Linear recurrence relations in $Q$-systems via lattice points in polyhedra

TL;DR

This paper proves that the characters of Kirillov-Reshetikhin modules follow linear recurrence relations using lattice point formulas, and explores their combinatorial and geometric properties, especially in exceptional Lie types.

Contribution

It establishes linear recurrence relations for KR module characters via lattice points, reduces complex formulas to linear algebra problems, and introduces a conjecture linking dimensions to Ehrhart quasipolynomials.

Findings

Characters satisfy linear recurrence relations in most types.

Dimension functions are quasipolynomials in the module parameter.

New proof of lattice point formulas in type G_2.

Abstract

We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR) modules $W_{m}^{(a)}, m\in \mathbb{Z}_{m\geq 0}$ associated to a node $a$ of the Dynkin diagram of a complex simple Lie algebra $\mathfrak{g}$ satisfies a linear recurrence relation except for some cases in types $E_7$ and $E_8$. To this end we use the $Q$-system and the existing lattice point summation formula for the decomposition of KR modules, known as domino removal rules when $\mathfrak{g}$ is of classical type. As an application, we show how to reduce some unproven lattice point summation formulas in exceptional types to finite problems in linear algebra and also give a new proof of them in type $G_2$, which is the only completely proven case when KR modules have an irreducible summand with multiplicity greater than 1. We also apply the recurrence to prove that the function $\dim W_{m}^{(a)}$ is a quasipolynomial in $m$ and establish its properties. We conjecture that there exists a rational polytope such that its Ehrhart quasipolynomial in $m$ is $\dim W_{m}^{(a)}$ and the lattice points of its $m$-th dilate carry the same crystal structure as the crystal associated with $W_{m}^{(a)}$.