Commuting families in Hecke and Temperley-Lieb Algebras

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1 Commuting families in Hece and Temperley-Lieb Algebras Tom Halverson Department of Mathematics Macalester College Saint Paul, MN Manuela Mazzocco Department of Mathematics University of Wisconsin Madison, WI Arun Ram Department of Mathematics University of Wisconsin Madison, WI and Department of Mathematics and Statistics University of Melbourne Parville VIC 3010 Australia Abstract We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations We show that they come from Jucys-Murphy elements in the affine Hece algebra of type A, which in turn come from the Casimir element of the quantum group U h gl n We also give the explicit specializations of these results to the finite Temperley-Lieb algebra 1 Introduction The Jucys-Murphy elements are a family of commuting elements in the group algebra of the symmetric group In characteristic 0, these elements have enough distinct eigenvalues to give a full analysis of the representation theory of the symmetric group [OV] Even in positive characteristic these elements are powerful tools [K] Similar elements are used in the Hece algebras of type A and, in a strong sense, it is these elements that control the beautiful connections between the modular representation theory of Hece algebras of type A and the Foc space representations of the affine quantum group (see [Ar] and [Gr]) Since the Temperley-Lieb algebra is a quotient of the Hece algebra of type A it inherits a commuting family of elements from the Hece algebra In order to use these elements for modular representation theory it is important to have good control of the expansion in terms AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20 1

2 of the standard basis of planar Brauer diagrams In this paper we study this question, in the more general setting of the affine Temperley-Lieb algebras Specifically, we analyze a convenient choice of a commuting family of elements in the affine Temperley-Lieb algebra Our main result, Theorem 28, is an explicit expansion of these elements in the standard basis The fact that, in the Templerley-Lieb algebra, these elements have integral coefficients is made explicit in Remar 29 The import of this result is that this commuting family can be used to attac questions in modular representation theory In Section 3 we review the Schur-Weyl duality setup of Orellana and Ram [OR] which (following the ideas in [Re]) explains how commuting families in centralizer algebras arise naturally from Casimir elements We explain, in detail, the cases that lead to commuting families in the affine Hece algebras of type A and the affine Temperley-Lieb algebra One new consequence of our analysis is an explanation of the special relation that is used in one of the Temperley- Lieb algebras of Graham and Lehrer [GL] In our context, this relation appears naturally from the Schur-Weyl duality (see Proposition 32) Using the nowledge of eigenvalues of Casimir elements we compute the eigenvalues of the commuting families in the affine Hece algebra and in the affine Temperley-Lieb algebra in the generic irreducible representations (analogues of the Specht, or Weyl, modules) The research of M Mazzocco was partially supported by a Mar Mensin Honors Research Award at the University of Wisconsin, Madison She was an undergraduate researcher participating in research and teaching initiatives partly supported by the National Science Foundation under grant DMS The research of A Ram was also partially supported by this award A significant portion of this research was done during a residency of A Ram at the Max-Planc- Institut für Mathemati (MPI) in Bonn He thans the MPI for support, hospitality, and a wonderful woring environment The research of T Halverson was partially supported by the National Science Foundation under grant DMS Affine braid groups, Hece and Temperley-Lieb algebras 21 The Affine Braid Group B The affine braid group is the group B of affine braids with strands (braids with a flagpole) The group B is presented by generators T 1, T 2,, T 1 and X ε 1, T i = i i+1 and X ε 1 = (21), with relations For 1 i define X ε 1 T 1 X ε 1 T 1 = T 1 X ε 1 T 1 X ε 1 X ε 1 T i = T i X ε 1, for i > 1, T i T j = T j T i, if i j > 1, T i T i+1 T i = T i+1 T i T i+1, if 1 i 2, (22) X ε i = T i 1 T i 2 T 2 T 1 X ε 1 T 1 T 2 T i 1 = i (23) 2

