Generating twisted Cherednik eigenfunctions

Generating t wisted Cherednik eigenfunctions A. Mirono v b,c,d, ∗ , A. Morozo v a,c,d, † , A. P op olito v a,c,d, ‡ FIAN/TD-06/26 to the memory of ITEP/TH-09/26 F rancesco Calogero I ITP/TH-09/26 MIPT/TH-09/26 a MIPT, Dolgoprudny, 141701, Russia b L eb e dev Physics Institute, Mosc ow 119991, Russia c NRC “Kur chatov Institute”, 123182, Mosc ow, Russia d Institute for Information T r ansmission Pr oblems, Mosc ow 127994, Russia Abstract Hamiltonians H a k of new in tegrable systems asso ciated with the in teger rays (1 , a ) (comm utative subal- gebras) of Ding-Iohara-Miki (DIM) algebra in the N -bo dy representation are closely related to comm uting t wisted Cherednik Hamiltonians C ( a ) i , H a k = P N i =1 ( C ( a ) i ) k . Moreo ver, symmetric combinations of eigenfunc- tions in the twisted Cherednik system were recen tly shown to pro duce the DIM Hamiltonian eigenstates. W e explicitly construct these twisted Cherednik eigenfunctions recurren tly by action of some (creation and p erm utation) op erations. It resembles of a far-going generalization of Kirillov-Noumi op erators, but exact relation remains to be sp ecified. 1 In tro duction Man y-b ody in tegrable systems of the Calogero-Moser-Sutherland-Ruijsenaars-Schneider type [1–3] attract at- ten tion already during more than half a century , since the first protot yp e was discov ered in the breakthrough pap er by F rancesco Calogero. They rev eal a lot of in teresting structures and ha v e opened man y in teresting new directions admitting v arious extensions. In particular, the Hamiltonians of the Ruijsenaars-Schenider (RS) system [3] form a commutativ e subalgebra of the Ding-Iohara-Miki (DIM) algebra [4, 5], or equiv alen tly , the elliptic Hall algebra [6–8] (whic h is basically the sam e [8, 9]), and the simplest of the RS Hamiltonians (ak a Macdonald op erator [10]) in the case of n particles is H RS 1 = N X i =1 Y j  = i tx i − x j x i − x j q ˆ D i where ˆ D i := x i ∂ ∂ x i . The eigenfunctions of the RS Hamiltonians are the celebrated symmetric Macdonald p olynomials [11]. In terms of the elliptic Hall algebra given b y the generating elements e  γ asso ciated with points of the integer t w o- d imensional lattice (i.e.  γ is a vector in this lattice), the RS Hamiltonians are lying on the v ertical ra y H RS k = H (0 , 1) k := e (0 ,k ) , see Fig.1 and form a commutativ e subalgebra. In this algebra, b y virtute of tw o automorphisms of the elliptic Hall algebra, O h and O v called Miki automorphisms [5, 12], which are generators of the S L (2 , Z ) group: O h : e ( p,r ) → e ( p,p − r ) O v : e ( p,r ) → e ( p + r,r ) ∗ mironov@lpi.ru,mirono v@itep.ru † morozov@itep.ru ‡ pop olit@gmail.com 1 one can generate many more commutativ e subalgebras, asso ciated with rays outcoming from the origin. In the elliptic Hall algebra formulation [7], commutativit y of these ra ys is one of the defining axioms [ e  γ , e k γ ] = 0 ∀  γ and k ∈ Z + and the S L (2 , Z ) symmetry allows one to connect these rays with each other. The S L (2 , Z ) automorphism O − 1 v acts in a trivial wa y [13]. In particular, it immediately maps the vertical ray (i.e. the RS Hamiltonians) to ray ( − 1 , 1): H ( − 1 , 1) k := e [ − k,k ] = q 1 2 P n i =1 (log q x i ) 2 · ˆ H (0 , 1) k · q − 1 2 P n i =1 (log q x i ) 2 Hence, the eigenfunctions of these Hamiltonians are just the same symmetric Macdonald polynomials m ultiplied b y the factor of q 1 2 P n i =1 (log q x i ) 2 . