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.

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