Linear recurrence relations in QQ-systems via lattice points in polyhedra

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 $W_{m}^{(a)}$.Comment: 26 pages. v2: minor changes, references added. v3: Conjecture 3.6 in v2 superseded by Proposition 3.5 in v3, Section 5 added, references adde

Positivity and periodicity of QQ-systems in the WZW fusion ring

We study properties of solutions of $Q$-systems in the WZW fusion ring obtained by the Kirillov-Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level $k$ restricted $Q$-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba-Nakanishi-Suzuki on the level $k$ restricted $Q$-systems.Comment: 29 pages;Table 1 reproduced from arXiv:math/9812022 [math.QA]; v2 : no changes in main results, paper reorganized, introduction rewritten, notations polished, typos corrected, references added; v3 : typos corrected; v4 : minor change

A Proof of the KNS conjecture : DrD_r case

We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum dimension solution of the $Q$-system of type $D_r$ obtained by a certain specialization of classical characters of the Kirillov-Reshetikhin modules. To this end, we use various symmetries of quantum dimensions. As a result, we obtain an explicit formula for the positive solution of the level $k$ restricted $Q$-system of type $D_r$ which plays an important role in dilogarithm identities for conformal field theories.Comment: 13 pages, v3. published version, minor update (references added, typos corrected