RSOS models and Jantzen-Seitz representations of Hecke algeb

Hecke algebras at roots of unity.

RSOS models and Jantzen-Seitz representations of Hecke algebras at roots of unity.Omar Foda, Bernard Leclercy Masato Okadoz,, x Jean-Yves Thibon and Trevor A. WelshAbstract

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of 1-dimensional con guration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras Hm . We provide an explanation of this coincidence by showing how the irreducible Hm -modules which remain irreducible under restriction to Hm?1 (Jantzen-Seitz modules) can be determined b from the decomposition of a tensor product of representations of sln .

1 IntroductionThe solution of a class of\restricted-solid-on-solid" (RSOS) models by the corner transfer matrix method leads to the evaluation of weighted sums of combinatorial objects called paths 1]. The Kyoto group realized that these combinatorial sums are branching functions b of the a ne Lie algebra sl2 6], and was able to de ne similar models associated with other b a ne Lie algebras, in particular with sln 16]. b b b For the models associated with the cosets (sln )1 (sln )1=(sln )2, a di erent description of the branching functions as generating series of certain sets of partitions has been obtained in 9], and was used to derive fermionic expressions for the con guration sums. It turns out that exactly the same partitions arise in the modular representation theory of the symmetric groups: as conjectured by Jantzen and Seitz 14] and established recently by Kleshchev 20], such partitions label the irreducible representations of a symmetric group Sm over a eld of characteristic n which remain irreducible under restriction to Sm?1. The aim of this Letter is to provide an explanation of this seemingly mysterious coincidence. The rst point is to replace symmetric groups in characteristic n by Hecke algebras of type p over C at an nth root of unity. Indeed, the representation theories of Fn Sm] and A Hm ( n 1) have many formal similarities, but the consideration of Hecke algebras removes the restriction of n being a prime, which does not appear on the statisticalp mechanics side. Moreover, a connection between the representation theory of Hm ( n 1) and the level 1 b representations of the quantum a ne algebra Uq (sln ) has been pointed out in 22]. Building on a conjecture of 22], recently proved by Ariki and Grojnowski, we show that the Jantzenp Seitz type problem for Hm ( n 1) is equivalent to the decomposition via crystal bases of tensor b products of level 1 sln -modules. Using the results of 9], we can then characterize the Hecke algebra modules of Jantzen-Seitz type and explain the occurence of their partition labels in the con guration sums of RSOS-models.Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia. y Departement de Mathematiques, Universite de Caen, BP 5186, 14032 Caen Cedex, France. z Department of Mathematical Sciences, Faculty o

f Engineering Science, Osaka University, Osaka 560, Japan. x Institut Gaspard Monge, Universite de Marne-la-Vallee, 93166 Noisy-le-Grand Cedex, France.

Hecke algebras at roots of unity.

As an application, we express the generating function of the Jantzen-Seitz partitions b having a given n-core in terms of branching functions of sln . This is to be compared with a p well-known result on blocks of Hecke algebras. Indeed, the blocks of Hm ( n 1) are labelled by n-cores, and the dimension of a block is the number of n-regular partitions of m with the corresponding n-core. Using a formula rst proved in the fties by Robinson (in the symmetric group case) one can compute the generating series of the dimensions of all blocks b labelled by a given n-core, and recognize the string function of the level 1 sln -modules. Our bln other than the level 1 string function arise result shows that some branching functions of s p in a natural way in the representation theory of Hm ( n 1).

2 Characters and branching functions of sblnUsing the notion of paths, it was shown in 5] that the characters of the integrable highest b weight modules of sln may be obtained by enumerating certain coloured multipartitions. In this Letter, we are interested only in the case when the highest weight is of level one, and therefore the multipartitions are simply partitions. As is usual, we de ne a partition of m to be a sequence= ( 1; 2;:::; k ) such that 1 2 k and 1+ 2++ k= m. If i> k, we understand that= 0. The set of all partitions of m is denoted (m) and we write i=m 0

(m):

Occasionally, it will be convenient to use the exponent notation= ( a1; a2;:::; ar ) r 1 2 where here 1> 2>> r> 0, and ai> 0 speci es the multiplicity of i in . If ai< n for 1 i r then we say that the partition is n-regular. The set of all n-regular partitions of m is denoted n (m), and we de nen= m 0 n (m):

The Young diagram associated with the partition is an array of k left-adjusted rows of nodes (or boxes) in which the ith row contains i nodes. For example if= (4; 3; 1), the corresponding Young diagram is:

F (4;3;1)=

:

We will not distinguish between a partition and its Young diagram. The partition 0= ( 01; 02;:::) conjugate to is de ned such that 0j is the length of the j th column of, reading from left to right. A coloured partition is a Young diagram in which each node is lled by its colour charge c( ) given by c( )= (j? i) mod n, when is the node at the intersection of the ith row and the j th column. For example, in the case n= 3, the coloured partition= (5; 5; 4; 1; 1) appears as follows:0 1 2 0 1 2 0 1 2 0 1 2 0 1: 0 2

Hecke algebras at roots of unity.

Let mi be the number of nodes of with colour charge i. The energy of is de ned by P? E ( )= m0, and its weight by wt ( )= 0? in=01 mi i . (We use freely the standard b notations for roots, we …… 此处隐藏：6240字，全部文档内容请下载后查看。喜欢就下载吧 ……