Speaker: Daniel Soskin (IAS)

Title: Multiplicative inequalities for totally matrices

Abstract: Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss multiplicative determinantal inequalities as a source of such functions. In the joint work with M. Gekhtman, we have shown that the set of multiplicative inequalities is finitely generated for matrices of any order. In the joint project with M. Gekhtman and Z. Greenberg we provide a list of all generators of multiplicative determinantal inequalities for the case of square matrices of order 4.
We also extend the problem of description of multiplicative inequalities in minors to inequalities in all cluster variables of finite type cluster algebras. We show that the generators of these sets of inequalities are in bijection with cluster variables, and are associated to sinks in fully sources/sinks orientation of the associated Dynkin diagram in the exchange graph of the finite type cluster algebras.