = ϕ ( K 0 ⊗ Q ) ∼ = Q as graded v ector spaces.  Giv en a graded Hopf algebra H , let O H ( x ) ∈ N [ [ x ] ] denote its Poinc ar´ e serie s — the ordinary generating func tion for the sequence of dimensions of its g raded components , O H ( x ) := X n ≥ 0 dim H n x n . Supp ose H is c onnected graded and K is a graded Hopf s ubalgebra. In this case, their P o incar ´ e se ries are of the form 1 + a 1 x + a 2 x 2 + · · · 4 MARCELO AGUIAR AN D AARON LAUVE with a i ∈ N and the quotien t O H ( x ) / O K ( x ) is a w ell-defined p o w er se ries in Z [ [ x ] ]. Corollary 4. L et H b e a c onne cte d gr ade d Hopf al g ebr a. If K ⊆ H is any gr ade d Hopf s ub algebr a, then the quotient O H ( x ) / O K ( x ) of Poin c ar´ e series is nonne gative, i.e., b elongs to N [ [ x ] ] . Pr o of. By Theorem 3 , H ∼ = K ⊗ Q a s graded ve ctor spaces, where Q = H /K + H . Hence O H ( x ) = O K ( x ) O Q ( x ) a nd the result follo ws.  Example 5. Consider the Ho pf algebra QSym of quasisymmetric functions in coun t- ably many v ariables, and the Hopf subalgebra Sym of symme tric functions. They are connected graded, so by Theorem 3 , QSym is a free mo dule o ver Sym . G a rsia and W allac h pro v e this same fact for the algebras QSym n and Sym n of (quasi) symmetric functions in n v ariables (where n is a finite num b er) [ 7 ]. While QSym n and Sym n are quotien t alg ebras of QSym and Sym , t hey are no t quotien t coalgebras. Since a Hopf algebra structure is needed in order to apply Theorem 3 , w e cannot deriv e the result of Garsia a nd W allac h in this m anner. The pap ers [ 7 ] and [ 12 ] pro vide information on the spac e Q n en t ering in the decomposition QSym n ∼ = Sym n ⊗ Q n . 2. Lagrange’s theorem for Hopf monoids in specie s W e first review the notion of Hopf monoid in the category of sp ecies, followin g [ 2 ], and then pro v e Lagrange’s theorem in this setting. W e restrict atten tio n to the case of connected Hopf monoids. 2.1. Hopf monoids in sp ecies. The notion of sp ecies w as introduced by Joy al [ 9 ]. It formalizes t he notion of combinatorial structure and pro vides a framew ork for studying the generating functions whic h en umerate these structures. The bo ok [ 4 ] b y Bergeron, Lab elle and Leroux exp ounds the theory of set species. Jo y a l’s w ork indicates that sp ecies ma y also b e regarded as a lgebraic ob jects; this is the p oin t of view adopted in [ 2 ] and in this work. T o this end, it is conv enien t to work with v ector sp ecies. A (ve ctor) sp e cies is a functor q from finite sets and bijections to v ector spaces and linear maps. Sp ecifically , it is a family of v ector spaces q [ I ], one for each finite set I , together with linear maps q [ σ ] : q [ I ] → q [ J ], one fo r eac h bijection σ : I → J , satisfying q [id I ] = id q [ I ] and q [ σ ◦ τ ] = q [ σ ] ◦ q [ τ ] whenev er σ and τ are composable bijections. The notation q [ a, b, c, . . . ] is shorthand for q [ { a, b, c, . . . } ] and q [ n ] is shorthand for q [1 , 2 , . . . , n ]. The space q [ n ] is an S n -mo dule via σ · v = q [ σ ]( v ) for v ∈ q [ n ] and σ ∈ S n . A sp ecies q is finite dimensional if eac h v ector space q [ I ] is finite dimensional. In this paper, all s p ecies are finite dimensional. A morphism of sp ecies is a nat ural transformatio n of functors. Let Sp denote the category of (finite dimensional) species. W e giv e t w o elemen ta ry examples that w ill b e use ful later. Example 6. Let E be the exp onential sp e cies , defined b y E [ I ] = k {∗ I } for all I . The sym b o l ∗ I denotes a n elemen t canonically asso ciated to the set I (for definiteness , w e LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 5 ma y take ∗ I = I ). Thu s, E [ I ] is a 1-dimensional space with a distinguished basis elemen t. A riche r example is provide d b y the sp ecies L of line ar or d e rs , defined by L [ I ] = k { linear orders o n I } for all I (a space of dimension n ! when | I | = n ). W e use · to de note the Cauchy pr o duct of t wo species. Sp ecifically ,  p · q  [ I ] := M S ⊔ T = I p [ S ] ⊗ q [ T ] for all finite se ts I . The nota t io n S ⊔ T = I indicates that S ∪ T = I and S ∩ T = ∅ . The sum runs o v er all suc h or der e d d e c omp osi tion s of I , or equiv alen tly ov er all subsets S of I : there is o ne term fo r S ⊔ T and another for T ⊔ S . The Cauc h y product turns Sp in to a symmetric monoidal category . The braiding simply s witc hes t he tens or fa ctors. The unit ob ject is the species 1 de fined b y 1 [ I ] := ( k if I is empt y , 0 otherwise. A monoid in the category ( Sp , · ) is a sp ecies m tog ether w ith a morphism of s p ecies µ : m · m → m , i.e., a family of maps µ S,T : m [ S ] ⊗ m [ T ] → m [ I ] , one for each ordered decomp osition I = S ⊔ T , satisfying appropriate asso ciativity and unital conditions, and naturality with resp ect t o bijections. Briefly , to eac h m -structure on S and m -structure on T , there is assigned a n m -structure on S ⊔ T . The a nalogous ob ject in the category of graded v ector spaces is a graded algebra. F or the sp ecie s E , a monoid structure is defined by sending the basis elemen t ∗ S ⊗ ∗ T to the basis elemen t ∗ I . F or L , a monoid structure is provided b y concatenation of linear orders: µ S,T ( ℓ 1 ⊗ ℓ 2 ) = ( ℓ 1 , ℓ 2 ). A c omonoid in the category ( Sp , · ) is a sp ecies c to gether with a morphism of sp ecies ∆ : c → c · c , i.e., a family of maps ∆ S,T : c [ I ] → c [ S ] ⊗ c [ T ] , one for eac h ordered decomp osition I = S ⊔ T , which ar e natural, coasso ciativ e and counital. F or the sp ecies E , a comonoid structure is defined b y sending the basis v ector ∗ I to the basis v ector ∗ S ⊗ ∗ T . F o r L , a comonoid structure is pro vided by restricting a to t a l order ℓ on I to total o r ders on S and T : ∆ S,T ( ℓ ) = ℓ | S ⊗ ℓ | T . W e assume that our sp ecies q are c onne cte d , i.e., q [ ∅ ] = k . In this case, the (co)unital conditions for a (co)monoid force the maps µ S,T (∆ S,T ) to b e the canonical identifica- tions if either S or T is empt y . Th us, in defining a connected (co)monoid structure one only nee ds to s p ecify the maps µ S,T (∆ S,T ) when b oth S and T are nonempt y . A Hopf monoid in the categor y ( Sp , · ) is a mono id and comonoid whose t w o structures are compatible in an appropriate sense, and whic h carries an an tip o de. In this pap er w e only consider connected Hopf monoids. F or suc h Ho pf monoids, the existence o f the an tip o de is guaranteed . The sp ecies E and L , with the structures outlined ab ov e, are t w o imp ort a n t example s of c onnected Hopf monoids. 