Periodicities of T-systems and Y-systems

TL;DR

This paper formulates a periodicity conjecture for restricted T-systems related to quantum affine algebras and provides partial proofs using various mathematical methods across different Lie algebra types.

Contribution

It introduces the periodicity conjecture for restricted T-systems and proves it partially for several Lie algebra types using multiple approaches.

Findings

Conjecture formulated for restricted T-systems.

Partial proofs achieved for types A, B, C, D.

Methods include cluster algebra, determinant, and direct approaches.

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).