3 By drawing pictures of the corresponding affine braids it is easy to chec that X ε i X ε j = X ε j X ε i, for 1 i, j, (24) so that the elements X ε 1,, X ε are a commuting family for B Thus X = X ε i 1 i is an abelian subgroup of B The free abelian group generated by ε 1,, ε is Z and X = {X λ λ Z } where X λ = (X ε 1 ) λ 1 (X ε 2 ) λ2 (X ε ) λ, (25) for λ = λ 1 ε λ ε in Z Remar 21 An alternate presentation of B can be given using the generators T 0, T 1,, T 1 and τ where τ = X ε 1 T1 1 T 1 1 = and T 0 = τ 1 T 1 τ = Remar 22 The affine braid group B is the affine braid group of type GL The affine braid groups of type SL and P GL are the subgroup B Q = T 0, T 1,, T 1 and the quotient BP = B τ, respectively Then τ = X ε 1 X ε2 X ε is a central element of B, τt i τ 1 = T i+1 (where the indices are taen mod n), and τx ε i τ 1 = X ε i+1 and Z( B ) = τ, B = τ B Q, BP = τ B Q In B we have τ = Z, and τ B P Z Z/Z so that τ = Z/Z is defined to be the image of τ under the homomorphism 22 The Temperley-Lieb algebra TL (n) A Temperley-Lieb diagram on dots is a graph with dots in the top row, dots in the bottom row, and edges pairing the dots such that the graph is planar (without edge crossings) For example, d 1 = and d 2 = are Temperley-Lieb diagrams on 7 dots The composition d 1 d 2 of two diagrams d 1, d 2 T is the diagram obtained by placing d 1 above d 2 and identifying the bottom vertices of d 1 with the top dots of d 2 removing any connected components that live entirely in the middle row If T is the set of Temperley-Lieb diagrams on dots then the Temperley-Lieb algebra TL (n) is the associative algebra with basis T, TL (n) = span{d T } with multiplication defined by d 1 d 2 = n l (d 1 d 2 ), where l is the number of blocs removed from the middle row when constructing the composition d 1 d 2 and n is a fixed element of the base ring For example, using the diagrams d 1 and d 2 above, we have d 1 d 2 = = n 3

4 The algebra TL (n) is presented by generators e i = i i+1, 1 i 1, (26) and relations e 2 i = ne i, e i e i±1 e i = e i, and e i e j = e j e i, if i j > 1 (27) (see [GHJ, Lemma 284]) Remar 23 In the definition of the Temperley-Lieb algebra, and for other algebras defined in this paper, the base ring could be any one of several useful rings (eg C, C(q), C[[h]], Z[q, q 1 ], Z[n] or localizations of these at special primes) The most useful approach is to view the results of computations as valid over any ring R with n, q, h R such that the formulas mae sense 23 The Surjection H (q) TL (n) The affine Hece algebra H is the quotient of the group algebra of the affine braid group C B by the relations T 2 i = (q q 1 )T i + 1, so that C B H (28) is a surjective homomorphism (q is a fixed element of the base ring) The affine Hece algebra H is the affine Hece algebra of type GL The affine Hece algebras of types SL and P GL are, respectively, the quotients H Q and H P of the group algebras of BQ and B P (see Remar 22) by the relations (28) The Iwahori-Hece algebra is the subalgebra H of H generated by T 1,, T 1 In the Iwahori-Hece algebra H, define e i = q T i, for i = 1, 2,, 1 (29) Direct calculations show that e 2 i = (q + q 1 )e i and that e 1 e 2 e 1 = e 1 and e 2 e 1 e 2 = e 2 if and only if q 3 q 2 T 1 q 2 T 2 + qt 1 T 2 + qt 2 T 1 T 1 T 2 T 1 = 0 (210) Thus, setting n = [2] = q + q 1, there are surjective algebra homomorphisms given by ψ : H (q) H (q) TL (n) X ε T i T i q e i (211) The ernel of ψ is generated by the element on the left hand side of equation (210) In the notation of Theorem 41, the representations of H correspond to the case when µ = Writing H λ/ as H λ, the element from (210) acts as 0 on the irreducible Iwahori-Hece algebra modules H 3 and H 3, and (up to a scalar multiple) it is a projection onto H 3 Remar 24 There is an alternative surjective homomorphism that instead sends T i e i q 1 This alternative surjection has ernel generated by q 3 + q 2 T 1 + q 2 T 2 + q 1 T 1 T 2 + q 1 T 2 T 1 + T 2 T 1 T 2 This element is 0 on H 3 and H 3, and (up to a scalar multiple) it is a projection onto H 3 4

5 Remar 25 A priori, there are two different inds of integrality for the Temperley-Lieb algebra: coefficients in Z[n] or coefficients in Z[q, q 1 ] (in terms of the basis of Temperley-Lieb diagrams) The relation between these is as follows If [2] = q + q 1 = n then q = 1 2 (n + n 2 4), q 1 = 1 2 (n (n 2 4), since q 2 nq + 1 = 0 Then [] = q q q q 1 = (+1)/2 m=1 so that [] is a polynomial in n The polynomials ( ) n 2m+1 (n 2 4) m 1 2m 1 n = (q + q 1 ) and {} = q + q and [] = q q q q 1, all form bases of the ring C[(q + q 1 )] The transition matrix B between the [] and the {} is triangular (with 1s on the diagonal) and the transition matrix C between the n and the {} is also triangular (the non zero entries are binomial coefficients) Hence, the transition matrix BC 1 between [] and n has integer entries and so [] is, in fact, a polynomial in n with integer coefficients 24 Affine Temperley-Lieb algebras The affine Temperley-Lieb algebra T a is the diagram algebra generated by i e 0 =, e i = (1 i 1), and τ = The generators of T a satisfy e2 i = ne i, e i e i±1 e i = e i, τe i τ 1 = e i+1 (where the indices are taen mod ) and τ 2 e 1 = = = = e 1 e 2 e 1 (212) (see [GL, 415(iv)]) In T a, we let Xε 1 = T1 1 T2 1 T 1 1 τ 1 (see Remar 21) Graham and Lehrer [GL, 43] define four slightly different affine Temperley-Lieb algebras, the diagram algebra T a and the algebras defined as follows: a Type GL : TL is H with the relation (210), Type SL : TL a is H Q with the relation (210), a Type P GL : TL is H P with the relation (210) For each invertible element α in the base ring there is a surjective homomorphism H TL a T a τ τ ατ T i q e i q e i (213) and every irreducible representation of TL a factors through one of these homomorphisms (see [GL, Prop 414(v)]) In Proposition 32 we shall see that these homomorphisms arise naturally in the Schur-Weyl duality setting 5

