  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).
