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 $\sl_n$, 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 $\sl_n$.