6 25 A commuting family in the affine Temperley-Lieb algebra View the elements X ε i in the affine Temperley-Lieb algebra TL a via the surjective algebra homomorphism of (213) Define (q q 1 )m 1 = q 1 X ε 1 and (q q 1 )m i = q i 2 (X ε i q 2 X ε i 1 ), (214) for i = 2, 3,, Since X ε i X ε j = X ε j X ε i for all 1 i, j, and the m i are linear combinations of the X ε i, Proposition 26 For 1 i, m i m j = m j m i in TL a, for all 1 i, j (a) X ε i = q (i 2) (q q 1 )(m i + q 1 m i 1 + q 2 m i q (i 1) m 1 ), (b) X ε X ε i = q (i 2) (q q 1 )(m i + [2]m i [i]m 1 ) Proof Rewrite (214) as and use induction, X ε i = q (i 2) (q q 1 )m i + q 2 X ε i 1 X ε i = q (i 2) (q q 1 )m i + q 2( q (i 1 2) (q q 1 )(m i 1 + q 1 m i q (i 2) m 1 ) ), to obtain the formula for X ε i i X ε j = j=1 in (a) Summing the formula in (a) over i gives ( ) i j 2 q (j 2) (q q 1 ) q l m j l = q (i 2) (q q 1 ) j=1 and, thus, formula (c) follows from i j 1 q i j l m j l = j=1 l=0 i j=1 r=1 l=0 j q i j (j r) m r = i r=1 j=r i q i+r 2j m r = i j 1 q i j l m j l j=1 l=0 i [i r + 1]m r r=1 The following Lemma is a transfer of the recursion X ε i following are the base cases of Lemma 27 = T i 1 X ε i 1 T i 1 to the m i The m 1 = q 1 q q 1 X ε 1 and m 2 = x q q 1 e 1 (e 1 m 1 + m 1 e 1 ) Lemma 27 Let x be the constant defined by the equation e 1 X ε 1 e 1 = xe 1 For 2 i, m i = qi 2 x q q 1 e i 2 i 1 (e i 1 m i 1 + m i 1 e i 1 ) ([i l] [i l 2])m l e i 1 l=1 6

7 Proof From (23) and (29) we have X ε i = (q 1 e i 1 )X ε i 1 (q 1 e i 1 ) Substituting this into the definition of m i gives (q q 1 )m i = q i 2 (X ε i q 2 X ε i 1 ) = q i 2 (q 1 e i 1 )X ε i 1 (q 1 e i 1 ) q i 4 X ε i 1 = q i 2 e i 1 X ε i 1 e i 1 q i 3 (e i 1 X ε i 1 + X ε i 1 e i 1 ) Use Proposition 26 (a) to substitute for X ε i 1, (q q 1 )m i = (q q 1 )q (m+i 3) (q i 2 e i 1 m i 1 e i 1 q i 3 (e i 1 m i 1 + m i 1 e i 1 )) which gives + (q q 1 )q (i 3) (q 1 m i q (i 2) m 1 )(q i 2 e 2 i 1 2q i 3 e i 1 ) = (q q 1 ) ( qe i 1 m i 1 e i 1 (e i 1 m i 1 + m i 1 e i 1 ) ) + (q q 1 )(m i q (i 3) m 1 )(q + q 1 2q 1 )e i 1 ( ) = (q q 1 qei 1 m ) i 1 e i 1 (e i 1 m i 1 + m i 1 e i 1 ) +(q q 1 )(m i q (i 3), m 1 )e i 1 m i = qe i 1 m i 1 e i 1 (e i 1 m i 1 + m i 1 e i 1 ) +(q q 1 )(m i 2 + q 1 m i 3 + q 2 m i q (i 3) m 1 )e i 1 (215) Using induction, substitute for the first m i 1 in this equation to get i 2 m i = (e i 1 m i 1 + m i 1 e i 1 ) + (q q 1 ) q (i 2 l) m l e i 1 + q ( q i 3 x q q 1 e i 1 2m i 2 e i 1 l=1 i 3 ) ([i l 1] [i l 3])m l e i 1 = qi 2 x q q 1 e i 2 i 1 (e i 1 m i 1 + m i 1 e i 1 ) ([i l] [i l 2])m l e i 1 l=1 l=2 26 Diagram Representation of Murphy Elements Label the vertices from left to right in the top row of a diagram d T with 1, 2,,, and label the corresponding vertices in the bottom row with 1, 2,, The cycle type of a diagram d T is the set partition τ(d) of {1, 2,, } obtained from d by setting 1 = 1, 2 = 2,, = If τ(d) is a set partition of the form {{1, 2,, γ 1 }, {γ 1 + 1, µ 1 + 2,, γ 1 + γ + 2},, {γ γ l 1 + 1,, }}, where (γ 1,, γ l ) is a composition of, then we simplify notation by writing τ(d) = (γ 1,, γ l ) For example d = has τ(d) = (5, 3, 4) There are diagrams whose cycle type cannot be written as a composition (for example d = has cycle type {{1, 4}, {2, 3}}) but all of the diagrams needed here have cycle types that are compositions 7

