Combinatorics of solvable lattice models, and modular representations of Hecke algebras
Omar Foda, Bernard Leclerc, Masato Okado, Jean-Yves Thibon, Trevor A. Welsh
Abstract
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
(
1
–
√
n
)
.
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
(
1
–
√
n
)
: irreducible representations that remain irreducible under restriction to
H
m−1
(
1
–
√
n
)
.
Using the above relationships, we characterise the JS representations of
H
m
(
1
–
√
n
)
and show that the generating series that count them are branching functions of affine $\sl_n$.