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Kostka-Foulkes polynomials and Macdonald spherical functions 4 Kendra Nelsen 0 Department of Mathematics 0 University of Wisconsin, Madison 2 Madison, WI 53706 USA n [email protected] a J 2 Arun Ram 2 Department of Mathematics ] University of Wisconsin, Madison T Madison, WI 53706 USA R [email protected] . h t a m Abstract. Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials (q-weight multiplicities) arise in many places in combinatorics, [ representation theory, geometry, and mathematical physics. This paper attempts to organize 1 the different definitions of these objects and prove the fundamental combinatorial results from v 8 “scratch”, in a presentation which, hopefully, will be accessible and useful for both the nonexpert 9 and researchers currently working in this very active field. The combinatorics of the affine Hecke 2 algebra plays a central role. The final section of this paper can be read independently of the 1 0 rest of the paper. It presents, with proof, Lascoux and Schu¨tzenberger’s positive formula for the 4 Kostka-Foulkes poynomials in the type A case. 0 / h 0. Introduction t a m The classical theory of Hall-Littlewood polynomials and the Kostka-Foulkes polynomials ap- : pearsin themonographof I.G.Macdonald[Mac]. The Hall-Littlewoodpolynomialsformabasisof v theringofsymmetricfunctionsandtheKostka-Foulkespolynomialsaretheentriesofthetransition i X matrix between the Hall-Littlewood polynomials and the Schur functions. r This theory enters in many different places in algebra, geometry and combinatorics. Many of a these connections appear in [Mac]: (a) [Mac, Ch. II] explains how this theory describesthe structure of the Hall algebra of finite o-modules, where o is a discrete valuation ring. (b) [Mac, Ch.IV] explainshow the Hall-Littlewood polynomialsenterinto therepresentation theory of GL (F ) where F is a finite field with q elements. n q q (c) [Mac, Ch, V] shows that the Hall-Littlewood polynomials arise as spherical functions for GL (Q ) where Q is the field of p-adic numbers. n p p ResearchpartiallysupportedbytheNationalScienceFoundation(DMS-0097977)theNationalSecurityAgency(MDA904- 01-1-0032)andbyEPSRCGrantGR K99015attheNewtonInstituteforMathematicalSciences. Keywords: symmetricfunctions,representationtheory,affineHeckealgebras,Kazhdan-Lusztigpolynomials. 2 k. nelsen and a. ram (d) [Mac, Ch. III §6 Ex. 6] explains how the Kostka-Foulkes polynomials relate to the inter- section cohomology of unipotent orbit closures for GL (C) and [Mac, Ch. III §8 Ex. 8] n explains how the Kostka-Foulkes polynomials describe the graded decomposition of the representations of the symmetric groups S on the cohomology of Springer fibers. n (e) [Mac, Ch. App. A §8 and Ch. III §6] gives that the Kostka-Foulkes polynomials are q-analogues of the weight multiplicities for representations of GL (C). n (f) [Mac, Ch.III(6.5)]explainshowtheKostka-Foulkespolynomialsencodeasubtlestatistic on column strict Young tableaux. Macdonald [Mac2, (4.1.2)] showed that there is a formula for the spherical functions for the Chevalley group G(Q ) which generalizes the formula for Hall-Littlewood symmetric functions. p This combinatorial formula is in terms of the root system data of the Chevalley group G. In [Lu] LusztigshowedthatMacdonald’ssphericalfunctionformulacanbeseenintermsoftheaffineHecke algebra and that the “q-weight multiplicities” or generalized Kostka-Foulkes polynomials coming from these spherical functions are Kazhdan-Lusztig polynomials for the affine Weyl group. Kato [Kt] proved the “partition function formula” for the q-weight multiplicities which was conjectured byLusztig. Thepartitionfunctionformulahasledtocontinuinganalysisoftheconnectionbetween the q-weight multiplicities, functions on nilpotent orbits, filtrations of weight spaces by the kernels of powers of a regular nilpotent element, and degrees in harmonic polynomials (see [JLZ] and the references there). The connection between Hall-Littlewood polynomials and o-modules has seen generalizations in the theory of representations of quivers, the classical case being the case where the quiver is a loopconsistingof onevertexandoneedge. Thistheoryhasbeengeneralizedextensivelyby