8 If γ = (γ 1,, γ l ) is a composition of define d γ = τ(d)=γ d (216) as the sum of the Temperley-Lieb diagrams on dots with cycle type γ Define d γ be the sum of diagrams obtained from the summands of d γ by wrapping the first edge in each row around the pole, with the orientation coming from X ε 1 as shown in the examples below When the first edge in the top row connects to the first vertex in the bottom row only one new diagram is produced, otherwise there are two For example, in TL a 4, d 31 = d 13 = d 22 = + d 31 = d 13 = + d 22 = + View d γ and d γ as elements of T L a by setting d γ = d γ1 i, if γ is a composition of i with i < With this notation, expanding the first few m i in terms of diagrams gives (q q 1 )m 1 = q 1 d 1, (q q 1 )m 2 = xd 2 q 1 d 2, (q q 1 )m 3 = qxd 1,2 q 1 [2]d 1,2 xd 3 + q 1 d 3, (q q 1 )m 4 = q 2 xd 1 2,2 q 1 ([3] [1])d 1 2,2 x[2]d 2,2 + q 1 [2]d 2,2 qxd 1,3 + q 1 [2]d 1,3 + xd 4 q 1 d 4, (q q 1 )m 5 = q 3 xd 1 3,2 q 1 ([4] [2])d 1 3,2 q2 xd 1 2,3 + q 1 ([3] [1])d 1 2,3 + qxd 1,4 q 1 [2]d 1,4 qx[2]d 1,2,2 + q 1 [2] 2 d 1,2,2 + qx[2]d 2,3 q 1 [2]d 2,3 x([3] [1])d 2,1,2 + q 1 ([3] [1])d 2,1,2 + x[2]d 3,2 q 1 [2]d 3,2 xd 5 + q 1 d 5, where, as in Lemma 27, x is the constant defined by the equation e 1 X ε 1 e 1 = xe 1 Theorem 28 Let x be the constant defined by the equation e 1 X ε 1 e 1 = xe 1 Then (q q 1 )m 1 = q 1 d 1, (q q 1 )m 2 = xd 2 q 1 d 2 and, for i 2, m i = (m i ) γ d γ + (m i ) γd γ, compositions γ where the sum is over all compositions γ = 1 b 1 r 1 1 b 2 r 2 1 b lr l of i with r l > 1, and (m i ) γ = ( 1) γ l(γ) 1 q b 1 x q q 1 (m i ) γ = ( 1) γ l(γ) q 1 with l(γ) = l + b b l b j 0,j>1 ([b j + 2] [b j ]), q q 1 ([b 1 + 1] [b 1 1]) b j 0,j>1 and ([b j + 2] [b j ]), 8

