Characters and chromatic symmetric functions

TL;DR

This paper explores the combinatorial and algebraic properties of chromatic symmetric functions associated with posets, extending interpretations of coefficients and linking them to characters, permanents, and immanants.

Contribution

It generalizes combinatorial interpretations of chromatic symmetric function coefficients to all posets using character evaluations.

Findings

Extended combinatorial interpretations to all posets.

Connected coefficients to character evaluations.

Derived new insights into permanents and immanants.

Abstract

Let $P$ be a poset, $inc(P)$ its incomparability graph, and $X_{inc(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in {\em Adv. Math.}, {\bf 111} (1995) pp.~166--194. Certain conditions on $P$ imply that the expansions of $X_{inc(P)}$ in standard symmetric function bases yield coefficients which have simple combinatorial interpretations. By expressing these coefficients as character evaluations, we extend several of these interpretations to {\em all} posets $P$. Consequences include new combinatorial interpretations of the permanent and other immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{inc(P),q}$ when $P$ is a unit interval order.