Ringel, Lusztig, Nakajima and many others and is developing quickly; fairly recent references are [Nak1] and [Nak2]. The connectionto Springer representationsof Weyl groups and the representationsof Cheval- ley groups over finite fields has been developed extensively by Lusztig, Shoji and others; a good survey of the current theory is in [Shj1] and the recent papers [Shj2] show how this theory is beginning to extend its reach outside Lie theory into the realm of complex reflection groups. Sincethe theoryof Macdonaldsphericalfunctions(thegeneralizationof Hall-Littlewood polynomials) and q-weight multiplicities (the generalizationof Kostka-Foulkes polynomials) appearsin so many important parts of mathematics it seems appropriate to give a survey of the basics of this theory. This paper is an attempt to collect together the fundamental combinatorial results analogous to those which are found for the type A case in [Mac]. The presentation here centers on the role played by the affine Hecke algebra. Hopefully this will help to illustrate how and why these objects arise naturally from a combinatorial point of view and, at the same time, provide enough underpinning to the algebra of the underlying algebraic groups to be useful to researchers in representation theory. Using the terms Hall-Littlewood polynomial and Macdonald spherical function interchangeably, and using the words Kostka-Foulkes polynomial and q-weight multiplicity interchangeably, the results that we prove in this paper are: (1) The interpretation of the Hall-Littlewood polynomials as elements of the affine Hecke algebra (via the Satake isomorphism), (2) Macdonald’s spherical function formula, (3) The expansion of the Hall Littlewood polynomial in terms of the standard basis of the affine Hecke algebra, (4) ThetriangularityoftransitionmatricesbetweenMacdonaldsphericalfunctionsandother bases of symmetric functions, kostka-foulkes polynomials 3 (5) The straightening rules for Hall-Littlewood polynomials, (6) The orthogonality of Macdonald spherical functions, (7) The raising operator formula for Kostka-Foulkes polynomials, (8) The partition function formula for q-weight multiplicities, (9) The identification of the Kostka-Foulkes polynomial as a Kazhdan-Lusztig polynomial. All of these results are proved here in general Lie type. They are all previously known, spread throughout various parts of the literature. The presentation here is a unified one; some of the proofs may (or may not) be new. Section4is designedso thatit canbereadindependentlyoftherestof thepaper. InSection4 we give the proof of Lascoux-Schu¨tzenberger’spositive combinatorial formula [LS] (see also [Mac, Ch. III (6.5)]) for Kostka-Foulkes polynomials in type A. Versions of this proof have appeared previously in [Sch] and in [Bt]. This proof has a reputation for being difficult and obscure. After finally getting the courage to attack the literature, we have found, in the end, that the proof is not so difficult after all. Hopefully we have been able to explain it so that others will also find it so. Acknowledgements. A portionof thispaperwaswrittenduringa stayof A. Ram attheNewton Institute for the Mathematical Sciences at Cambridge University. A. Ram thanks them for their hospitalityandsupportduringSpring2001. Thepreparationofthispaperhasbeengreatlyaidedby handwritten lecture notes of I.G. Macdonald from lectures he gave at the University of California, San Diego, in Spring 1991. In several places we have copied rather unabashedly from them. Over many years Professor Macdonald has generously given us lots of handwritten notes. We cannot thank him enough, these notes have opened our eyes to many beautiful things and shown us the “right way” many times when we were going astray. 