2016/02/07 by Chulhee Lee, Chul-hee Lee, Lee, Chul-hee
Mathematics · #52C07 #81R50 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.CO #math.QA #math.RT #msc:52C07 #msc:81R50
paper · pdf · doi:10.48550/arxiv.1602.02347
26 pages. v2: minor changes, references added. v3: Conjecture 3.6 in v2 superseded by Proposition 3.5 in v3, Section 5 added, references added
openalex publication_date 2016/02/07 · arxiv created 2017/04/24 · arxiv updated 2017/04/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR) modules Wm(a), m∈ ℤm≥ 0 associated to a node a of the Dynkin diagram of a complex simple Lie algebra \mathfrakg satisfies a linear recurrence relation except for some cases in types E7 and E8. 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 \mathfrakg 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 G2, 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 Wm(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 Wm(a) and the lattice points of its m-th dilate carry the same crystal structure as the crystal associated with Wm(a).