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • H H H H H H H H H H H H H H H H H H H H H H H H H H H                            A A A A A A A A A A A A A A A A A A A A A A A A A A A                            (1,1) ray (2,1) ray (-1,1) ray (-2,1) ray (1,2) ray (-1,2) ray e ( − 1 , 0) e ( − 2 , 0) e (1 , 0) e (2 , 0) e ( − 1 , 1) e (1 , 1) e (0 , 1) e (0 , 2) e (1 , 2) e (2 , 2) @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @                            U O v * O h Figure 1: 2 d in teger lattice of generators of the elliptic Hall/DIM algebra. Eac h ra y (p,r) gives rise to a comm utative subalgebra, and each pair of rays (p,r) and (-p,-r) form a Heisenberg subalgebra. Ho w ev er, action of the second automorphism O h is far less trivial, and the commutativ e Hamiltonians asso ciated with ra ys ( − 1 , a ), which can b e obtained by action of this automorphism are not that simple. One of the wa ys to generate them explicitly is describ ed in [14]. The simplest Hamiltonian at a = 2 is [14, 15] ˆ H ( − 1 , 2) 1 := e [ − 1 , 2] = N X i =1 1 q 1 2 x i Y j  = i ( tx i − x j ) ( x i − x j ) ( q tx i − x j ) ( q x i − x j ) q ˆ D i q ˆ D i + q 1 2 ( t − q )( t − 1) q − 1 X i  = j Y k  = i,j ( tx i − x k )( tx j − x k ) ( x i − x k )( x j − x k ) 1 ( q x i − x j ) q ˆ D i q ˆ D j As we already noted, eigenfunctions of the Hamiltonians ˆ H ( − 1 , 1) k asso ciated with ray (-1,1) are the same sym- metric Macdonald p olynomials m ultiplied by the factor of q 1 2 P n i =1 (log q x i ) 2 . Eigenfunctions of the Hamiltonians ˆ H ( − 1 ,a ) k are [15, 16] the so called twisted Baker-Akhiezer functions in tro duced by O. Chalykh [18, 30] multiplied b y the factor q 1 2 a P n i =1 (log q x i ) 2 . The Bak er-Akhiezer function is a (quasi)p olynomial. Hence, in order to deal 2 with p olynomials, we will use the “rotated” Hamiltonians defined as ˆ H ( a ) k = q − 1 2 a P n i =1 (log q x i ) 2 · ˆ H ( − 1 ,a ) k | {z } e [ − k,ak ] · q 1 2 a P n i =1 (log q x i ) 2 (1) The next imp ortan t step is to note that the Hamiltonians of all these commutativ e subalgebras asso ciated with ra ys can be realized in terms of pow er sums of expressions made of Cherednik op erators and acting on the space of symmetric functions 1 [13] 2 . W e call these expressions twisted Cherednik op erators, and they giv e rise to new systems of comm uting Hamiltonians and, hence, we arrive at new many-bo dy integrable systems. They are of our main in terest here. Note that these commutativ e t wisted Cherednik op erators are generally acting on spaces of non-symmetric functions. In particular, these Hamiltonians asso ciated with the simplest ray ( − 1 , 1) are just the Cherednik Hamilto- nians C i : H ( − 1 , 1) k = P n i =1 C k i . In the case of n = 2, they are C 1 = ( tx 1 − x 2 ) t ( x 1 − x 2 ) q ˆ D 1 + (1 − t ) x 2 t ( x 1 − x 2 ) σ 1 , 2 q ˆ D 1 C 2 = q ˆ D 2 ( x 1 − tx 2 ) t ( x 1 − x 2 ) − q ˆ D 2 (1 − t ) x 2 t ( x 1 − x 2 ) σ 1 , 2 where σ 1 , 2 p erm utes x 1 and x 2 . Eigenfunctions of these Hamiltonians are the non-symmetric Macdonald p olynomials [19–21]. Similarly , the Hamiltonians asso ciated with rays ( − 1 , a ) are the twisted Cherednik Hamiltonians C ( a ) i : H ( − 1 ,a ) k = P n i =1  C ( a ) i  k . In the case of n = 2, a = 2, they are (see details in sec.2.2) C (2) 1 = q 1 2 t 2 ( tx 1 − x 2 )( q tx 1 − x 2 ) ( x 1 − x 2 )( q x 1 − x 2 ) q 2 ˆ D 1 + q 1 2 t 2 (1 − t ) x 2 ( x 1 − q tx 2 ) ( x 1 − x 2 )( x 1 − q x 2 ) q 2 ˆ D 2 σ 1 , 2 + q t 2 (1 − t )( tx 1 − x 2 ) x 1 2 1 x 1 2 2 ( x 1 − x 2 )( q x 1 − x 2 ) q ˆ D 1 + ˆ D 2 σ 1 , 2 − q t 2 (1 − t ) 2 x 1 2 1 x 3 2 2 ( x 1 − x 2 )( x 1 − q x 2 ) q ˆ D 1 + ˆ D 2 C (2) 2 = q ˆ D 2 q 1 2 t 2 ( x 1 − tx 2 )( x 1 − q tx 2 ) ( x 1 − x 2 )( x 1 − q x 2 ) q ˆ D 2 − q ˆ D 2 q 3 2 t 2 (1 − t )( x 1 − tx 2 ) x 2 ( x 1 − x 2 )( x 1 − q x 2 ) q ˆ D 2 σ 1 , 2 − − q t 2 (1 − t )( tx 1 − x 2 ) x 1 2 1 x 1 2 2 ( x 1 − x 2 )( x 1 − q x 2 ) q ˆ D 1 + ˆ D 2 σ 1 , 2 − q t 2 (1 − t ) 2 x 3 2 1 x 1 2 2 ( x 1 − x 2 )( x 1 − q x 2 ) q ˆ D 1 + ˆ D 2 The eigenfunctions of these twisted Cherednik Hamiltonians are then naturally called non-symmetric twisted Macdonald p olynomials [22–24]. Quite non-trivially , the sums of these eigenfunctions with prop er co efficien ts pro duce symmetric functions, which are eigenfunctions of the original DIM Hamiltonians ˆ H ( a ) k [24], whic h is in complete