Richard P. Stanley Seminar in Combinatorics

Speaker:  Alejandro Morales  (UMass, Amherst)

Title:  Linear relations and Lorentzian property of chromatic symmetric functions

Abstract:

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian Wachs q-analogue (q-CSF) have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet 2013.

In the first part of the talk, I will discuss a linear relation of the q-CSF for abelian paths in terms of the Garsia--Remmel q-rook and q-hit numbers originally due to Guay-Paquet and its relation to the e-positivity conjecture of Stanley--Stembridge and Shareshian--Wachs. This is joint work with Colmenarejo and Panova. In the second part of the talk, I will discuss the Newton polytope of CSFs of Dyck paths, whether it is saturated, and a conjectured Lorenztian property for these CSFs that is true for the abelian case. This is joint work with Matherne and Selover.