Wojciech J. Gajda

Abstract:

A classical and important question in algebraic combinatorics of the symmetric functions is the following: given a symmetric function, is it Schur-positive? The celebrated Haiman's result on Schur-positivity of Macdonald polynomials relates this combinatorial problem to the geometry of Hilbert schemes of $n$ points in the plane. Given a finite family of Macdonald polynomials we transform it into a new symmetric function using the notion of cumulants, and we conjecture that it is Schur-positive. The trivial cumulant of only one variable corresponds to the Haiman's result. We discuss the combinatorial structure of Macdonald cumulants and we provide some partial results towards resolving the higher order Macdonald positivity conjecture. The talk will be elementary and assume no prior knowledge in the discussed fields.