analogy with the standard fact that the non-symmetric Macdonald p olynomials b eing summed to the symmetric ones are eigenfunctions of the RS Hamiltonians. In this pap er, we study the twisted Cherednik in tegrable system, and our goal is to describ e the eigenfunctions of the twisted Cherednik Hamiltonians. W e start in section 2 with description of the Cherednik and twisted Cherednik Hamiltonians, and describ e a set of algebraic relations giving them. Then, in sections 3 and 4, we obtain eigenfunctions of the Cherednik and twisted Cherednik Hamiltonians corresp ondingly . In section 5, we describ e essen tial properties of the twisted eigenfunctions, and, in section 6, an algorithmic pro cedure, which allo ws one to generate these eigenfunctions effectiv ely . Section 7 contains a summary and some concluding remarks. W e attach to this submission a MAPLE file that allows one to generate the twisted and non-twisted Macdonald p olynomials, b oth non-symmetric and symmetric, its short description can be found in the App endix. Notation. Throughout the pap er, we use the notation α for the w eak comp osition (of length n ) of an integer | α | := P n i =1 α i (w e admit some α i = 0 may stand at the end of the w eak comp osition), the notation α + for the corresp onding partition (Y oung diagram), i.e. the weak comp osition when α i are ordered: α 1 ≥ α 2 ≥ . . . ≥ α n ≥ 0, and the notation α − for the inv ersely ordered case 0 ≤ α 1 ≤ α 2 ≤ . . . ≤ α n . W e also regularly use the follo wing quantities: the eigenv alues of the Cherednik Hamiltonians Λ ( i ) α = q α i t − ζ ( α ) i 1 At the algebraic level, it expresses the rel a tion [17] between the DIM (Elliptic Hall) algebra and spherical DAHA [25]. 2 Note that all formulas in [13] differ from those in the present pap er by the rotation as in (1). 3 where ζ ( α ) i = # { k < i | α k ≥ α i } + # { k > i | α k > α i } , and their ratios r α,i := Λ ( i +1) α Λ ( i ) α = q α i +1 − α i t ζ ( α ) i − ζ ( α ) i +1 Note that ζ ( α + ) i = i − 1. Throughout the pap er, the term “p olynomial” implies a p olynomial in x 1 a i ’s. 2 Twisted Cherednik Hamiltonians W e start with description of operators and their comm utation relations used in constructing the Cherednik system. 2.1 Basic op erators Cherednik op erators C k and Demazure-Lustig op erators T i are defined [25–27] R ij : = 1 + (1 − t − 1 ) x j x i − x j (1 − σ i,j ) (2) R − 1 ij = 1 + (1 − t ) x j x i − x j (1 − σ i,j ) T i : = R i,i +1 σ i,i +1 = σ i,i +1 + ( t − 1 − 1) x i +1 x i − x i +1 (1 − σ i,i +1 ) = 1 + x i − t − 1 x i +1 x i − x i +1 ( σ i,i +1 − 1) , i = 1 , . . . , n − 1 T − 1 i = σ i,i +1 R − 1 i,i +1 C i : = t 1 − i   n Y j = i +1 R i,j   q ˆ D i   i − 1 Y j =1 R − 1 j,i   B : = T n − 1 . . . T 2 T 1 x 1 where ˆ D i := x i ∂ ∂ x i and σ i,j p erm utes x i and x j . The pro ducts in C i are obtained so that the smaller index stands to the left. One can also introduce the op eration π : π F ( x 1 , x 2 , . . . , x n ) := F ( q x n , x 1 , . . . , x n − 1 ) (3) so that C i = t 1 − i T i T i +1 . . . T n − 1 π T − 1 1 T − 1 2 . . . T − 1 i − 1 (4) These quantities satisfy a set of relations: • At i = 1 , . . . , n − 2 (Heck e algebra): ( T i − 1)( T i + t − 1 ) = 0 [ T i , T j ] = 0 , | i − j | ≥ 2 (5) T i T i +1 T i = T i +1 T i T i +1 (6) • At i = 1 , . . . , n − 1: tT i C i +1 T i = C i [ T i , C j ] = 0 i  = j, j − 1 (7) tT i x i T i = x i +1 [ T i , x j ] = 0 i  = j, j + 1 (8) C i B = B C i +1 (9) 4 • At i = 1 , . . . , n − 2: π − 1 T i π = T i +1 (10) • At i, j = 1 , . . . , n : [ C i , C j ] = 0 (11) 2.2 Twisted Hamiltonians W e introduce a -twisted Hamiltonians C ( a ) i , which reduce to the standard Cherednik ones at a = 1, and hav e zero grading 3 in x i ’s at any a : C ( a ) i = t 1 − i   n Y j = i +1 R i,j   1 x a − 1 a i q ˆ D i   i − 1 Y j =1 R − 1 j,i   C ( a ) i : = 1 x i  x i C ( a ) i  a (12) The corresp onding twisted “intert wining” op erator