6 MARCELO AGUIAR AN D AARON LAUVE F or further details on Hopf monoids in sp ecies, see Chapter 8 o f [ 2 ]. The theory o f Hopf monoids in sp ecies is dev elop ed in P art I I of t his reference; sev eral examples ar e discusse d in Chapters 12 and 13. 2.2. Lagrange’s theorem for connected Hopf monoids. Giv en a connected Hopf monoid k in species, w e let k + denote the sp ecies defined b y k + [ I ] = ( k [ I ] if I 6 = ∅ , 0 if I = ∅ . If k is a s ubmonoid of a monoid h , then k + h denotes the right ideal of h gene rated b y k + . In other w ords, ( k + h )[ I ] = X S ⊔ T = I S 6 = ∅ µ S,T  k [ S ] ⊗ h [ T ]  . Theorem 7. L et h b e a c onne cte d Hopf monoi d i n the c ate gory of sp e cie s . If k is a Hopf submonoid of h , then h is a fr e e left k -mo dule. Mor e over, h ∼ = k · ( h / k + h ) as left k -mo dules ( a nd as sp e cies). The pro of is given af ter a series of preparato r y results. Our argumen t parallels Radford’s pro of of the first statemen t in Theorem 2 [ 22 , Section 4]. The main ingredien t is a result of Larson and Sw eedler [ 11 ] kno wn as the fundamen tal theorem of Hopf mo dules [ 18 , Theorem 1.9.4]. It states that if ( M , ρ ) is a left Hopf mo dule ov er K , then M is free as a left K -mo dule and in fact is isomorphic to the Hopf mo dule K ⊗ Q , where Q is the space o f c oinvariants for the coaction ρ : M → K ⊗ M . T ak euc hi extends this result to the con text of Hopf monoids in a braided monoidal category with equalizers [ 33 , Theorem 3.4]; a similar res ult (in a more r estrictive setting) is giv en b y Lyubashenk o [ 14 , Theorem 1.1]. A s a sp ecial case of T ak euchi’s result, we hav e the follo wing. Prop osition 8. L et m b e a left Hopf mo dule over a c onne c te d Hopf monoid k in sp e cies. Ther e is a n isomorphism m ∼ = k · q of le f t Hopf mo dules, whe r e q [ I ] :=  m ∈ m [ I ] | ρ S,T ( m ) = 0 for S ⊔ T = I , T 6 = I  . In p articular, m is fr e e as a left k -mo dule. Here ρ : m → k · m de notes the comodule structure, whic h consists of maps ρ S,T : m [ I ] → k [ S ] ⊗ m [ T ] , one for eac h ordered decomposition I = S ⊔ T . Giv en a comonoid h and t w o subsp ecies u , v ⊆ h , the we dge of u and v is the subspecies u ∧ v of h defined b y u ∧ v := ∆ − 1 ( u · h + h · v ) . The remaining ingredien ts needed for the pro of are supplied b y the follo wing lemmas. Lemma 9. L et h b e a c omono i d i n s p e cies. If u and v ar e sub c omonoid s of h , then: LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 7 (i) u ∧ v is a sub c omonoid of h and u + v ⊆ u ∧ v ; (ii) u ∧ v = ∆ − 1  u · ( u ∧ v ) + ( u ∧ v ) · v  . Pr o of. (i) The pro ofs of the analogo us statemen ts for coalgebras given in [ 1 , Section 3.3 ] extend to this setting. (ii) F rom the definition, ∆ − 1  u · ( u ∧ v ) + ( u ∧ v ) · v  ⊆ u ∧ v . Now, since u ∧ v is a s ub comonoid, ∆( u ∧ v ) ⊆  ( u ∧ v ) · ( u ∧ v )  ∩ ( u · h + h · v ) = u · ( u ∧ v ) + ( u ∧ v ) · v , since u , v ⊆ u ∧ v . This prov es the con vers e inclusion.  Lemma 10. L et h b e a Hopf m onoid in sp e cies and k b e a s ubm o noid. L et u , v ⊆ h b e subsp e ci e s which ar e left k -submo dules of h . Then u ∧ v is a left k -submo dule of h . Pr o of. Since h is a Hopf monoid, the copro duct ∆ : h → h · h is a morphism of left h -mo dules, w here h acts on h · h via ∆. Hence it is also a morphism of left k - mo dules. By hy p othesis, u · h + h · v is a left k -submo dule of h · h . Hence, u ∧ v = ∆ − 1 ( u · h + h · v ) is a left k -submo dule of h .  Lemma 11. L et h b e a Hopf m onoid in sp e cies and k a Hopf submon o id. L e t c b e a sub c omonoi d of h and a left k -submo dule of h . Then ( k ∧ c ) / c is a left k -Hopf mo dule. Pr o of. By L emma 10 , k ∧ c is a left k -submo dule of h . Therefore, the quotien t ( k ∧ c ) / c b y the left k -submo dule c is a le ft k -mo dule. W e next argue that ( k ∧ c ) / c is a k -como dule. Consider the composite k ∧ c ∆ − → k · ( k ∧ c ) + ( k ∧ c ) · c ։ k ·  k ∧ c ) / c , where the first map is gran ted b y Lemma 9 and the second is the pro jection mo dulo c on t he second co ordinate. Since c is a sub comonoid, t he compo site factors throug h c and induces ( k ∧ c ) / c → k ·  k ∧ c ) / c . This define s a le ft k -como dule structure on ( k ∧ c ) / c . Finally , the compatibilit y betw een the mo dule and como dule structures on ( k ∧ c ) / c is inhe rited from the compatibility b et we en the pro duct and copro duct of h .  W e are nearly ready for the pro of o f the main result. First, recall the c or adic al filtr ation of a connected comonoid in sp ecies [ 2 , Se ction 8.10]. Giv en a connected comonoid c , define subspecies c ( n ) b y c (0) = 1 and c ( n ) = c (0) ∧ c ( n − 1) for all n ≥ 1 . W e then hav e c (0) ⊆ c (1) ⊆ · · · ⊆ c ( n ) ⊆ · · · c and c = [ n ≥ 0 c ( n ) . Pr o of o f The or em 7 . W e show that there is a sp ecies q suc h that h ∼ = k · q as left k -mo dules. As in the pro of o f Theorem 3 , it then follo ws that q ∼ = h / k + h . Define a sequence k ( n ) of s ubsp ecies of h by k (0) = k and k ( n ) = k ∧ k ( n − 1) for all n ≥ 1 . 