Combinatorics of solvable lattice models, and modular representations of Hecke algebras

Details Combinatorics of solvable lattice models, and modular representations of Hecke algebras Combinatorics of solvable lattice models, and modular representations of Hecke algebras

1997-01-18

arxiv.org/abs/q-alg/9701021v1

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX, 54 pages, including eepic and eps figures

Scientific paper

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 $\sl_n$, 2. The irreducible representations of type-A Hecke algebras at roots of unity: $H_m(\sqrt[n]{1})$. 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 $(sl(n)^_1 \times sl(n)^_1)/ sl(n)^_2$. 2. Jantzen-Seitz (JS) representations of $H_m(\sqrt[n]{1})$: irreducible representations that remain irreducible under restriction to $H_{m-1}(\sqrt[n]{1})$. Using the above relationships, we characterise the JS representations of $H_m(\sqrt[n]{1})$ and show that the generating series that count them are branching functions of affine $\sl_n$.

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