is B ( a ) =  T n − 1 . . . T 2 T 1 π − 1 x − 1 a n  a − 1 · B = q  T n − 1 . . . T 2 T 1 x − 1 a 1 π − 1  a π x a +1 a 1 (13) They celebrate the following prop erties: • At i = 1 , . . . , n − 1: tT i C ( a ) i +1 T i = C ( a ) i [ T i , C ( a ) j ] = 0 i  = j, j − 1 (14) B ( a ) C ( a ) i +1 = C ( a ) i B ( a ) (15) • At i, j = 1 , . . . , n : [ C ( a ) i , C ( a ) j ] = 0 (16) 3 Eigenfunctions of the Cherednik Hamiltonians 3.1 Non-symmetric Macdonald p olynomials The common p olynomial eigenfunctions of C i ’s are enumerated by w eak comp ositions and are called non- symmetric Macdonald p olynomials E wλ , where λ is the Y oung diagram, i.e. λ 1 ≥ λ 2 ≥ . . . ≥ λ n ≥ 0, and w is a p erm utation: C i · E wλ = Λ ( i ) wλ · E wλ The eigenv alues are Λ ( i ) λ = q λ i t 1 − i (17) and Λ ( i ) wλ = Λ w ([1 ,n ]) i λ (18) where we denoted through w ([1 , n ]) i the i -th element of the sequence w ([1 , n ]). If one denotes α := w λ , it can b e also written in the form Λ ( i ) α = q α i t − ζ ( α ) i (19) where ζ ( α ) i := # { k < i | α k ≥ α i } + # { k > i | α k > α i } . The non-symmetric Macdonald p olynomials are graded with the grading | α | := P n i =1 α i . 3 W e “rotate” the a -twisted Hamiltonians as compared with [13, Sec.6] so that they hav e zero grading and may admit polynomial solutions. This is the origin of p eculiar x 1 − a a i -factor in front of q ˆ D i , in contrast with the naive x − 1 i . 5 3.2 T riangular structure The non-symmetric Macdonald p olynomials are E α = x α + X β <α C αβ x µ (20) where α is a weak comp osition with n parts (unordered, and some of the parts may be zero). If there are tw o w eak comp ositions , α and β , α > β if α + > β + (e.g., in accordance with the lexicographic order), and if the ordered partitions coincide, one compares the minimal length of p erm utations of the symmetric group S n that allo w one to mak e an ordered partition. The less is the length, the larger is weak comp osition . In other words, the largest one is the Y oung diagram. This is called Bruhat or der [28]. One of the wa ys to unambiguously restore the co efficien ts C αβ in (20) is to use an orthogonality condition with resp ect to the Cherednik sc alar pro duct: D f , g E = n Y i =1 I dx i x i f ( x i ; q , t ) g ( x − 1 i ; q − 1 , t − 1 ) Y i>j ( x i /x j ; q ) ∞ ( q x j /x i ; q ) ∞ ( tx i /x j ; q ) ∞ ( tq x j /x i ; q ) ∞ 3.3 Prop erties of non-symmetric Macdonald p olynomials The op erator B acts on the non-symmetric Macdonald p olynomials in the following wa y [27]: B · E [ α 1 ,...,α n ] ( x 1 , . . . , x n ) = t − # { α i ≤ α 1 } E [ α 2 ,...,α n ,α 1 +1] ( x 1 , . . . , x n ) (21) One can immediately see this from (9), (19) and the fact that action of B raises the grading up by unity , and the co efficien t is restored from the analysis of the leading term in the triangular expansion (20). Acting on the b oth parts of (21) by C n in the form (4) and using (19), one comes to the symmetry property of the non-symmetric Macdonald p olynomials (Knop–Sahi recurrence) [28, 29] E [ α 2 ,...,α n ,α 1 +1] ( x 1 , x 2 , . . . , x n ) = q − α 1 x n E [ α 1 α 2 ,...,α n ] ( q x n , x 1 , x 2 , . . . , x n − 1 ) (22) Other essen tial prop erties of the non-symmetric Macdonald p olynomials are v alid also for the t wisted p oly- nomials (this is because commutation relations of C ( a ) i and C i with T i are the same, see (7) and (14)), w e discuss them in sec.5 for arbitrary a , and do not rep eat them here. They include: • The stability prop ert y , i.e. reduction under x n = 0, see (34). • Their b eha viour under the action of op erators T i ’s, see (36). • Construction of the symmetric p olynomials, see (38). 