8 MARCELO AGUIAR AN D AARON LAUVE Eac h k ( n ) is a sub comonoid and a left k -submodule of h . This follo ws from Lemmas 9 and 10 b y induction on n . Then Lemma 1 1 pro vides a left k -Hopf mo dule structure on the quotien t sp ecies k ( n ) / k ( n − 1) for all n ≥ 1. Hence k ( n ) / k ( n − 1) is a free left k -mo dule, b y P rop osition 8 . W e claim that there ex ists a se quence of s p ecies q n suc h that k ( n ) ∼ = k · q n as left k -mo dules for a ll n ≥ 0; that is , eac h k ( n ) is a free left k -mo dule. Moreo v er, w e claim that the q n can be c ho sen so that q 0 ⊆ q 1 ⊆ · · · ⊆ q n ⊆ · · · and the ab ov e isomorphisms are compatible with the inclusions q n − 1 ⊆ q n and k ( n − 1) ⊆ k ( n ) . W e pro v e the claims b y induction on n ≥ 0. W e start by letting q 0 = 1 . F or n ≥ 1, w e hav e k ( n − 1) ∼ = k · q n − 1 and k ( n ) / k ( n − 1) ∼ = k · q ′ n for some sp ecies q ′ n (the former b y induction hy p othesis a nd the latter b y the ab ov e argumen t). Let q n = q n − 1 ⊕ q ′ n . By c ho osing an arbitrary k - mo dule section o f the map k ( n ) ։ k ( n ) / k ( n − 1) ∼ = k · q ′ n (p ossible b y freeness), w e obtain an isomorphism k ( n ) ∼ = k · q n extending the isomorphism k ( n − 1) ∼ = k · q n − 1 . This pro v es the claims. W e utilize the coradical filtration of h t o finish the pro of. Since h is connected, h (0) = 1 ⊆ k = k (0) , and by induction, h ( n ) ⊆ k ( n ) for all n ≥ 0 . Hence, h = [ n ≥ 0 h ( n ) = [ n ≥ 0 k ( n ) ∼ = [ n ≥ 0 k · q n ∼ = k · q where q = [ n ≥ 0 q n . Th us h is free as a left k -mo dule.  Let π : h → k b e a morphism of Hopf monoids. The right Hopf kernel of π is the sp ecies define d b y (1) Hk er ( π ) = k er  h ∆ − → h · h π + · id − − − → k + · h  , where π + : h → k + is π follow ed b y the canonical pro jection k ։ k + . F or the f ollo wing result, w e emplo y dualit y for Hopf monoids [ 2 , Section 8.6 .2 ]. (W e assume all sp ecies are finite dimensional.) Theorem 12. L et h b e a c onne cte d Hopf monoi d in the c ate gory of sp e cie s and k a quotient Hopf mo n oid via a mo rp h ism π : h ։ k . Then h is a c of r e e left k -c omo dule. Mor e over, h ∼ = k · Hk er( π ) as left k -c omo dules (and as sp e cies). LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 9 Pr o of. By duality , k ∗ is a Hopf submonoid of h ∗ , so h ∗ ∼ = k ∗ · ( h ∗ / k ∗ + h ∗ ) by Theorem 7 . Dualizing again w e obtain the result, sinc e h ∗ / k ∗ + h ∗ = cok er  k ∗ + · h ∗ π ∗ + · i d − − − → h ∗ · h ∗ ∆ ∗ − → h ∗  .  3. Applica tions and examples 3.1. A test for H opf submonoids. Tw o in v ariants asso ciated to a (finite dimen- sional) sp ecies q a re the exp onential gener ating series E q ( x ) and the typ e gener ating series T q ( x ). They are giv en b y E q ( x ) = X n ≥ 0 dim q [ n ] x n n ! and T q ( x ) = X n ≥ 0 dim q [ n ] S n x n , where q [ n ] S n = q [ n ] / k { v − σ · v | v ∈ q [ n ] , σ ∈ S n } . Both are specializations of the cycle index series Z q ( x 1 , x 2 , . . . ); see [ 4 , Section 1.2] for the defin ition. Sp ecifically , E q ( x ) = Z q ( x, 0 , 0 , . . . ) and T q ( x ) = Z q ( x, x 2 , x 3 , . . . ) . The cycle index series is m ultiplicativ e under Cauc h y pro duct [ 4 , Section 1.3]: if h = k · q , then Z h ( x 1 , x 2 , . . . ) = Z k ( x 1 , x 2 , . . . ) Z q ( x 1 , x 2 , . . . ). By specialization, the same is true for the exp onen tial and t yp e generating series . Let Q ≥ 0 denote the nonnegative rational n um b ers. An immediate conseque nce of Theorems 7 and 12 is t he follow ing. Corollary 13. L et h and k b e c onne cte d Hopf monoids in sp e c i e s. Supp ose k is either a Hop f submonoid or a quotient Hopf monoid of h . Then the quotient of cycle index series Z h ( x 1 , x 2 , . . . ) / Z k ( x 1 , x 2 , . . . ) is nonne gative, i.e., b elongs to Q ≥ 0 [ [ x 1 , x 2 , . . . ] ] . I n p articular, the quotients E h ( x ) / E k ( x ) and T h ( x ) / T k ( x ) ar e also nonne gative. Giv en a connected Hopf monoid h in sp ecies, w e ma y use Corollary 13 to determine if a give n species k ma y be a Hopf submonoid (or a quotien t Hopf m onoid). Example 14. A p artition o f a set I is a n unordered collection of disjoin t nonempt y sub- sets of I w hose union is I . The no tation ab . c is shorthand f or the partition  { a, b } , { c }  of { a, b, c } . Let Π b e the sp ecies of set partitions, i.e., Π [ I ] is t he ve ctor space with basis the set of all partitio ns of I . Let Π ′ denote the subsp ecies linearly spanned by set partitions with dis tinct blo ck sizes. F or e xample, Π [ a, b, c ] = k  abc, a . bc, ab . c, a . bc, a . b . c  and Π ′ [ a, b, c ] = k  abc, a . bc, ab . c, a . bc  . The sequences ( dim Π [ n ]) n ≥ 0 and ( dim Π ′ [ n ]) n ≥ 0 app ear in [ 29 ] as A000110 and A007837, res p ectiv ely . W e hav e E Π ( x ) = exp  exp( x ) − 1  = 1 + x + x 2 + 5 6 x 3 + 5 8 x 4 + · · · and E Π ′ ( x ) = Y n ≥ 1  1 + x n n !  = 1 + x + 1 2 x 2 + 2 3 x 3 + 5 24 x 4 + · · · . 10 MARCELO AGUIAR AN D AARON LAUVE A Hopf monoid structure on Π is defined in [ 2 , Section 12.6]. There are ma ny linear bases of Π indexed by set partitions, and man y w a ys to embed Π ′ as a subsp ecies of Π . Is it possible to e m b ed Π ′ as a Hopf submonoid of Π ? W e ha ve E Π ( x )  E Π ′ ( x ) = 1 + 1 2 x 2 − 1 3 x 3 + 1 2 x 4 − 11 30 x 5 + · · · , so the answ er is negativ e b y Corollary 13 . In fa ct, it is no t p ossible to embed Π ′ as a Hopf s ubmonoid of Π for an y p o ten tial Ho pf monoid structure on Π . W e remark that the t yp e generating series quotien t fo r the pair of species in Exam- ple 14 is nonnegativ e: T Π ( x ) = 1 + x + 2 x 2 + 3 x 3 + 5 x 4 + 7 x 5 + 11 x 6 + 15 x 7 + · · · , T Π ′ ( x ) = 1 + x + x 2 + 2 x 3 + 2 x 4 + 3 x 5 + 4 x 6 + 5 x 7 + · · · , T Π ( x )  T Π ′ ( x ) = 1 + x 2 + 2 x 4 + 3 x 6 + 5 x 8 + 7 x 10 + · · · . This can b e understo o d b y a pp ealing to the Hopf algebra Sy m of symmetric func tions. A basis fo r its homogenous compo nen t of degree n is indexed b y in teger partitions of n , so O Sym ( x ) = T Π ( x ). Mor eov er, T Π ′ ( x ) enume rates the in t eger part it io ns with o dd part sizes and Sym do es indeed contain a Hopf su balgebra w ith this P oincar´ e serie s. It is the algebra of Sch ur Q -functions. See [ 15 , I I I.8]. Thu s T Π ( x )  T Π ′ ( x ) is nonnegativ e b y C orollary 4 . 