1. Weyl groups, affine Weyl groups, and the affine Hecke algebra This section sets up the definitions and notations. Good references for this preliminary mate- rial are [Bou], [St] and [Mac4]. The root system and the Weyl group Let h∗ be a real vector space with a nondegenerate symmetric bilinear form h, i. The basic R data is a reduced irreducible root system R (defined below) in h∗. Associated to R are the weight R lattice 2α P = {λ∈ h∗ | hλ,α∨i ∈ Z for all α ∈ R} where α∨ = , (1.1) R hα,αi and the Weyl group s : h∗ −→ h∗ W =hs | α ∈ Ri generated by the reflections α R R (1.2) α λ 7−→ λ−hλ,α∨iα in the hyperplanes H = {x∈ h∗ | hx,α∨i = 0}, α ∈ R. (1.3) α R With these definitions R is a reduced irreducible root system if it is a subset of h∗ such that R (a) R is finite, 0 6∈ R and h∗ = R-span(R), R (b) W permutes the elements of R, i.e. wα ∈ R for w ∈ W and α ∈ R, (c) W is finite, 4 k. nelsen and a. ram (d) R ⊆P, (e) if α ∈ R then the only other multiple of α in R is −α, (f) h∗ is an irreducible W-module. R The choice of a fundamentalregion C for the actionof W on h∗ is equivalent to a choice of positive R roots R+ of R, R+ ={α ∈ R | hx,α∨i > 0 for all x ∈ C} and C ={x ∈ h∗ | hx,α∨i >0 for all α ∈ R+}. R Example 1.4. If h∗ = R2 with orthonormal basis ε =(1,0) and ε =(0,1), P =Z-span{ε ,ε }, R 1 2 1 2 and W = {1,s ,s ,s s ,s s ,s s s ,s s s ,s s s s } is the group of order 8 generated by the 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 reflections s and s in the hyperplanes H and H , respectively, where 1 2 α1 α2 α =2ε , α∨ = ε , 1 1 1 1 then R ={±α ,±α ,±(α +α ),±(α +2α )}. α =ε −ε , α∨ = α , 1 2 1 2 1 2 2 2 1 2 2 H ss2s1Hs1.....s2α...........C.2.1..........C+...............α.............2............•••••.....................................................................................................s..............................................1...........................................s............................................2.............................••••••..............s...........s...................................1..........1..............................s...........C...................................2...........................................C....................................................................................•••••••.................................................α.................................................2....................................................................................................................α.................................................1........••••••............................................................................................................................................................................................................................................................................................................................................................................................................................................................α.............+......................................1...................2......................................α.................................................2...........................................................α......................................•••••••.............1......................................+.................................................s........α................C......................1...........2................................s............................................2.....................α...................•••••...s...........................1...................1...........................................C....................................................................................................................................................................................•••••.........................H.....................αs..........s..2.2.........2.C..........s........1HCα1+2α2 This is the root system of type C . 2 For each α ∈ R+ define the raising operator R :P → P by R µ = µ+α. The dominance α α order on P is given by µ≤ λ if λ= R ···R µ (1.5) β1 βℓ for some sequence of positive roots β ,...,β ∈ R+. 