4 Eigenfunctions of the t wisted Cherednik Hamiltonians The p olynomial solutions of the t wisted Cherednik Hamiltonians C ( a ) i are av ailable at t = q − m with m b eing natural. Hence, hereafter we alwa ys imply t of this form. 4.1 Ground state solution The simplest of the eigenfunctions is the ground state, which is a symmetric function, as usual for the ground state eigenfunctions. Since symmetric eigenfunctions of the twisted Cherednik Hamiltonians C ( a ) i are simultane- ously eigenfunctions of the DIM Hamiltonians H ( a ) k [13] 4 , w e start with constructing the proper solution to these latter. It is a v ailable at t = q − m in the p olynomial form and is called t wisted Bak er-Akhiezer function [15, 16, 30]. The twisted Bak er-Akhiezer function, which is a function of 2 N complex parameters x i and y i , i = 1 , . . . , N , is defined as a sum B ( a ) m (  x,  y ) = N Y i =1 x log q y i a + mρ i i ! · ma X k ij =0 Y i α i +1 (36) where σ i α p ermutes the i -th and ( i + 1)-th parts of α , and C (1) i , C (2) i are some constants of q and t : C (1) i,α : = − (1 − t )Λ ( i,a ) α t (Λ ( i,a ) α − Λ ( i +1 ,a ) α ) = − (1 − t )Λ ( i ) α t (Λ ( i ) α − Λ ( i +1) α ) = − (1 − t ) t (1 − r α,i ) (37) C (2) i,α : = (Λ ( i,a ) α − Λ ( i +1 ,a ) α t )(Λ ( i,a ) α − t − 1 Λ ( i +1 ,a ) α ) t (Λ ( i,a ) α − Λ ( i +1 ,a ) α ) 2 = (Λ ( i ) α − Λ ( i +1) α t )(Λ ( i ) α − t − 1 Λ ( i +1) α ) t (Λ ( i ) α − Λ ( i +1) α ) 2 = (1 − tr α,i )(1 − t − 1 r α,i ) t (1 − r α,i ) 2 9 where we introduced the quantit y r α,i := Λ ( i +1) α Λ ( i ) α = q α i +1 − α i t ζ ( α ) i − ζ ( α ) i +1 F ormulas (36)-(37) follow from (14). Since quantities (37) do not c hange when all the eigenv alues are rescaled at once, they do not dep end on a at all. F or instance: T 1 E ( a ) [0 , 0 , 1] = E ( a ) [0 , 0 , 1] T 1 E ( a ) [0 , 1 , 0] = − 1 − t t (1 − q t ) E ( a ) [0 , 1 , 0] + E ( a ) [1 , 0 , 0] T 1 E ( a ) [1 , 0 , 0] = − 1 − t t (1 − q − 1 t − 1 ) E ( a ) [1 , 0 , 0] + (1 − q )(1 − q t 2 ) t (1 − q t ) 2 E ( a ) [0 , 1 , 0] T 2 E ( a ) [0 , 0 , 1] = − 1 − t t (1 − q t 2 ) E ( a ) [0 , 0 , 1] + E ( a ) [0 , 1 , 0] T 2 E ( a ) [0 , 1 , 0] = − 1 − t t (1 − q − 1 t − 2 ) E ( a ) [0 , 1 , 0] + (1 − q t )(1 − q t 3 ) t (1 − q t 2 ) 2 E ( a ) [0 , 0 , 1] T 2 E ( a ) [1 , 0 , 0] = E ( a ) [1 , 0 , 0] 5.3 Symmetric t wisted Macdonald p olynomials Symmetric t wisted Macdonald p olynomials asso ciated with the dominant integral w eights can be obtained from the non-symmetric Macdonald polynomials by summing up ov er the W eyl group W = S n , i.e. ov er all p erm utations of the partition λ : M ( a ) λ = X α = w · λ w ∈ W E ( a ) α ·    Y ( i,j ): α j >α i j >i Λ ( i ) α − t − 1 Λ ( j ) α Λ ( i ) α − Λ ( j ) α    (38) where λ is a partition (Y oung diagram), the pro duct in the summand runs o ver pairs of ( i, j ) such that α i < α j at i < j , and the sum runs o v er all p ermutations w from the symmetric group S n . These formulas follow from (36), and the coefficients are independent of a , as in (36), which pro ves the corresp onding conjecture in [22, Sec.6.4] . Hence, they can be read off from the a = 1 case when they giv e [20, 31] the symmetric Macdonald p olynomials in the standard normalization of the P polynomials [11]. 6 Algorithmic construction of t wisted non-symmetric Macdonald p olynomials 6.1 Algorithmic pro cedure Note that acting with op erators T i and using (35) (which we denote through “B-op eration” b elow) allo ws one to construct the t wisted non-symmetric Macdonald p olynomials recursively muc h similarly to how it is done in the non-twisted case [28, Lemma 2.1.2]. The pro cedure consists of tw o steps, and is as follo ws. • Assume one has all p olynomials at level | α | − 1. Then, one pic ks up prop er p olynomials at that level in order to generate all p ossible α − ’s at level | α | with the B-op eration. • Then, one uses the T i -op erators in order to generate all weak compositions from these α − ’s. F or instance, in order to get the solution en umerated b y the weak composition [0 , 0 , 3], one uses the following sequence: [0 , 0 , 0] B − → [0 , 0 , 1] T 2 − → [0 , 1 , 0] T 1 − → [1 , 0 , 0] B − → [0 , 0 , 2] T 2 − → [0 , 2 , 0] T 1 − → [2 , 0 , 0] B − → [0 , 0 , 3] Here by action of T i on E ( a ) α w e mean that one has to subtract from T i ( E ( a ) α ) the prop er contribution of E ( a ) α in accordance with (36). 