Báo cáo khoa học 演講訊息

Abstract: By tensor multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simple connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope. Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970’s, around 1999 Knutson and Tao invented their hive model for the type A cases, which led to the solution of the saturation conjecture. Outside type A, Berenstein and Zelevinsky’s models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model. In this talk, I will explain how to use cluster algebras, an interesting class of commutative algebras introduced by Fomin and Zelevinsky, to discover new polyhedral models for all Dynkin types. Those new models improve Berenstein-Zelevinsky’s models, or in some sense generalize the hive model. In the first hour of this 2-hour talk I will explain the main idea and result; In the second hour I will introduce the main machinary – the upper cluster algebra and its quiver with potential model. The talk is based on the preprint arXiv:1603.02521. The talk is friendly to a general audience.