1 ℓ The various fundamental chambers for the action of W on h∗ are the w−1C, w ∈ W. The R inversion set of an element w ∈ W is R(w) ={α ∈ R+ | H is between C and w−1C} and ℓ(w) =Card(R(w)) (1.6) α is the length of w. If R− = −R+ ={−α | α ∈ R+} then R = R+∪R− and R(w) = {α ∈ R+ | wα ∈ R−}, for w ∈ W. kostka-foulkes polynomials 5 The weight lattice, the set of dominant integral weights, and the set of strictly dominant integral weights, are P ={λ∈ h∗ | hλ,α∨i ∈ Z for all α ∈ R}, R P+ = P ∩C ={λ∈ h∗ | hλ,α∨i ∈ Z for all α ∈ R+}, (1.7) R ≥0 P++ = P ∩C ={λ∈ h∗ | hλ,α∨i ∈ Z for all α ∈ R+}, R >0 where C = {x∈ h∗ | hx,α∨i≥ 0 for all α ∈ R+} is the closure of the fundamental chamber C. R Thesimplerootsarethepositiverootsα ,...,α suchthatthehyperplanesH ,1 ≤ i ≤n,are 1 n αi the walls of C. The fundamental weights, ω ,...,ω ∈ P, are given by hω ,α∨i = δ , 1 ≤i,j ≤ n, 1 n i j ij and n n n P = Zω , P+ = Z ω , and P++ = Z ω . (1.8) i ≥0 i >0 i i=1 i=1 i=1 X X X The set P+ is an integral cone with vertex 0, the set P++ is a integral cone with vertex n P+ −→ P++ ρ = ω = 1 α, and the map (1.9) i 2 λ 7−→ λ+ρ i=1 α∈R+ X X is a bijection. In Example 1.4, with the root system of type C , the picture is 2 H H C ......................................s.........2s..........1s.........1s...............2.s......................2.C..........................C...........................................s....................................1.....................................s...........s........................2...........1...........................s...........C.........................1......................................s.....................................2...................................C.................ω................................................2.............0............................••••••......................................................................................................................................................................................................................................................................................................................................................................................α.................................................1............................................................s........•••••............................1.......................ω.................s................................1...2.................................••••.....s...................................1.....................................C.C..........................................................•••.........................................................s...................................2......................................s.......••.......................1.......s..................C....2........................C...............•................................................................................................Hα2 .......................................s..........2s...........1s..........1s................2.s........................2C............................C...........................................s....................................1........................................s...........s..........................2..........1............................s...........C............................1.......................................s....................................2.......................................C................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................α...................................................1............................ρ....................................s......•••••............................................................1.......................................s....................................2...................................••••...s.....................................1.........................................C...........................................................•••...........................................................s....................................2.......................................s.......••........................1......s...................C.....2.........................C...............