Third Year Candidacy Requirement/Combinatorics Seminar: "Hook-Partition Immanant Characters from Stanley-Stembridge Characters"

Speaker: Nathan Lesnevich, Washington University in Saint Louis

Abstract: Stanley and Stembridge's well-known conjecture that the chromatic symmetric functions of incomparability graphs of 3+1-free posets are e-positive itself is equivalent (up to a Guay-Paquet reduction) to a certain character (Gamma) being a nonegative integral sum of permutation characters. This Gamma character is itself a special case of a more general construction indexed by partitions whose analogous conjecture would prove Schur positivity for monomial immanants of Jacobi-Trudi matrices.

In this talk we will give a direct combinatorial proof for a special case of  the general conjecture that the immanant-related characters can be written as nonnegative integral sums of the characters in the Stanley-Stembridge conjecture. The special case we give is for those general characters indexed by hook-partitions. Additionally, we will give some computational simplifications to the general conjecture and a characterization of the problem via Bruhat order.

Host: Martha Precup