Quantum periodicity and Kirillov-Reshetikhin modules

Abstract

We give a proof of the periodicity of quantum $T$-systems of type $A_n\times A_{\ell }$ with certain spiral boundary conditions. Our proof is based on categorification of the $T$-system in terms of the representation theory of quantum affine algebras, more precisely on relations between classes of Kirillov-Reshetikhin modules and of evaluation modules.

13 pages; v2 : typos corrected, boundary conditions in Theorem 2.7 recalled. Accepted for publication in Progress in Mathematics (vol. in Honor of N. Reshetikhin)