10 The action of B-op eration is very simple. F or instance, [0 , 0 , 0] B − → [0 , 0 , 1] B − → [0 , 1 , 1] B − → [1 , 1 , 1] B − → [1 , 1 , 2] B − → . . . corresp onds to Ξ ( a ) [0 , 0 , 0] − → Ξ ( a ) [0 , 0 , 1] − → Ξ ( a ) [0 , 1 , 1] − → Ξ ( a ) [1 , 1 , 1] − → Ξ ( a ) [1 , 1 , 2] − → . . . This procedure generates the w eak composition α − maximally remote from the Y oung diagram. Other weak comp ositions are obtained b y moving parts of the w eak comp osition to the left by action of T i ’s, and each p erm utation increases the n umber of Ξ ( a ) β emerging in expansion of E ( a ) α , and it increases the n um b er of fractions in front of them b y one, which pro ves the conjecture of [22] that the nu m b er of fractions N α in F α,β (  x ) in formula (28) (whic h is the same for all β ) is N α − plus the minimal length of permutation that brings the weak comp osition α to α − . On one hand, this pro cedure constructively pro ves the conjecture of [22] that the co efficien ts in fron t of Ξ ( a ) β are rational functions that do not dep end on a . On the other hand, these co efficien ts are not that simple. Let us lo ok at them in detail. As so on as we need to mov e the larger num b ers to the left, we need only the second line in (36). Then, we obtain for E ( a ) σ i α using (28): E ( a ) σ i α = T i E ( a ) α − C (1) i,α E ( a ) α = 1 t n tx i x i +1 o n x i x i +1 o · σ i E ( a ) α +   (1 − t − 1 ) n x i x i +1 o − C (1) i,α   E ( a ) α where σ i acts on E ( a ) α p erm uting x i and x i +1 . Using (37), we obtain finally E ( a ) σ i α = 1 t n tx i x i +1 o n x i x i +1 o · σ i E ( a ) α + 1 t (1 − t ) (1 − r α,i ) n r α,i x i +1 x i o n x i +1 x i o E ( a ) α (39) or E ( a ) σ i α = 1 t X β   n tx i x i +1 o n x i x i +1 o · [ σ i F α,β (  x )] · Ξ ( a ) σ i β + (1 − t ) (1 − r α,i ) n r α,i x i +1 x i o n x i +1 x i o F α,β (  x ) · Ξ ( a ) β   (40) Note that this implies that the co efficients in fron t of pro ducts in F α,β (  x ) en tering F λµ ( x ) are ratios of q -num b ers (as was conjectured in [22]). 6.2 Examples of algorithmic pro cedure F or instance, acting by T 1 on E ( a ) [0 , 1 , 1] , in accordance with this formula, we obtain E ( a ) [1 , 0 , 1] : E ( a ) [1 , 0 , 1] = T 1 E ( a ) [0 , 1 , 1] − C (1) 1 , [0 , 1 , 1] | {z } 1 − t − 1 1 − qt 2 E ( a ) [0 , 1 , 1] = 1 t n tx 1 x 2 o n x 1 x 2 o Ξ ( a ) [1 , 0 , 1] + (1 − t ) t (1 − q t 2 ) n q t 2 x 2 x 1 o n x 2 x 1 o Ξ ( a ) [0 , 1 , 1] whic h agrees with formula (40). More in teresting is the case when β i = β i +1 . F or instance, one obtains E ( a ) [0 , 2 , 0] from E ( a ) [0 , 0 , 2] b y the action of T 2 . E ( a ) [0 , 0 , 2] con tains the term with β = [0 , 1 , 1]: F [0 , 0 , 2] , [0 , 1 , 1] = (1 − t ) (1 − q t ) n tx 2 x 3 o n x 2 q x 3 o n tx 2 x 1 o n x 2 x 1 o 11 In accordance with (40), it results in tw o terms in F [0 , 2 , 0] , [0 , 1 , 1] , i.e. in front of Ξ ( a ) [0 , 1 , 1] in E ( a ) σ 2 [0 , 0 , 2] : (1 − t ) (1 − q t )   n tx 2 x 3 o n x 2 x 3 o n tx 3 x 2 o n x 3 q x 2 o n tx 3 x 1 o n x 3 x 1 o + (1 − t ) (1 − q 2 t 2 ) n q 2 t 2 x 3 x 2 o n x 3 x 2 o n tx 2 x 3 o n x 2 q x 3 o n tx 2 x 1 o n x 2 x 1 o   (41) This combination is lo oking different from that in [22, Eq.