  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 $\sl_n$.