3.2. T ests for Hopf monoid s. Let ( a n ) n ≥ 0 b e a sequence of nonnegat iv e integers with a 0 = 1 . Do es there exist a connected Hopf monoid h with dim h [ n ] = a n for all n ? The next result pr ovides conditions that the sequence ( a n ) n ≥ 0 m ust satisfy in order for this to b e the case. The pro o f mak es use of the Hadamar d pr o duct of Ho pf monoids [ 2 , Sections 8.1 and 8.13]. If h and k are Hopf monoids, so is h × k , with ( h × k )[ I ] = h [ I ] ⊗ k [ I ] for eac h finite set I . The exp onential sp ecies E is the unit elemen t for the H adamard pro duct. Corollary 15 (The (ord/exp)-test) . F or an y c onne cte d Hopf m onoid in sp e cies h ,  X n ≥ 0 dim h [ n ] x n . X n ≥ 0 dim h [ n ] n ! x n  ∈ Q ≥ 0 [ [ x ] ] . Pr o of. Consider the canonical mo r phism o f Ho pf monoids L ։ E [ 2 , Section 8.5]; it maps any linear order ℓ ∈ L [ I ] to the basis elemen t ∗ I ∈ E [ I ]. The Hadamard pro duct then yie lds a morphism of Hopf monoids L × h ։ E × h ∼ = h . By Corollary 13 , E L × h ( x ) / E h ( x ) ∈ Q ≥ 0 [ [ x ] ]. Since E L × h ( x ) = P n ≥ 0 dim h [ n ] x n , the result follo ws.  Let a n = dim h [ n ]. Corollary 15 states that the r a tio o f the ordinary to the exp onen- tial generating function of the sequence ( a n ) n ≥ 0 m ust b e nonnegativ e. T his t r a nslates in to a sequence of p olynomial ine qualities, the first of whic h are as follo ws: (2) 5 a 3 ≥ 3 a 2 a 1 , 23 a 4 + 12 a 2 a 2 1 ≥ 20 a 3 a 1 + 6 a 2 2 . LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 11 In particular, not ev ery no nnegat iv e sequence ar ises as t he dimension sequ ence of a Hopf m onoid. The following test is of a similar nature, but in v olv es the t yp e instead of the ex- p onen tial generating function. The conditions then dep end not just o n the dimension sequence of h , but also on its sp ecies structure. Corollary 16 (The (ord/t yp e)-test) . F or any c onne cte d Hopf m onoid in sp e cies h ,  X n ≥ 0 dim h [ n ] x n . X n ≥ 0 dim h [ n ] S n x n  ∈ N [ [ x ] ] . Pr o of. W e argue as in the pro of of Corollar y 15 , using ty p e generating functions instead. Since w e ha ve T L × h ( x ) = P n ≥ 0 dim h [ n ] x n , t he result follo ws.  R emark. The previous result may also b e deriv ed as follo ws. According to [ 2 , Chapter 15], associat ed to the Hopf monoid h there are t wo gra ded Hopf algebras K ( h ) and K ( h ), as w ell as a surjectiv e morphism K ( h ) ։ K ( h ) . Moreo v er, the P o incar ´ e se ries for the se Hopf a lgebras are O K ( h ) ( x ) = X n ≥ 0 dim h [ n ] x n and O K ( h ) ( x ) = X n ≥ 0 dim h [ n ] S n x n . Corollary 16 no w follo ws f rom (the dual form of ) Corollary 4 . 3.3. Additional tests for Hopf monoids. The metho d o f Section 3.2 can b e applied in m ultiple situations in order to deduce additional inequalities that the dimension sequence of a connected Hopf monoid m ust satisfy . W e illustrate this next. Let k b e a fixed nonnegativ e in teger. Let E · k denote the k - th Cauc h y p ow er of the exp o nen t ia l sp ecies E . The space E · k [ I ] has a basis consisting of func tions f : I → [ k ]. The sp ecie s E · k carries a Hopf monoid structure [ 2 , Examples 8.17 and 8.18] and an y fixed inclusion [ k ] ֒ → [ k +1] giv es rise to an injectiv e morphism of Hopf monoids E · k ֒ → E · ( k +1) . Employing the Hadamard pro duct as in Section 3.2 , we obt a in an injectiv e morphism of Hopf monoids E · k × h ֒ → E · ( k +1) × h where h is an arbitr a ry connected Hopf monoid. F rom the nonnegativity of the first co efficien ts of E E · ( k +1) × h ( x )  E E · k × h ( x ) w e obtain (2 k + 1 ) a 2 ≥ 2 k a 2 1 and (3 k 2 + 3 k + 1) a 3 ≥ 3(3 k 2 + k ) a 2 a 1 − 6 k 2 a 3 1 . These ine qualities hold for eve ry k ∈ N . Letting k → ∞ w e deduce (3) a 2 ≥ a 2 1 and a 3 ≥ 3 a 2 a 1 − 2 a 3 1 . Example 17. Consider the sp ecies e of elements . The set I is a basis of the space e [ I ], so the dimension sequence of e is a n = n . This sequence do es not satisfy the second inequalit y in ( 3 ). Therefore, the species e does not carry any Hopf monoid s tructure. 12 MARCELO AGUIAR AN D AARON LAUVE 3.4. A test for Hopf monoids o v er E. Our next result is a nec essary condition for a Hopf monoid in s p ecies to con tain or s urject on to the exp onen tial sp ecies E . Giv en a sequence ( a n ) n ≥ 0 , it s binomial tr ansform ( b n ) n ≥ 0 is de fined b y b n := n X i =0  n i  ( − 1) i a n − i . Corollary 18 (The E - test) . Supp ose h is a c onne cte d Hopf monoid that ei ther c ontains the sp e cies E or surje cts onto E (in b oth c a s es as a Hopf mono id). L et a n = dim h [ n ] and a n = dim h [ n ] S n . T h en t he binomial tr a nsform of ( a n ) n ≥ 0 must b e nonne gative and ( a n ) n ≥ 0 must b e nonde cr e asin g. More plainly , in this setting, we must h av e the follo wing ineq ualities: a 1 ≥ a 0 , a 2 ≥ 2 a 1 − a 0 , a 3 ≥ 3 a 2 − 3 a 1 + a 0 , . . . and a n ≥ a n − 1 for all n ≥ 1 . Pr o of. By Corollary 13 , the q uotien t E h ( x ) / E E ( x ) is nonne gative . But E E ( x ) = exp( x ), so the quotien t is giv en b y b 0 + b 1 x + b 2 x 2 2 + b 3 x 3 3! + · · · , where ( b n ) n ≥ 0 is the binomial tra nsform of ( a n ) n ≥ 0 . Replacing expo nen t ia l for t yp e generating functions yields the result f o r ( a n ) n ≥ 0 , sinc e T E ( x ) = 1 1 − x .  