Periodicities of T-systems and Y-systems

Abstract

Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or 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). §1. IntroductionThe Y-system was introduced as a system of functional relations concerning the solutions of the thermodynamic Bethe ansatz equations for the factorizable scattering theory and the solvable lattice models [Z] [KNT]. As a side remark, it may be worth mentioning at this point that "T" stands for transfer matrix and "Q" stands for quantum character [Ki2] in the original literature.As a more recent development, a connection between the Q-systems and cluster algebras is clarified by [Ked], [DiK]. Also, a connection between the T-systems (or q-characters) and cluster algebras is studied while seeking a natural categorification of cluster algebras by abelian monoidal categories [HL].Having these results as a background, we make three simple observations:(1) There are actually two classes of the Y-systems (resp. T-systems), namely, the unrestricted and restricted Y-systems (resp. T-systems). The latter is obtained by a certain reduction from the former. The periodicity property above mentioned is for the restricted Y-systems. (2) The cluster algebra structure is simpler in the T-systems than in the Y-systems. (3) The representation theory of quantum affine algebras is more directly connected with the T-systems than with the Y-systems.These observations motivate us to ask if there is a similar periodicity property for the restricted T-systems, and indeed, there is.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. We remark that the restricted T-systems are relations in certain quotients of the Grothendieck ring Rep U q (ĝ), while the T-systems studied in [HL] are relations in certain subrings of Rep U q (ĝ). Accordingly, the correspondence between the T-systems for the simply laced case and cluster algebras considered here and the one in [HL] are close but slightly different. We also note that the correspondence between the unrestricted T-systems for the simply laced case and cluster algebras is described in [DiK, Appendix B].Let us explain the outline of the paper, whose contents are roughly divided into three parts.In the first part (Section 2) we introduce the unrestricted T-systems toget...

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