Combinatorics Seminar: "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). We find a simple determinantal formula for K-L immanants indexed by 1324-, 2143-avoiding permutations, and use this formula to determine a sufficient condition on k such that these K-L immanants are positive on k-positive matrices.  

Host: Laura Escobar Vega