Periodicities of T-systems and Y-systems
Abstract
The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).
Cite
@article{arxiv.0812.0667,
title = {Periodicities of T-systems and Y-systems},
author = {Rei Inoue and Osamu Iyama and Atsuo Kuniba and Tomoki Nakanishi and Junji Suzuki},
journal= {arXiv preprint arXiv:0812.0667},
year = {2010}
}
Comments
83 pages with 8 figures; references added, Prop. 3.4 added, proofs for Theorem 2.12 and Prop. 7.3 shortened + minor corrections