Melissa Sherman-Bennett: Kazhdan-Lusztig immanants and k-positive matrices

Abstract

Immanants are functions on square matrices which generalize the determinant and permanent. This talk will focus on positivity properties of Kazhdan-Lusztig (K-L) immanants, which are immanants defined using q=1 specializations of Kazhdan-Lusztig polynomials. Rhoades and Skandera (2006) showed, using work of Haiman (1993) and Stembridge (1991), that K-L immanants are nonnegative on matrices whose minors are all nonnegative. I will discuss joint work with S. Chepuri investigating what can be said about the positivity of K-L immanants if you loosen the positivity constraints on the matrices, passing to k-positive matrices (minors of size at most k x k are positive). The most straightforward K-L immanants are indexed by 123-, 2143-avoiding permutations; I'll give a sufficient condition on k such that these K-L immanants are positive on k-positive matrices. I'll also discuss a larger class of K-L immanants for which we conjecture our result holds.