9 Proof From our computations above, m 1 = Ad 1 and m 2 = Bd 2 Ad 2, where A = q 1 q q 1 and B = Let m 1 = Ad 1 For i > 2 the recursion in Lemma 27 gives x q q 1 i 2 m i = q i 2 Be i 1 (e i 1 m i 1 + m i 1 e i 1 ) ([i l] [i l 2])m l e i 1 = q i 2 Bd 1 i 2,2 ([i 1] [i 3])Ad 1 i 2,2 ( (m i 1 ) γ rd γ,r+1 + (m i 1 ) ) γ r d γ,r+1 i 2 + ([i l] [i l 2]) ( (m l ) γ d γ 1 i 2 l 2 + (m l ) γ d γ 1 2) i 2 l l=2 So if d has cycle type γ = 1 b 1 r 1 1 b 2 r 2 1 b lr l with r l > 0, then (a) Each part of size r (r > 1) contributes ( 1) r 1 to the coefficient Thus, there is a total contribution of ( 1) γ l(γ) from these parts (b) Each inner 1 b (b 0) contributes a factor of [b + 2] [b] to the coefficient (c) The first 1 b (b > 0) contributes a q b B in a nonstarred class, (c ) The first 1 b (b = 0) contributes a B in a nonstarred class, which is the same as case (c) with b = 0 (d) The first 1 b (b > 0) contributes a ([b + 1] [b 1])A in a starred class (d ) The first 1 b (b = 0) contributes an A in a starred class, which is the same as case (d) with b = 0 assuming [ 1] = 0 l=1 Remar 29 To view m 1,, m in the (nonaffine) Temperley-Lieb algebra TL (n) (via (211)) let X ε 1 = 1 so that x = q + q 1 If b 1 > 1 then d γ = d γ and if b 1 = 0 then d γ = 2d γ In both cases the coefficients in Theorem 28 specialize to (m i ) γ + (m i ) γ = ( 1) γ l(γ) 1 [b 1 + 1] ([b j + 2] [b j ]) b j 0,j>1 and ( (mi ) γ + (m i ) ) γ dγ, m i = γ where the sum is over compositions γ = 1 b 1 r 1 1 b 2 r 2 1 b lr l of i with r l > 1 examples are The first few m 1 = q 1 q q 1 = q 1 q q 1 d 1, m 2 = e 2 = d 2, m 3 = [2]d 12 d 3, m 4 = [3]d 1 2,2 [2]d 2,2 [2]d 1,3 + d 4, m 5 = [4]d 1 3,2 [3]d 1 2,3 + [2]d 1,4 [2] 2 d 1,2,2 + [2]d 2,3 ([3] [1])d 2,1,2 + [2]d 3,2 d 5 9

10 3 Schur functors 31 R-matrices and quantum Casimir Elements Let U h g be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra g We shall use the notations and conventions for U h g as in [LR] and [OR] There is an invertible element R = a i b i in (a suitable completion of) U h g U h g such that, for two U h g modules M and N, the map Ř MN : M N N M m n b i n a i m M N N M is a U h g module isomorphism In order to be consistent with the graphical calculus these operators should be written on the right The element R satisfies quasitriangularity relations (see [LR, (21-23)]) which imply that, for U h g modules M, N, P and a U h g module isomorphism τ M : M M, M N M N P M (N P ) = (N P ) M N P M Ř M,N P = (ŘMN id P )(id N ŘMP ) which, together, imply the braid relation M N τ M τ M = N M N M Ř MN (id N τ M ) = (τ M id N )ŘMN, M N P P N M = M N P (M N) P = P (M N) P M N Ř M N,P = (id M ŘNP )(ŘMP id N ), M N P P N M (ŘMN id P )(id N ŘMP )(ŘNP id M ) = (id M ŘNP )(ŘMP id N )(id P ŘMN) Let ρ be such that ρ, α i = 1 for all simple roots α i As explained in [LR, (214)] and [Dr], there is a quantum Casimir element e hρ u in the center of U h g and, for a U h g module M we define a U h g module isomorphism and the elements C M satisfy C M : M M m (e hρ u)m M C M N = (ŘMNŘNM) 1 (C M C N ), and C M = q λ,λ+2ρ id M (31) M C M 10

11 if M is a U h g module generated by a highest weight vector v + of weight λ (see [LR, Prop 214] or [Dr, Prop 32]) Note that λ, λ + 2ρ = λ + ρ, λ + ρ ρ, ρ are the eigenvalues of the classical Casimir operator [Dx, 785] From the relation (31) it follows that if M = L(µ), N = L(ν) are finite dimensional irreducible U h g modules then ŘMNŘNM acts on the λ isotypic component L(λ) cλ µν of the decomposition L(µ) L(ν) = λ L(λ) cλ µν by the constant q λ,λ+2ρ µ,µ+2ρ ν,ν+2ρ (32) 32 The B module M V Let U h g be a quantum group and let M and V be U-modules such that the operators ŘMV, Ř V M and ŘV V are well defined Define Ři, 1 i 1, and Ř2 0 in End U(M V ) by Ř i = id M id (i 1) V ŘV V id ( i 1) V and Ř 2 0 = (ŘMV ŘV M) id ( 1) V Then the braid relations Ř i Ř i+1 Ř i = = = Ři+1ŘiŘi+1 and Ř 2 0Ř1Ř2 0Ř1 = imply that there is a well defined map = = = = Ř1Ř2 0Ř1Ř2 0 Φ: B End U (M V ) T i Ř i, 1 i 1, X ε 1 Ř 2 0, (33) which maes M V into a right B module By (31) and the fact that Φ(X ε i ) = ŘM V (i 1),V ŘV,M V (i 1) = i (34) the eigenvalues of Φ(X ε i ) are related to the eigenvalues of the Casimir The Schur functors are the functors FV λ : {U-modules} { B -modules} M Hom U (M(λ), M V (35) ) where Hom U (M(λ), M V ) is the vector space of highest weight vectors of weight λ in M V 11