•.................................................................................................Hα2 The set P+ The set P++ The simple reflections are s = s , for 1 ≤ i ≤ n. The Weyl group W has a presentation by i αi generators s ,...,s and relations 1 n s2 =1, for 1 ≤ i ≤n, i sisjsj··· = sjsisj···, i 6=j, (1.10) mij factors mij factors | {z } | {z } where π/m is the angle between the hyperplanes H and H . A reduced word for w ∈ W is ij αi αj an expression w = s ···s for w as a product of simple reflections which has p minimal. The i1 ip following lemma describes the inversion set in terms of the simple roots and the simple reflections and shows that if w =s ···s is a reduced expression for w then p =ℓ(w). i1 ip 6 k. nelsen and a. ram Lemma 1.11. [Bou VI §1 no. 6 Cor. 2 to Prop. 17] Let w =s ···s be a reduced word for w. i1 ip Then R(w) ={α ,s α ,...,s ···s α }. ip ip ip−1 ip i2 i1 The Bruhat order, or Bruhat-Chevalley order, (see [St, §8 App., p. 126]) is the partial order on W such that v ≤ w if there is a reduced word for v, v = s ···s , which is a subword of a reduced j1 jk word for w, w =s ···s , (i.e. s ,...,s is a subsequence of the sequence s ,...,s ). i1 ip j1 jk i1 ip The affine Weyl group For λ∈ P, the translation in λ is t : h∗ −→ h∗ λ R R (1.12) x 7−→ x+λ. The extended affine Weyl group W˜ is the group W˜ = {wt | w ∈ W,λ ∈ P}, (1.13) λ with multiplication determined by the relations t t = t , and wt =t w, (1.14) λ µ λ+µ λ wλ for λ,µ ∈ P and w ∈ W. The group W˜ is the group of transformations of h∗ generated by the R s , α ∈ R+, and t , λ ∈ P. The affine Weyl group W is the subgroup of W˜ generated by the α λ aff reflections s :h∗ → h∗ in hyperplanes H ={x∈ h∗ | hx,α∨i =k}, α ∈ R+,k ∈ Z. (1.15) α,k R R α,k R The reflections s can be written as elements of W˜ via the formula α,k s =t s =s t . (1.16) α,k kα∨ α α −kα∨ The highest root of R is the unique element ϕ ∈ R+ such that the fundamental alcove A =C ∩{x∈ h∗ | hx,ϕ∨i < 1} (1.17) R is a fundamental region for the action of W on h∗. The various fundamental chambers for the aff R action of W on h∗ are w−1A, w ∈ W . The inversion set of w ∈ W˜ is aff R aff R(w) ={H | H is between A and w−1A} and ℓ(w) = Card(R(w)) α,k α,k is the length of w. If w ∈ W and λ∈ P then ℓ(wt ) = |hλ,α∨i+χ(wα)|, (1.18) λ α∈R+ X where, for a root β ∈ R, set χ(β)= 0, if β ∈ R+, and χ(β)= 1, if β ∈ R−. kostka-foulkes polynomials 7 Continuing Example 1.4, we have the picture H = H H ϕHHHHHααααα11111+++++αHHHHHααααα122222ααααα+,,,,,22222−−−−−α,,,,,54321212345 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.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................HHHHHHHHHααααααααα111111121+++++++,+02222222α=ααααααα22222222,1,,,,,,,H−−−3210=321α=2HHϕα,01+=2αH2α0 Let H =H and s = s =t s =s t , (1.19) α0 ϕ,1 0 ϕ,1 φ∨ φ φ −φ∨ and let H ,...,H and s ,...,s be as in (1.10). Then the walls of A are the hyperplanes α1 αn 1 n H ,H ,...,H and the groupW hasa presentationby generatorss ,s ,...,s and relations α0 α1 αn aff 0 1 n s2 =1, for 0 ≤ i ≤n, i sisjsj··· = sjsisj···, i 6=j, (1.20) mij factors mij factors | {z } | {z } where π/m is the angle between the hyperplanes H and H . ij αi αj Let w be the longest element of W and let w be the longest element of the subgroup W = 0 i ωi {w ∈ W | wω =ω }. Let ϕ∨ =c α∨ +···c α∨. Then (see [Bou, VI §2 no. 3 Prop. 6]) i i 1 1 n n Ω = {g ∈ W˜ | ℓ(g)= 0}= {g | c =1}, where g =t w w . (1.21) i i i ωi i 0 Each element g ∈ Ω sends the alcove A to itself and thus permutes the walls H ,H ,...,H of α0 α1 αn A. Denote the resulting permutation of {0,1,...,n} also by g. Then gs g−1 =s , for 0 ≤i ≤n, (1.22) i g(i) and the group W˜ is presented by the generators s ,s ,...,s and g ∈ Ω with the relations (1.18) 0 1 n and (1.20). 8 k. nelsen and a. ram The affine Hecke algebra Let K = Z[q,q−1]. The affine Hecke algebra H˜ is the algebra over K given by generators T , i 1 ≤i ≤ n, and xλ, λ∈ P, and relations T T T ··· =T T T ···, for all i 6=j, i j i j i j mij factors mij factors |Ti2{=z (q}−q−|1)T{iz+1}, for all 1 ≤i ≤ n, (1.23) xλxµ =xµxλ =xλ+µ, for all λ,µ ∈ P, xλ−xsiλ xλT = T xsiλ+(q−q−1) , for all 1 ≤i ≤ n, λ ∈ P. i i 1−x−αi An alternative presentation of H˜ is by the generators T , w ∈ W˜ , and relations w T T = T , if ℓ(w w ) =ℓ(w )+ℓ(w ), w1 w2 w1w2 1 2 1 2 T T = (q−q−1)T +T , if ℓ(s w) <ℓ(w) (0 ≤i ≤n). si w w siw i With notations as in (1.12-1.20) the conversion between the two presentations is given by the relations T =T ···T , if w ∈ W and w = s ···s is a reduced word, w i1 ip aff i1 ip T =xωiT−1 , for g ∈ Ω as in (1.19), gi w0wi i (1.24) xλ =T T−1, if λ= µ−ν with µ,ν ∈ P+, tµ tν T =T x−φ∨, where φ is the highest root of