R emark. Myhill’s theory of c ombinatorial function s [ 6 , 19 ] provides necessary and suffi- cien t conditions that a seque nce ( a n ) n ≥ 0 m ust satisfy in order for its binomial transform to b e nonnegativ e: the sequence m ust arise f r om a particular t yp e of op erator defined on finite sets. W ork of Menni [ 1 7 ] expands o n this from a categor ical p ersp ectiv e. It w o uld be in teresting to relate these ideas to the ones of this paper. W e mak e a further remark regarding connected line arize d Hopf monoids. These are Hopf monoids of a set theoretic nature. See [ 2 , Section 8.7 ] f o r details. Br iefly , there are set m aps µ S,T : H[ S ] × H[ T ] → H[ I ] and ∆ S,T : H[ I ] → H[ S ] × H[ T ] whic h pro duce a Hopf monoid in (v ector) species when the set species H is linearized. It follows that if h is a linearized Hopf monoid other than the trivial Hopf mono id 1 , then there is a morphism of Hopf monoids f r om h onto E . Th us, Corollary 18 pro vides a t est for existence of a linearized Hopf monoid structure on h . Example 19. W e return to the sp ecies Π ′ of set partitio ns into distinct blo ck sizes. W e might a sk if this can b e made in to a linearized Hopf monoid in some w ay (after Example 14 , this w ould not b e as a Hopf submonoid of Π ). With a n and b n as abov e, w e hav e: ( a n ) n ≥ 0 = 1 , 1 , 1 , 4 , 5 , 16 , 82 , 169 , 54 1 , . . . , ( b n ) n ≥ 0 = 1 , 0 , 0 , 3 , − 8 , 25 , − 9 , − 119 , 736 , . . . . Th us Π ′ fails the E - test and the answ er to the ab o ve question is negativ e. LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 13 3.5. A test for Hopf monoids ov er L. Let h b e a connected Hopf mo no id in sp ecies. Let a n = dim h [ n ] and a n = dim h [ n ] S n . Note that the analogous integers for the species L of linear orders are b n = n ! and b n = 1. Now suppo se that h con tains L or surjects on to L as a Hopf monoid. An obvious necessary condition for this situation is that a n ≥ n ! and a n ≥ 1. O ur next result provide s stronger conditions. Corollary 20 (The L -test) . Supp ose h is a c onne c te d Hopf mono i d that either c ontains the sp e cies L or surje cts onto L (in b oth c a s e s as a Hopf monoid). If a n = dim h [ n ] and a n = dim h [ n ] S n , then a n ≥ na n − 1 and a n ≥ a n − 1 ( ∀ n ≥ 1) . Pr o of. It follows fro m Corolla r y 13 that b oth E h ( x ) / E L ( x ) and T h ( x ) / T L ( x ) are non- negativ e. Thes e yield the first and s econd set of inequalities, respective ly .  Before giving an application of the corollary , w e in tro duce a new Hopf monoid in sp ecies. A c omp osition of a se t I is an ordered collection of disjoin t nonempt y subsets of I whose union is I . The notation ab | c is shortha nd for the comp osition  { a, b } , { c }  of { a, b, c } . Let P al denote the sp ecies of set comp o sitions whose sequenc e of blo c k sizes is palin- dromic. So, for instance, P al [ a, b ] = k  ab, a | b, b | a  and P al [ a, b, c, d, e ] = k  abcde, a | bcd | e, ab | c | de, a | b | c | d | e, . . .  . The latter space has dimension 171 = 1 + 5  4 3  +  5 2  3 + 5! while dim P al [5] S 5 = 4 (accoun ting for the four t yp es of palindromic s et compositions show n ab ov e). Giv en a palindromic set comp osition F = F 1 | · · · | F r , we view it as a tr iple F = ( F − , F 0 , F + ), where F − is the initial sequ ence of blocks , F 0 is the cen tral blo c k if this exists (if the num b er of blo c ks is o dd) and otherwise it is the em pt y set, and F + is the final seq uence of block s. That is, F − = F 1 | · · · | F ⌊ r/ 2 ⌋ , F 0 = ( F ⌊ r/ 2 ⌋ +1 if r is o dd, ∅ if r is ev en, F + = F ⌈ r/ 2+1 ⌉ | · · · | F r . Giv en S ⊆ I , let F | S := F 1 ∩ S | F 2 ∩ S | · · · | F r ∩ S , where empt y in tersections a re deleted. Then F | S is a comp o sition of S . Let us sa y that S is admissible for F if #  F i ∩ S  = #  F r +1 − i ∩ S  for all i = 1 , . . . , r . In this case, b oth F | S and F | I \ S are palindromic. W e employ the ab o v e nota tion to define pro duct a nd copro duct op erations o n Pal . Fix a decomp osition I = S ⊔ T . Pr o duct. Giv en palindromic set comp ositions F ∈ P al [ S ] a nd G ∈ P al [ T ], w e put µ S,T ( F ⊗ G ) :=  F − | G − , F 0 ∪ G 0 , G + | F +  . 14 MARCELO AGUIAR AN D AARON LAUVE In other w ords, we concatenate the initial seque nces of blo c ks of F and G in that order, merge their cen tr a l blo c ks, and concatenate the ir final sequen ces in the o pp osite order. The result is a palindromic c omp osition of I . F or ex ample, with S = { a, b } and T = { c, d, e, f } , ( a | b ) ⊗ ( c | d e | f ) 7→ a | c | de | f | b. Copr o duct. Giv en a palindromic set composition F ∈ P al [ I ], w e put ∆ S,T ( F ) := ( F | S ⊗ F | T if S is admis sible for F , 0 otherwise. F or example, w ith S and T as abov e, ad | b | e | cf 7→ 0 and e | abcd | f 7→ ( ab ) ⊗ ( e | cd | f ) . These op erations endo w Pa l with the structure of Hopf monoid, as ma y b e easily c heck ed. Example 21. A linear order may b e seen as a palindro mic set comp osition (with single- ton blo c ks). Both Hopf monoids P al and L are cocomm utativ e and not comm uta tiv e. W e ma y then ask if P al con tains (or surjects onto) L as a Hopf monoid. W riting a n = dim Pa l [ n ], w e ha ve: ( a n ) n ≥ 0 = 1 , 1 , 3 , 7 , 43 , 171 , 158 1 , 8793 , 1 08347 , . . . . Ho w eve r, ( a n − na n − 1 ) n ≥ 1 = 0 , 1 , − 2 , 15 , − 44 , 555 , − 2274 , 3800 3 , . . . , so P al fails the L -test and the answ er to the ab ov e ques tion is negativ e. 3.6. Examples of nonnegativ e quotien ts. W e commen t on a f ew examples where the quotien t p o w er series E h ( x ) / E k ( x ) is no t only nonnegativ e but is kno wn to ha v e a com binatorial in terpretatio n as a generating function. Example 22. Consider the Hopf monoid Π of set partitions. It con t ains E as a Hopf submonoid via the map that sends ∗ I to the partition of I into singletons. W e ha v e E Π ( x ) / E E ( x ) = exp  exp( x ) − x − 1  , whic h is the exp o nen tia l generating function fo r the num b er of set partitions into blo cks of size strictly big g er than 1. This fact ma y also b e understo o d with the a id of Theo- rem 7 , as follows . Giv