Some applications of Rees products of posets to equivariant gamma-positivity

TL;DR

This paper extends the theory of Rees products of posets to equivariant settings, proving formulas for homology dimensions and demonstrating Schur gamma-positivity of related symmetric functions in algebraic combinatorics.

Contribution

It introduces equivariant generalizations of Rees product formulas and applies them to establish Schur gamma-positivity in symmetric functions.

Findings

Proved equivariant homology formulas for Rees products with automorphism groups.

Established Schur gamma-positivity of certain symmetric functions.

Extended combinatorial theory to include group actions on posets.

Abstract

The Rees product of partially ordered sets was introduced by Bj\"orner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the dimension of the homology of the Rees product of a graded poset $P$ with a certain $t$-analogue of the chain of the same length as $P$. Equivariant generalizations of these formulas are proven in this paper, when a group of automorphisms acts on $P$, and are applied to establish the Schur gamma-positivity of certain symmetric functions arising in algebraic and geometric combinatorics.