(107)], but is actually the same. In order to generate further E ( a ) [2 , 0 , 0] = E ( a ) σ 1 [0 , 2 , 0] , one acts by T 1 on E ( a ) [0 , 2 , 0] . There is a conspiracy there. F or instance, the co efficien t in front of Ξ ( a ) [0 , 1 , 1] in E ( a ) σ 2 [2 , 0 , 0] comes from tw o terms (41) and also from the co efficient in fron t of Ξ ( a ) σ 1 [1 , 0 , 1] . The sum of these three terms miraculously pro duces a single term in front of Ξ ( a ) [0 , 1 , 1] in E ( a ) [2 , 0 , 0] , (31): F [2 , 0 , 0] , [0 , 1 , 1] = (1 − t ) (1 − q t )   n tx 2 x 3 o n x 2 x 3 o n tx 3 x 2 o n x 3 q x 2 o n tx 3 x 1 o n x 3 x 1 o + (1 − t ) (1 − q 2 t 2 ) n q 2 t 2 x 3 x 2 o n x 3 x 2 o n tx 2 x 3 o n x 2 q x 3 o n tx 2 x 1 o n x 2 x 1 o   × (1 − t ) (1 − q 2 t ) n q 2 tx 2 x 1 o n x 2 x 1 o + + (1 − t 2 ) (1 − q t )(1 − q 2 t 2 ) n tx 1 x 2 o n x 1 x 2 o n q 2 t 2 x 3 x 1 o n x 2 x 1 o n tx 2 x 3 o n x 2 q x 3 o n tx 2 x 1 o n x 2 x 1 o = q (1 + q )(1 − t ) 2 (1 − q t )(1 − q 2 t ) n tx 3 x 2 o n q x 3 x 2 o n tx 2 x 3 o n q x 2 x 3 o n q tx 3 x 1 o n x 3 x 1 o n q tx 2 x 1 o n x 2 x 1 o 7 Conclusion 7.1 Summary Let us briefly summarize the emerging pattern of eigenfunctions of the twisted Cherednik Hamiltonians described in the previous sections. • The eigenfunctions pro duce a system of excitations ov er the ground state. • The ground state eigenfunction Ω ( a ) m (  x ) of the ground state is still a rather complicated function, which is explicitly known only in particular cases. • Excitations E ( a ) α (  x ) instead are reasonably simple, they dep end on the twist a only through Ω ( a ) m (  x ). ⋄ They are lab eled by w eak comp ositions α = { α 1 , . . . , α n } , α = s ( λ ) b eing an arbitrary p ermutation s of lines in the Y oung diagram (partition) λ = { λ 1 , . . . , λ n } (with λ 1 ≥ λ 2 ≥ . . . ≥ λ n ≥ 0). ⋄ At t = q − m , the eigenfunctions are p olynomials of  x 1 a , decomp osed into p eculiar sums E ( a ) α (  x ) = X β ≤ α F α,β (  x ) · Ξ ( a ) β (42) where Ξ ( a ) β are directly made from Ω ( a ) m , and co efficients F ( m ) αβ are indep endent of the t wisting a and are r ational (!) functions of  x , com bined in to p olynomials due to p eculiar identities for the polynomials Ω ( a ) m , which hav e not yet b een tamed in any systematic wa y . ⋄ The eigenfunctions can be constructed by action of the permutation op erators ˆ T i pro ducing E σ i α from E α , and of the creating op erators B ( a ) , pro ducing E α from the low er level E ′ α , | α | = | α ′ | + 1 by adding a b ox to the weak comp osition, whic h is a direct generalization of the Knop-Sahi recursion. ⋄ T ec hnically effective is first to build the eigenfunctions E α − with 0 ≤ α n ≤ α n − 1 ≤ . . . ≤ α 1 b y the creation operators and then to use the p ermutation op erators ˆ T i in order to generate all other E α with the same α − . ⋄ In fact, there are many wa ys to reach a giv en α starting from the ground state { 0 , . . . , 0 } , but, nev ertheless, they pro duce unam biguous result. The system of relations b etw een these different com binations of creation and p ermutation op erators that ensures it is not yet studied in detail. ⋄ There is not yet any clear reason for the coefficients F ( m ) α,β (  x ) to b e factorized: the p ermutation op erators pro duce them in the form of long sums. But in the so-far studied examples the sums b ecome very short if exist at all. 12 • Due to the independence of a , man y features of the coefficients F ( m ) α,β can already b e observ ed for non- t wisted non-symmetric Macdonald p olynomials (at a = 1) and even for non-symmetric Jack p olynomials. Ho w e v er, at a = 1, there is no clear wa y to see the v ery need of decomp osing polynomials in to com binations of rational functions lik e (42), nor a wa y to unam biguously do so, this explains why this essen tial structure has not b een well-kno wn for researchers so far. 7.2 Concluding remarks T o conclude, in this pap er, we prov ed the three conjectures that w e prop osed in [22]: • The co efficien ts F α,β (  x ) in front of Ξ ( a ) β in form ula (28) are rational functions that do not dep end on a . • The num b er of fractions in F α,β (  x ) is equal to the minimal length of p ermutation that brings the weak comp osition α to α − . • Symmetric p olynomials are made from E ( a ) α b y formula (38). W e ha ve built an algorithmic procedure that allows one to generate arbitrary twisted (non-p olynomial) eigenfunctions basing