12 33 The quantum group U h gl n Although the Lie algebra gl n is reductive, not semisimple, all of the general setup of Sections 31 and 32 can be applied without change The simple roots are α i = ε i ε i+1, 1 i n 1, and ρ = (n 1)ε 1 + (n 2)ε ε n 1 (36) The dominant integral weights of gl n are λ = λ 1 ε λ n ε n, where λ 1 λ 2 λ n, and λ 1,, λ n Z and these index the simple finite dimensional U h gl n -modules L(λ) A partition with n rows is a dominant integral weight with λ n 0 If λ n < 0 and denotes the 1-dimensional determinant representation of U h gl n then (see [FH, 155]) L(λ) = λn L(λ + ( λ n,, λ n )) with λ + ( λ n,, λ n ) a partition (37) Identify each partition λ with the configuration of boxes which has λ i boxes in row i For example, λ = = 5ε 1 + 5ε 2 + 3ε 3 + 3ε 4 + ε 5 + ε 6 (38) If µ and λ are partitions with µ λ (as collections of boxes) then the sew shape is the collection of boxes of λ that are not in µ For example, if λ is as in (38) and µ = then = If b is a box in position (i, j) of λ the content of b is c(b) = j i = the diagonal number of b, so that (39) 4 5 are the contents of the boxes for the partition in (38) If ν is a partition and V = L( ) then L(ν) V = λ ν + L(λ), (310) where the sum is over all partitions λ with n rows that are obtained by adding a box to ν [Mac, I App A (84) and I (516)], Hence, the U h gl n -module decompositions of L(µ) V = λ L(λ) H, Z 0, (311) 12

13 are encoded by the graph Ĥ/µ with vertices on level : edges: labels on edges: {sew shapes with boxes} γ/µ, if γ is obtained from λ by adding a box content of the added box (312) For example if µ = (3, 3, 3, 2) =, then the first 4 rows of Ĥ/µ are (313) The following result is well nown (see [Ji] or [LR, (44)]) Proposition 31 If U = U h gl n and V = L(ε 1 ) = L( ) is the n-dimensional standard representation of gl n then the map Φ of (33) factors through the surjective homomorphism (28) to give a representation of the affine Hece algebra For a sew shape with boxes identify paths from µ to in Ĥ/µ with standard tableaux of shape by filling the boxes, successively, with 1, 2,, as they appear In the example graph Ĥ/µ above 1 corresponds to the path

14 34 The quantum group U h gl 2 In the case when n = 2, U = U h gl 2 and the partitions which appear in (311) and in the graph Ĥ /µ all have 2 rows For example if µ = (42) = then the first few rows of Ĥ/µ are = 0 : = 1 : = 2 : = 3 : = 4 : (314) and this is the graph which describes the decompositions in (311) Proposition 32 If U = U h gl 2, M = L(µ) where µ is a partition of m with 2 rows, and V = L(ε 1 ) = L( ) is the 2-dimensional standard representation of gl 2 then the map Φ of (33) factors through the surjective homomorphism of (213) with α 2 = q 2m 1 to give a representation of the affine Temperley-Lieb algebra T a Proof The proof that the ernel of Φ contains the element (212) is exactly as in the proof of [OR, Thm 61(c)]: The element e 1 in T2 a acts on V 2 as (q + q 1 ) pr where pr is the unique U h g-invariant projection onto L( ) in V 2 Using e 1 T 1 = q 1 e 1 and the pictorial equalities = = q 1 it follows that Φ 2 (e 1 X ε 1 T 1 X ε 1 ) acts as (q +q 1 )q 1 ŘL( ),L(µ)ŘL(µ),L( ) (id L(µ) pr) By (31), this is equal to q 1 (C L(µ) C L( ) )C 1 L(µ) L( ) Φ 2(id L(µ) e 1 ) = q 1 q µ,µ+2ρ q ε 1+ε 2,ε 1 +ε 2 +2ρ C 1 L(µ+ε 1 +ε 2 ) Φ 2(id L(µ) e 1 ) and the coefficient q 1 q µ,µ+2ρ q ε 1+ε 2,ε 1 +ε 2 +2ρ C 1 L(µ+ε 1 +ε 2 ) simplifies to q 1 q µ,µ+2ρ q ε 1+ε 2,ε 1 +ε 2 +2ρ q µ+ε 1+ε 2,µ+ε 1 +ε 2 +2ρ = q 1 q 2(µ 1+µ 2 ) = q 2m 1, where m = µ 1 + µ 2 = µ 14