R, s0 sφ The Kazhdan-Lusztig basis The algebra H˜ has bases {xλT | w ∈ W,λ ∈ P} and {T xλ | w ∈ W,λ ∈ P}. w w The Kazhdan-Lusztig basis {C′ | w ∈ W˜ } is another basis of H˜ which plays an important role. It w is defined as follows. The bar involution on H˜ is the Z-linear automorphism :H˜ → H˜ given by q =q−1 and T =T−1 , for w ∈ W˜ . w w−1 For 0 ≤ i ≤ n, T = T−1 = T −(q−q−1) and the bar involution is a Z-algebra automorphism of i i i H˜. If w = s ···s is a reduced word for w then, by the definition of the Bruhat order (defined i1 ip after Lemma 1.11), T =T ···T = T ···T =T−1···T−1 w i1 ip i1 ip i1 ip =(T −(q−q−1))···(T −(q−q−1))= T + a T , i1 ip w vw v v<w X kostka-foulkes polynomials 9 with a ∈ Z[(q−q−1)]. vw Setting τ =qT and t = q2, the second relation in (1.21) i i T2 =(q−q−1)T +1 becomes τ2 =(t−1)τ +t. (1.25) i i i i The Kazhdan-Lusztig basis {C′ | w˜ ∈ W˜ } of H˜ is defined [KL] by w C′ =C′ and C′ =t−ℓ(w)/2 P τ , (1.26) w w w  yw y y≤w X   subject to Pyw ∈ Z[t21,t−12], Pww =1, and degt(Pyw) ≤ 12(ℓ(w)−ℓ(y)−1). If p =q−(ℓ(w)−ℓ(y))P (1.27) yw yw then C′ =q−ℓ(w) P qℓ(y)T = P q−(ℓ(w)−ℓ(y))T = p T , (1.28) w yw y yw y yw y y≤w y≤w y≤w X X X with p ∈ Z[q,q−1], p = 1, and p ∈ q−1Z[q−1], (1.29) yw ww yw since deg (P (q)q−(ℓ(w)−ℓ(y))) ≤ℓ(w)−ℓ(y)−1−(ℓ(w)−ℓ(y))=−1. The following proposition q yw establishes the existence and uniqueness of the C′ and the p . w yw Proposition 1.30. Let (W˜ ,≤) be a partially ordered set such that for any u,v ∈ W˜ the interval [u,v] = {z ∈ W˜ | u ≤ z ≤ v} is finite. Let M be a free Z[q,q−1]-module with basis {T | w ∈ W˜ } w and with a Z-linear involution :M → M such that q =q−1 and T =T + a T . w w vw v v<w X Then there is a unique basis {C′ |w ∈ W˜ } of M such that w (a) C′ =C′ , w w (b) C′ =T + p T , with p ∈ q−1Z[q−1] for v < w. w w vw v vw v<w X Proof. The p are determined by induction as follows. Fix v,w ∈ W with v < w. If v = w vw then p = p = 1. For the induction step assume that v < w and that p are known for all vw ww zw v <z ≤ w. The matrices A = (a ) and P = (p ) are upper triangular with 1’s on the diagonal. The vw vw equations T = T = a T = a a T and w w vw v uv vw u v u,v X X p T = C′ = C′ = p T = p a T , uw u w w vw v vw uv u u v v X X X imply AA =Id and P = AP. Then f = a p =((A−1)P) = (AP −P) = (P −P) = p −p , uz zw uw uw uw uw uw u<z≤w X 10 k. nelsen and a. ram is a known element of Z[q,q−1]; f = f qk such that f =(p −p ) =p −p =−f. k uw uw uw uw k∈Z X Hence f = −f for all k ∈ Z and p is given by p = f qk. k −k uw uw k k∈XZ<0 The finite Hecke algebra H and the group algebra of P are the subalgebras of H˜ given by H =(subalgebra of H˜ generated by T ,...,T ), and 1 n (1.31) K[P]=K-span {xλ | λ ∈ P}, where K=Z[q,q−1], respectively. The Weyl group W acts on K[P] by wf = c xwµ, for w ∈ W and f = c xµ ∈ K[P]. (1.32) µ µ µ∈P µ∈P X X Theorem 1.33. The center of the affine Hecke algebra is the ring Z(H˜) =K[P]W ={f ∈ K[P] | wf =f for all w ∈ W} of symmetric functions in K[P]. Proof. If z ∈ K[P]W thenby the fourthrelationin (1.23)T z =(s z)T +(q−q−1)(1−x−αi)−1(z− i i i s z) =zT +0, for 1 ≤i ≤n, and by the third relation in (1.23) zxλ =xλz, for all λ ∈ P. Thus z i i commutes with all the generators of H˜ and so z ∈ Z(H˜). Assume z = c xλT ∈ Z(H˜). λ,w w λ∈P,w∈W X Let m ∈ W be maximal in Bruhat order subject to c 6=0 for some γ ∈ P. If m 6=1 there exists γ,m a dominant µ∈ P such that c =0 (otherwise c 6= 0 for every dominant µ∈ P, γ+µ−mµ,m γ+µ−mµ,m which is impossible since z is a finite linear combination of xλT ). Since z ∈ Z(H˜) we have w z =x−µzxµ = c xλ−µT xµ. λ,w w λ∈P,w∈W X Repeated use of the third relation in (1.21) yields T xµ = d xνT w ν,v v ν∈P,v∈W X where d are constants such that d =1, d =0 for ν 6=wµ, and d =0 unless v ≤w. So ν,v wµ,w ν,w ν,v z = c xλT = c d xλ−µ+νT λ,w w λ,w ν,v v λ∈P,w∈W λ∈P,w∈W ν∈P,v∈W X X X and comparing the coefficients of xγT gives c = c d . Since c = 0 it m γ,m γ+µ−mµ,m mµ,m γ+µ−mµ,m follows that c =0, which is a contradiction. Hence z = c xλ ∈ K[P]. γ,m λ∈P λ The fourth relation in (1.23) gives P zT =T z = (s z)T +(q−q−1)z′ i i i i where z′ ∈ K[P]. Comparing coefficients of xλ on both sides yields z′ = 0. Hence zT = (s z)T , i i i and therefore z =s z for 1 ≤ i ≤n. So z ∈ K[P]W. i