Periodicities of T-systems and Y-systems
Mon, Jan. 12th 2009, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
The (unrestricted) T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional representations of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The T-system initially appeared as the relations among the `transfer matrices' of the fusion vertex models hierarchy. (This is why it is called so.) The T-system is also considered to be a difference analogue of 2D Toda lattice as a discrete dynamical system. The T-system admits a reduction called the restricted T-system. \par In this talk I present the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the Y-systems by Zamolodchikov, Ravanini-Tateo-Valleriani, Kuniba-Nakanishi in 90's. Then, I explain various methods concerning the periodicity such as the cluster algebra and cluster category method, the determinant method, and the direct method. \\ \\ This talk is based on the recent joint work with R. Inoue, O. Iyama, A. Kuniba, and J. Suzuki.