15 35 The quantum group U h sl 2 The restriction of an irreducible representation L(λ) of U h gl n to U h sl n is irreducible and all irreducible representations of U h sl n are obtained in this fashion Since the determinant representation is trivial as an U h sl n module it follows from (37) that the irreducible representations L sln (λ) of U h sl n are indexed by partitions λ = (λ 1,, λ n ) with λ n = 0 Hence, the graph which describes the U h sl 2 -module decompositions of L(µ) V = λ L(λ) T, Z 0 (315) is exactly the same as the graph for U h gl 2 except with all columns of length 2 removed from the partitions More precisely, the decompositions are encoded by the graph ˆT /µ with vertices on level : {µ 1 µ 2 +, µ 1 µ 2 + 2,, µ 1 µ 2 } Z 0 edges: l l ± 1 For example if m = 7 and µ 1 µ 2 = 3 then the first few rows of ˆT /µ are = 0 : (316) = 1 : = 2 : = 3 : = 4 : (317) Paths in (317) correspond to paths in (314) which correspond to standard tableaux T of shape 4 Eigenvalues 41 Eigenvalues of the X ε i in the affine Hece algebra Recall, from (28), that the affine Hece algebra H is the quotient of the group algebra of the affine braid group C B by the relations T 2 i = (q q 1 )T i + 1 (41) As observed in Proposition 32 the map Φ in (33) maes the module L(µ) V in (311) into an H module Thus the vector spaces in (311) are the H -modules given by H H = F λ V (L(µ)), where F λ V The following theorem is well nown (see, for example, [Ch]) 15 are the Schur functors of (35)

16 Theorem 41 (a) The X ε i, 1 i, mutually commute in the affine Hece algebra H (b) The eigenvalues of X ε i are given by the graph Ĥ/µ of (313) in the sense that if for Ĥ/µ, then Ĥ /µ = {sew shapes with boxes} and Ĥ = {standard tableaux T of shape } Ĥ /µ is an index set for the simple H modules H appearing in L(µ) V, and H has a basis {v T T Ĥ } with X ε i v T = q 2c(T (i)) v T, where c(t (i)) is the content of box i of T (c) κ = X ε1 X ε is a central element of H and κ acts on H by the constant q 2 b c(b) Proof (a) is a restatement of (24) (b) Since the H action and the U h gl n action commute on L(µ) V it follows that the decomposition in (311) is a decomposition as (U h gl n, H ) bimodules, where the H are some H -modules Comparing the L(λ) components on each side of L(λ) H l = L(µ) V l = L(µ) V (l 1) V = ( ν/µ L(ν) H l 1) V gives λ = λ λ/ν= L(ν) H l H ν/µ l 1 = λ = λ/ν= ν (L(λ) ( λ/ν= H ν/µ ) ) l 1 H ν/µ l 1, (42) for any l Z 0 and sew shape with l boxes Iterate (42) (with l =, 1, ) to produce a decomposition H = H 1 T, T Ĥ where the summands H 1 T are 1-dimensional vector spaces This determines a basis (unique up to multiplication of the basis vectors by constants) {v T T Ĥ } of H which respects the decompositions in (42) for 1 l Combining (31), (32) and (34) gives that X ε i acts on the L(λ) component of the decomposition (310) by the constant q λ,λ+2ρ ν,ν+2ρ ε 1,ε 1 +2ρ = q 2c(λ/ν) 16

17 since if λ = ν + ε j, so that λ is the same as ν except with an additional box in row j, then ν λ, λ/ν = and Hence, λ, λ + 2ρ ν, ν + 2ρ ε 1, ε 1 + 2ρ = ν + ε j, ν + ε j + 2ρ ν, ν + 2ρ (1 + 2(n 1)) = 2ν j + ε j, ε j + 2ρ 2n + 1 = 2ν j + (1 + 2(n j)) 2n + 1 = 2(ν j + 1) 2j = 2c(λ/ν) X ε i v T = q 2c(T (i)) v T, for 1 i, where T (i) is the box containing i in T The remainder of the proof, including the simplicity of the H -modules H, is accomplished as in [R, Thm 41] (c) The element X ε1 X ε is central in B (it is a full twist) and hence its image is central in H The constant describing its action on H follows from the formula X ε i v T = q 2c(T (i)) v T 42 Eigenvalues of the m i in T a Let m 1, m 2,, m be the commuting family in the affine Temperley-Lieb algebra as defined in (214) We will use the results of Theorem 41 to determine the eigenvalues of the m i in the (generically) irreducible representations Theorem 42 (a) The elements m i, 1 i, mutually commute in T a (b) The eigenvalues of the elements m i are given by the graph ˆT /µ of (317) in the sense that if the set of vertices on level is ˆT /µ for = {µ 1 µ 2 +, µ 1 µ 2 + 2,, µ 1 µ 2 } Z 0, and ˆT = {paths p = (µ = p (0) p (1) p () = ) to in ˆT /µ }, ˆT /µ then ˆT /µ is an index set for the simple T a modules T appearing in L(µ) V, and with m i v p = T has a basis {v p p ˆT } { ±[p (i 1) + 1]v p, if p (i 1) ± 1 = p (i 2) = p (i), 0, otherwise where p (i) is the partition (a single part in this case) on level i of the path p (c) κ = m + [2]m []m 1 is a central element of T a constant and κ acts on T by the [] q (µ 1+µ 2 ) (µ 1 µ 2 )+1 q q 1 + q (µ 1+µ 2 ) ([λ 1 λ 2 + 2] + [λ 1 λ 2 + 4] + + [µ 1 µ 2 + ]) 17

