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Lie Group Quantum Mathematics Seminar

Plucker inequalities and bounded Laurent monomials on the positive loci

Daniel Soskin, Lehigh University

Location:  Hill 705
Date & time: Friday, 01 December 2023 at 12:10PM - 1:10PM

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 several sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). We extend the problem in this project to the description of bounded Laurent monomials in cluster variables on the positive loci (joint work in progress with M.Gekhtman and Z.Greenberg). Another source deals with certain partial sums of Plucker relations (joint work with P.K.Vishwakarma 2023). The main tools we employed are network parametrization and Temperley-Lieb immanants.