on the ground state solution Ω ( a ) (  x ). Ho wev er, this solution is known only in the case of t = q − m with natural m due to its relation with the twisted Baker-Akhiezer function (23). Contin uation to an arbitrary t is generally unknown. It can b e sometimes done in the case when one has an explicit formula at hands. F or instance, in the simplest twisted n = 2, a = 2 case, formula (24) can b e directly extended [22] to Ω (2) ( x 1 , x 2 ; q , t ) = t − 1 2 log q x 1  − q q tx 2 x 1 ; q  ∞  − q q x 2 tx 1 ; q  ∞ Ho w ev er, even having explicit sums in the case of arbitrary twist a at n = 2 (see [22, Eq.(94)]), it is not that clear how to extend them to an arbitrary t . Generally , one needs a counterpart of the Noumi-Shiraishi p ow er series [32] in the twisted case, i.e. a twisted coun terpart of the triad [33], which is unknown y et. Another unclear point is the conspiracy illustrated at the v ery end of the previous section: t ypically the co efficien ts in front of each Ξ ( a ) α , which are sums of many terms, either factorize, or at least can b e presented as a sum of a muc h smaller num b er of factorized terms. The mechanism b ehind this phenomenon and the o verall answ er for these co efficients remain unclear. It should b e noted that symmetric Macdonald polynomials are kno wn to be constructed from a trivial p olynomial by applying prop er creation op erators [34, 35]. It is natural to ask what are counterparts of these op erators in the twisted case. W e are planning to return to these problems elsewhere. Ac kno wledgemen ts The w ork w as partially funded within the state assignmen t of the NRC Kurchato v Institute, w as partly supp orted b y the grant of the F oundation for the Adv ancement of Theoretical Physics and Mathematics “BASIS” and by Armenian SCS grants 24WS-1C031. References [1] F. 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In fact, the non-t wisted Macdonald polynomials are certainly a v ailable with muc h more effective Maple pac k ages [36]. Here they remain as a sp ecial case of twisted p olynomi- als, since our co de is designed to work with twisted polynomials. The t wisted polynomials are expressed through the ground state Ω ( a ) (  x ) in the co de, and the only difficult p oint remaining here is an explicit expression for Ω ( a ) (  x ). A t the moment, there is a closed formula for this function only at n = 2, any a in [22, Eqs.(74),(94)], and it is listed for other particular small v alues of parameters in the file Omega3.txt attac hed to the arXiv v ersion of [23]. Here is the example of how to use the MAPLE file. First of all load the file with simple read command read("md-calc.mpl"); The usual symmetric Macdonald p olynomials are given b y the mac function: mac([2,0]); M [2 , 0] ( x 1 , x 2 ) = x 2 1 + ( q + 1)( t − 1) ( q t − 1) x 1 x 2 + x 2 2 The non-symmetric counterparts of usual Macdonald p olynomials are given by the ns mac function: ns mac([2,0]); E (1) [2 , 0] ( x 1 , x 2 ) = x 2 1 + q ( t − 1)( q + 1) ( q 2 t − 1) x 1 x 2 + ( t − 1) q 2 ( q 2 t − 1) x 2 2 The non-symmetric t wisted Macdonald p olynomials, the main sub ject of this paper, are calculated with help of the twisted ns mac function twisted ns mac([2,0]); q t 2 E ( a ) [2 , 0] = n q tx 1 x 2 o n q x 1 x 2 o n tx 1 x 2 o n x 1 x 2 o Ξ ( a ) [2 , 0] + (1 − t ) (1 − q 2 t ) n q tx 2 x 1 o n q x 2 x 1 o n q 2 tx 2 x 1 o n x 2 x 1 o Ξ ( a ) [0 , 2] + + q (1 + q )(1 − t ) (1 − q 2 t ) n tx 1 x 2 o n q x 1 x 2 o n q tx 2 x 1 o n q x 2 x 1 o Ξ ( a ) [1 , 1] and, similarly , the twisted mac function gives the symmetric t wisted Macdonald p olynomials, according to (38). Of note, there are also help er functions slurp xi and deslurp xi which switch b etw een hiding the x - rescaling inside Ξ-functions and manifestly writing the rescaled Ω(  x ), and the underlying twisted mac1 function, of which all the ab ov e mac -functions are just the conv enience wrapp ers. 15