18 Proof (a) The elements X ε i commute with one another in the affine Hece algebra (see (24) and the m j are by definition linear combinations of the X ε i (see 214), so they commute (b) Let p be a path to in ˆT µ and let T be the corresponding standard tableau on 2 rows If p (i) = p (i 1) 1 = p (i 2) 2 or If p (i) = p (i 1) + 1 = p (i 2) + 2 then c(t (i 1)) = c(t (i)) 1 and, from (214) and Theorem 41(b), m i v T = q i 2 q 2c(T (i)) q 2 q 2c(T (i 1)) q q 1 v T = q i 2 q 2c(T (i)) q 2 q 2c(T (i))+2 q q 1 v T = 0 If p (i) = p (i 2) = p (i 1) 1 with T (i 1) = (a, b) then c(t (i)) = a and c(t (i 1)) = b 2 and ( m i v T = q i 2 q 2a q 2 q 2b+4 q q 1 v T = q i q (a+b+1) q (a b+1) q (a b+1)) q q 1 = q m [a b + 1]v T, where m = µ = a + b i + 1 If p (i) = p (i 2) = p (i 1) + 1 with T (i 1) = (a, b) then c(t (i 1)) = a 1 and c(t (i)) = b 1 and ( m i v T = q i 2 q 2b+2 q 2 q 2a+2 q q 1 v T = q i q (a+b+1) q (a b+1) q (a b+1)) q q 1 = q m [a b + 1]v T, where m = µ = a + b i + 1 (c) Let = The identity q λ 2 1 λ 1+λ 2 2 q 2c(b) = [λ 1 + λ 2 2i](q q 1 ) + []q µ 2 µ 1 +1, i=µ 2 b is best visible in an example: With λ = (10, 6) and µ = (4, 2), q 16 2 ( q 8 +q 10 +q 12 +q 14 +q 16 +q q 2 +q 4 +q 6 +q 8 = q6 +q 4 +q 2 +q 0 +q 2 +q q 12 +q 10 +q 8 +q 6 ( = (q 12 q 12 ) +(q 10 q 10 ) +(q 8 q 8 ) +(q 6 q 6 ) ( q 4 +q + 2 +q 0 +q 2 +q 4 +q 6 ) q 12 q 10 +q 8 +q 6 = ( 6 1 ) q 16 2i q (16 2i) + [10]q i=2 ) ) Then Proposition 26 says X ε X ε = q ( 2) (q q 1 )(m + [2]m []m 1 ), 18

19 and so m + [2]m []m 1 acts on T by the constant (q q 1 ) 1 q 2 q 2c(b) = (q q 1 ) 1 q (µ 1+µ 2 ) q λ 1+λ 2 2 b = (q q 1 ) 1 q (µ 1+µ 2 ) []q µ 2 µ q m p(0) +1 λ 2 1 = [] q q 1 + q m [m + 2i] i=µ 2 = [] q m p(0) +1 q q 1 λ 2 1 b q 2c(b) [λ 1 + λ 2 2i](q q 1 ) i=µ 2 + q m ([p () + 2] + [p () + 4] + + [p (0) + 2] + [p (0) + ]), since µ 1 + µ 2 = m, µ 1 µ 2 = p (0), λ 1 + λ 2 = m + and λ 1 λ 2 = p () References [Ar] S Arii, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series 26 Amer Math Soc, Providence, RI, 2002 ISBN: , MR (2004b:17022) [Ch] I Cheredni, A new interpretation of Gel fand-tzetlin bases, Due Math J 54 no 2 (1987) [Dr] VG Drinfel d, Almost cocommutative Hopf algebras, Leningrad Math J 1 (1990), [Dx] [FH] [GHJ] [GL] [Gr] [Ji] [K] J Dixmier, Enveloping algebras, Graduate Studies in Mathematics 11 Amer Math Soc, Providence, RI, 1996 ISBN: , MR (97c:17010) W Fulton and J Harris, Representation theory A first course, Graduate Texts in Mathematics 129 Springer-Verlag, New Yor 1991, ISBN: ; , MR (93a:20069) F Goodman, P de la Harpe, and V Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications 14 Springer-Verlag, New Yor, 1989 J Graham and G Lehrer, Cellular and diagram algebras in representation theory, in Representation theory of algebraic groups and quantum groups, Adv Stud Pure Math 40 Math Soc Japan, Toyo (2004) MR (2005i:20005) I Grojnowsi, Affine sl p controls the representation theory of the symmetric group and related Hece algebras, arxiv: mathrt/ M Jimbo, A q-analogue of U(gl(N + 1)), Hece algebra, and the Yang-Baxter equation, Lett Math Phys 11 (1986), no 3, , MR (87:17011) A Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press, Cambridge, 2005 ISBN: , MR (2007b:20022) 19

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