en I = S ⊔ T , the pro duct of a partition π ∈ Π [ S ] and a partition ρ ∈ Π [ T ] is the pa r tition π · ρ ∈ Π [ I ] eac h of whose blo c ks is either a blo c k of π or a blo ck of ρ . (In the nota tion of [ 2 , Section 12.6 ], w e are emplo ying the h -basis of Π .) Now, the I -comp onen t of the righ t ideal E + Π is linearly spanned b y elemen ts of the form ∗ S · π where I = S ⊔ T and π is a part ition of T . Then, since ∗ S = ∗ { i } · ∗ S \{ i } (for any i ∈ S ), w e ha v e that E + Π [ I ] is linearly spanned by elemen ts o f the form ∗ { i } · π where i ∈ I and π is a partition of I \ { i } . But the ab ov e description of the pro duct sho ws that these are precisely the partitions with at least one singleton block. LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 15 Example 23. Consider the Ho pf monoids L and E and the surjectiv e morphism π : L ։ E (as in the proof of C orollary 15 ). W e ha v e E L ( x ) = 1 1 − x and E E ( x ) = exp( x ) . It is w ell-kno wn [ 4 , Example 1 .3.9] that exp( − x ) 1 − x = X n ≥ 0 d n n ! x n where d n is the n um b er of der angemen ts of [ n ]. T ogether with Theorem 12 , this suggests the exis tence of a basis for the Hopf k ernel of π indexed b y derangemen ts. W e construct suc h a basis and expand on this discussion in Section 5.3 . Example 24. Let Σ b e the Hopf monoid of set comp ositions defined in [ 2 , Section 12.4]. It con tains L as a Hopf submonoid via the map that views a linear order a s a comp osition in to singletons. (In the nota t ion of [ 2 , Section 12.4 ], we are emplo ying the H -basis of Σ .) This and other morphisms relating E , L , Π and Σ , as well as other Hopf monoids, are discussed in [ 2 , Se ction 12.8]. The sequence (dim Σ [ n ]) n ≥ 0 is A000670 in [ 29 ]. W e ha v e E Σ ( x ) = 1 2 − exp( x ) . Moreo v er, it is kno wn from [ 31 , Ex ercise 5.4.(a)] that 1 − x 2 − exp( x ) = X n ≥ 0 s n n ! x n where s n is the n um b er of thr eshold graphs with vertex set [ n ] and no isolated v ertices. T ogether with Theorem 7 , this suggests the existence o f a ba sis f or Σ / L + Σ indexed b y suc h graphs. W e do not pursue this p ossibilit y in this pap er. 4. The dimension s equence of a connected Hopf monoid Let h b e a connected Hopf monoid and a n = dim h [ n ] for n ∈ N . T he results of Sections 3.2 and 3.3 , deriv ed from Theorem 7 , imp ose restrictions on the sequence a n in the form of p olynomial inequalities. The results of this section are neither w eak er nor stronger tha n those of Section 3 , but prov ide supplemen tary information on the dimension sequenc e a n . They do not mak e use of Theorem 7 . In this s ection, the base field k is of ar bitr ary c haracteristic. Prop osition 25. F or any n, i and j such that n = i + j , (4) a n ≥ a i a j . Pr o of. Since h is connected, the compatibilit y axiom for Hopf monoids (diagr a m (8.18 ) in [ 2 , Section 8.3.1]) implies tha t the comp osite h [ S ] ⊗ h [ T ] µ S,T − − → h [ I ] ∆ S,T − − − → h [ S ] ⊗ h [ T ] is the iden tity . The r esult follo ws b y c ho osing an y decomp osition I = S ⊔ T with | I | = n , | S | = i , and | T | = j .  16 MARCELO AGUIAR AN D AARON LAUVE R emark. The second inequalit y in ( 3 ) ma y b e com bined with ( 4 ) to obtain a 3 − a 2 a 1 ≥ 2 a 1 ( a 2 − a 2 1 ) ≥ 0 . Considerations o f this t yp e sho w that neither set of inequalities ( 2 ), ( 3 ) or ( 4 ) follows from the others. As a first consequence of Prop osition 25 , w e deriv e a result on the growth of the dimension seq uence. Corollary 26. If a 1 ≥ 1 , then the se quenc e a n is w e akly inc r e asing. If mor e over ther e exists k ≥ 1 such that a k ≥ 2 and a i ≥ 1 ∀ i = 0 , . . . , k − 1 , then a n = O (2 n/k ) . Pr o of. The first statemen t follow s from a n ≥ a 1 a n − 1 . No w fix k a s in the second statemen t. Given n ≥ k , write n = q k + r with q ∈ N and 0 ≤ r ≤ k − 1. F rom ( 4 ) w e obtain a n ≥ a q k a r ≥ 2 q = 2 − r /k 2 n/k . Th us a n = O (2 n/k ).  Define the supp ort of h to be the su pp ort of its dimension sequence; namely , supp( h ) = { n ∈ N | a n 6 = 0 } . W e turn t o c onsequences of P rop osition 25 on the support. Corollary 27. The s e t supp( h ) is a submonoid of ( N , +) . Pr o of. By (co)unitalit y o f h , 0 ∈ supp ( h ). (In fact, a 0 = 1 b y connectedness.) By Prop osition 25 , the set supp( h ) is closed under a ddition.  W e men tion that, conv ersely , giv en any submonoid S o f ( N , +), there exists a con- nected Hopf monoid h suc h that supp( h ) = S . Indeed, let Π S [ I ] b e the s pace spanned b y the set of pa rtitions of I whose blo ck sizes b elong to S \ { 0 } . Then Π S is a q uotien t Hopf monoid o f Π and supp( Π S ) = S . (The former follows from the form ulas in [ 2 , Section 12.6.2]; we employ the h -basis of Π .) Example 28. Consider the sp ecial case of the previous paragra ph in whic h S is the submonoid of ev en n um b ers. Then Π S is the sp ecies of set partitions in to blo c ks of ev en size. In part icular, a n = 0 for all o dd n , so the dimension sequence is neither increasing nor of exponential growth. This example shows that the hy p otheses of Coro llary 26 cannot be remo ve d. Corollary 29. The s e t supp ( h ) is either { 0 } o r infinite. The set N \ supp( h ) is fin i te if and only if gcd  supp( h )  = 1 . Pr o of. These statemen ts hold for all submonoids of N , hence for supp( h ) b y Corol- lary 27 . F or the sec ond statemen t , see [ 26 , Lemma 2.1].  LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 17 R emark. W e commen t on coun terparts for connected graded Hopf algebras of the results of this section. Consider the polynomial Hopf algebra H = k [ x 1 , . . . , x k ], in whic h the generators x i are primitiv e and of degree 1. The dimension seque nce is a n =  n + k − 1 k − 1  . In con tra st to Corollary 26 , this sequenc e is p olynomial ev en if a 1 > 1. It follo ws that Prop osition 25 has no coun terpart for connected graded Hopf algebras H , and that the multiplication map H i ⊗ H j → H i + j is not injectiv e in g eneral. Corollaries 27 and 29 fail for connected Hopf algebras ov er a field of p ositiv e c har- acteristic. In characteristic p , a coun terexample is prov ided by H = k [ x ] / ( x p ) with x primitiv e and of degree 1. On the other hand, if the field c haracteristic is 0, then the set S = { n ∈ N | H n 6 = 0 } is a submonoid of ( N , +). This follow s from the fa ct that in this case any connected Hopf algebra is a d omain. W e expand o n this p oint in the Appendix. 5. Hopf kernels for cocommut a tive Hopf monoids Hopf k ernels en ter in the decomp o sition of Theorem 12 (and in dual form, in The- orem 7 ). F or co comm utative Hopf monoids, Hopf k ernels and Lie kerne ls a r e closely related, as discussed in this section. W e prov ide a simple result that allow s us to de- scrib e the Hopf k ernel in certain situations and w e illustrate it with the case of the canonical morphism L ։ E . 5.1. Hopf and Lie kernel s. The sp ecies P ( h ) of primitive elem ents of a connected Hopf m onoid h is defined b y P ( h )[ I ] = { x ∈ h [ I ] | ∆( x ) = 1 ⊗ x + x ⊗ 1 } for eac h nonempt y finite set I , and P ( h )[ ∅ ] = 0. Equiv alen tly , P ( h )[ I ] = \ S ⊔ T = I S,T 6 = ∅ k er  ∆ S,T : h [ I ] → h [ S ] ⊗ h [ T ]  . It is a Lie submonoid of h under the comm uta tor brack et. See [ 2 , Sections 8.10 and 11.9] for more informatio n o n primitiv e elemen ts. Let π : h → k b e a mo r phism of connected Ho pf monoids. It restricts to a morphism of Lie monoids P ( h ) → P ( k ), whic h we still denote b y π . W e define the Lie kernel of π as the species Lk er( π ) = k er  π : P ( h ) → P ( k )  . It is a Lie ideal of P ( h ). The Hopf k ernel Hk er ( π ) is defined in ( 1 ). Lemma 30. L et π : h → k b e a morphi s m of c onne cte d Hopf monoids. The n Lk er( π ) ⊆ Hk er ( π ) . Pr o of. Let x ∈ Lker( π ). Then ( π + · id)∆( x ) = ( π + · id)(1 ⊗ x + x ⊗ 1) = 0 , since π + (1) = 0 and π ( x ) = 0. Thus x ∈ Hke r( π ).  Lemma 31. L et π : h → k b e a mo rphism of c onne cte d Hopf monoids. Then Hk er ( π ) is a submonoid o f h . 18 MARCELO AGUIAR AN D AARON LAUVE Pr o of. By defi nition, Hk er ( π ) = ∆ − 1  Eq( π · id , ι ǫ · id)  , where ι : 1 → k is the unit of k , ǫ : h → 1 is the counit of h , and Eq denotes the equalizer of tw o maps. Since π and ιǫ are morphisms of monoids h → k , the ab o ve equalizer is a submonoid of h · h . Since ∆ is a morphism of monoids, Hk er( π ) is a submonoid of h .  The follo wing result pro vides the a nno unced connection b et w een Lie and Hopf k ernels for co comm utativ e Hopf monoids. It makes use of the P o incar ´ e-Birkhoff-Witt and Cartier-Milnor-Mo or e theorems for sp ecies, whic h are dis cussed in [ 2 , Se ction 11.9.3]. Prop osition 3 2. L e t π : h → k b e a surje ctive morphism of c onne cte d c o c ommutative Hopf monoid s . Then Hk er( π ) is the submonoid of h gener ate d by Lk er( π ) . Pr o of. Lemmas 30 and 31 imply one inclusion. T o conclude the equalit y , it suffices to c heck that the dimensions agree, or equiv alen tly , that t he exp onential generating series are t he same. (W e are assuming finite dim ensionalit y throughout.) First of all, from Theorem 12 , w e ha v e E Hke r( π ) ( x ) = E h ( x ) / E k ( x ) . No w, s ince h is co commutativ e, w e ha ve h ∼ = U  P ( h )  ∼ = S  P ( h )  = E ◦ P ( h ) . The first is an isomorphism of Hopf monoids (the Cartier-Milnor-Mo ore theorem), the second is an isomorphism of comonoids (the P oincar ´ e-Birkhoff-Witt theorem), and t he third is the definition o f the sp ecies underlying S  P ( h )  [ 2 , Section 11.3]. It fo llo ws that E h ( x ) = exp  E P ( h ) ( x )  . F or the same reason, E k ( x ) = exp  E P ( k ) ( x )  , and therefore E Hke r( π ) ( x ) = exp  E P ( h ) ( x ) − E P ( k ) ( x )  . On the other hand, since the functors U and P define an adjoint equiv alence, they preserv e surjectivit y of maps. Th us, the induced map π : P ( h ) → P ( k ) is surjectiv e, and w e ha ve an exact seque nce 0 → Lk er( π ) → P ( h ) → P ( k ) → 0 . Hence, E Lk er( π ) ( x ) = E P ( h ) ( x ) − E P ( k ) ( x ) . Since Lk er( π ) is a Lie submonoid of P ( h ), the submonoid of h generated by Lk er( π ) iden tifies with U  Lk er( π )  . Therefore, as ab o v e, the generating series for the la tter submonoid is exp  E Lk er( π ) ( x )  = exp  E P ( h ) ( x ) − E P ( k ) ( x )  = E Hke r( π ) ( x ) , whic h is the de sired equalit y .  LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 19 R emark. The results of this section hold also for connected (not necess arily graded or finite dimensional) Hopf alg ebras. See [ 5 , Example 4.2 0] for a pro of of Prop osition 32 in this setting. The pro o f ab ov e used finite dimensionality of the (comp onen ts of the) sp ecies, but this hypothesis is not necess ary . The pro of in [ 5 ] ma y b e adapted to y ield the res ult for arbitrary sp ecies. 5.2. The Lie k ernel of π : L ։ E . W e return to the discus sion in Example 23 . The primitiv e elemen ts of the Hopf monoids E and L a re describ ed in [ 2 , Example 1 1 .44]. W e ha ve tha t P ( E ) = X and P ( L ) = Lie where X is the s p ecies of singletons , X [ I ] = ( k if | I | = 1, 0 otherwise, and Lie is the species underlying the Lie op er ad . It follo ws that the Lie k ernel of the canonical morphism π : L ։ E is giv en b y (5) Lk er( π ) [ I ] = ( Lie [ I ] if | I | ≥ 2, 0 otherwise. Before mov ing on to the Hopf ke rnel of π , w e pro vide s ome more information o n the sp ecies Lie . Let I b e a finite nonempt y set and n = | I | . It is know n that the space Lie [ I ] is o f dimension ( n − 1)!. W e pro ceed to describ e a linear basis indexed b y cyclic or ders on I . A cyclic or der on I is a n equiv alence class of line ar orders on I mo dulo t he action i 1 | · · · | i n − 1 | i n 7→ i n | i 1 | · · · | i n − 1 of the cyclic group of o rder n . Eac h class ha s n elemen ts so there are ( n − 1)! cyclic orders on I . W e us e ( b, a, c ) to denote the equiv alence class of the linear order b | a | c . W e fix a finite n onempt y set I and c ho ose a linear order ℓ 0 on I , sa y ℓ 0 = i 1 | i 2 | · · · | i n . The basis of Lie [ I ] will dep end on this c hoice. Giv en a cyclic order γ on I , let S b e the subset of I consisting of the elemen ts encounte red when tra v ersing the cycle from i 1 to i 2 clo c kwise, inclu ding i 1 but excluding i 2 (these are the first and second e lemen t s in ℓ 0 , respectiv ely). Let T consist of the rem aining elemen ts (fro m i 2 to i 1 ). Not e that i 1 ∈ S and i 2 ∈ T , so b oth S and T are nonempt y . The cyclic order γ on I induces cyclic orders o n S and T . W e denote them b y γ | S and γ | T . An elemen t p γ ∈ L [ I ] is defined recurs iv ely b y p γ := [ p γ | S , p γ | T ] = p γ | S · p γ | T − p γ | T · p γ | S . The elemen ts p γ | S ∈ L [ S ] and p γ | T ∈ L [ T ] are t hemselv es defined with resp ect to the induced linear orders ( ℓ 0 ) | S and ( ℓ 0 ) | T . The recursion starts with the case when I is a singleton { a } . In this case, w e set p ( a ) := a ∈ L [ a ] (the unique linear order). 20 MARCELO AGUIAR AN D AARON LAUVE Clearly a ∈ L [ a ] is a primitiv e elemen t. Since the primitiv e elemen ts are closed under comm uta tors, w e ha v e p γ ∈ Lie [ I ]. Moreov er, w e ha v e the follo wing. Prop osition 33. F or fixe d I and ℓ 0 as ab o v e , the set  p γ | γ is a cyclic or der on I  is a line a r b asis of Lie [ I ] . Pr o of. The construction of the elemen ts p γ is a reformu lation of the familiar construction of the Lyndon basis of a free Lie alg ebra [ 13 , 24 , 2 5 ]. Reading the elemen ts of t he cyclic order γ clo ck wise starting at the minimum of ℓ 0 giv es rise to a Lyndon w ord on I (without rep eated letters). The cyclic orders γ | S and γ | T giv e rise to the Lyndon w ords in the canonical factorization of this Lyndon w ord.  F or ex ample, supp o se that I = { a, b, c, d } , ℓ 0 = a | b | c | d and γ = ( b, a, c, d ). Then p ( b,a,c,d ) = [ p ( a,c,d ) , p ( b ) ] =  [ p ( a ) , p ( c,d ) ] , p ( b )  = h  p ( a ) , [ p ( c ) , p ( d ) ]  , p ( b ) i = h  a, [ c, d ]  , b i = a | c | d | b − a | d | c | b − c | d | a | b + d | c | a | b − b | a | c | d + b | a | d | c + b | c | d | a − b | d | c | a. R emark. The v ector sp ecies Lie is not the linearization of the set species of cycles. Note also that, for a g eneral bijection σ : I → J , the p - basis of Lie [ I ] will not map to the p -basis of Lie [ J ] unde r L [ σ ]. 5.3. The Hopf kernel of π : L ։ E. The a b o ve description ( 5 ) of the Lie k ernel o f π : L ։ E together w ith Prop osition 32 imply that the Hopf ke rnel of π is giv en b y Hk er ( π )[ I ] = X k ≥ 1 X S 1 ⊔···⊔ S k = I | S r |≥ 2 ∀ r Lie [ S 1 ] · · · Lie [ S k ] . An elemen t in Lie [ S 1 ] · · · Lie [ S k ] is a k -fold pro duct of primitive elemen ts x r ∈ Lie [ S r ]; eac h S r m ust ha v e at least 2 elemen ts. W e pro ceed to describ e a linear basis fo r Hk er ( π )[ I ]. As in Section 5.2 , w e fix a linear or der ℓ 0 = i 1 | i 2 | · · · | i n on I . The basis will b e indexed by der angemen ts of ℓ 0 . A derangemen t of ℓ 0 is a linear order ℓ = j 1 | j 2 | · · · | j n on I such tha t i r 6 = j r for all r = 1 , . . . , n . View linear orders as bijections [ n ] → I and define σ := ℓ ◦ ℓ − 1 0 . Then σ is a p erm utation of I and ℓ is a de rangemen t of ℓ 0 precisely when σ has no fixed p oints. Let ℓ b e a derangemen t of ℓ 0 and σ the asso ciated p ermutation. Let S 1 , . . . , S k b e the orbits of σ on I lab eled so that min S 1 < · · · < min S k according to ℓ 0 , and let γ r b e the cyclic o rder on S r induced b y σ . In other w ords, σ = γ 1 · · · γ k is the factorization of σ in to cycles , ordered in this sp ecific manner. Emplo ying the p -basis of Lie from Section 5.2 (defined with resp ect to ℓ 0 and the orders induc ed b y ℓ 0 on s ubsets of I ), w e define an elemen t p ℓ ∈ L [ I ] b y p ℓ := p γ 1 · · · p γ k . By ass umption, | S r | ≥ 2 f or all r . Hence p γ r ∈ Lk er( π ) [ S r ] a nd p ℓ ∈ Hk er ( π )[ I ]. LAGRANGE’S THEOREM FOR H OPF MONOIDS IN SPECIES 21 F or ex ample, let I = { e, i, m, s, t } , ℓ 0 = s | m | i | t | e and ℓ = i | t | e | m | s . Then σ = ( s, i, e )( m, t ) , S 1 = { i, e, s } , S 2 = { m, t } , and p ℓ = p ( s,i,e ) p ( m,t ) =  p ( s ) , p ( i,e )  p ( m,t ) =  s, [ i, e ]  [ m, t ] . Prop osition 34. F or fixe d I and ℓ 0 as ab o v e , the set  p ℓ | ℓ is a der angement of ℓ 0  is a line a r b asis of Hker( π )[ I ] . Pr o of. This follo ws from P rop osition 33 and the P o incar ´ e-Birkhoff-Witt theorem.  Example 35. W e describe t he p -basis of Hk er ( π )[ I ] in low cardinalities. Throughout, w e choose ℓ 0 = a | b | c | · · · . The space Hk er( π )[ a, b ] is 1-dimensional, linearly spanne d b y p b | a = p ( a,b ) = [ a, b ] . The space Hk er( π )[ a, b, c ] is 2-dimensional, linearly spanned by p b | c | a = p ( a,b,c ) = [ p ( a ) , p ( b,c ) ] =  a, [ b, c ]  , p c | a | b = p ( a,c,b ) = [ p ( a,c ) , p ( b ) ] =  [ a, c ] , b  . The space Hke r( π )[ a, b, c, d ] is 9- dimensional. There are 6 basis elemen ts corresponding to 4-cycle s, suc h as p c | a | d | b = p ( a,c,d,b ) = h  a, [ c, d ]  , b i , and 3 basis elemen ts corresp onding to pro ducts of t w o 2-cycles, suc h as p b | a | d | c = p ( a,b ) p ( c,d ) = [ a, b ] · [ c, d ] . Appendix The f ollo wing fact w as referred to in the last remark in Se ction 4 . Prop osition 36. L e t H b e a c onne cte d (not ne c essarily gr ade d) Hopf alge br a over a field of cha r acteristic 0 . T hen H is a domain. This result is pro v en in [ 34 , Lemma 1.8(a)], where it is attributed to Le Bruyn. W e pro vide a differen t pro of here. Pr o of. 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