ar X iv : q - a lg / 9 70 10 21 v 1 1 8 Ja n 19 97 Combinatorics of solvable lattice models , and modular representations of Hecke algebras

We review and motivate recently-observed relationships between exactly solvable lattice models and modular representations of Hecke algebras. Firstly, we describe how the set of n-regular partitions label both of the following classes of objects: 1. The spectrum of unrestricted solid-on-solid lattice models based on level-1 representations of the affine algebras ŝln, 2. The irreducible representations of type-A Hecke algebras at roots of unity: Hm( n √ 1). Secondly, we show that a certain subset of the n-regular partitions label both of the following classes of objects: 1. The spectrum of restricted solid-on-solid lattice models based on cosets of affine algebras C[n, 1, 1] = (ŝln)1 × (ŝln)1 / (ŝln)2, 2. Jantzen-Seitz (JS) representations of Hm( n √ 1): irreducible representations that remain irreducible under restriction to Hm−1( n √ 1). Using the above relationships, we characterise the JS representations of Hm( n √ 1) and show that the generating series that count them are branching functions of ŝln. Dept. of Mathematics, Univ. of Melbourne, Parkville, Victoria 3052, Australia. Dépt. de Mathématiques, Univ. de Caen, BP 5186, 14032 Caen cedex, France. Dept. of Math Sciences, Fac. of Eng. Science, Osaka U., Osaka 560, Japan. Inst. G. Monge, Univ. de Marne-la-Vallée, 93166 Noisy-le-Grand cedex, France.