RSOS models and Jantzen-Seitz representations of Hecke algeb(2)

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

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