On Equivariant flag $f$-vectors for balanced relative simplicial complexes

TL;DR

This paper investigates the properties of equivariant flag vectors in balanced relative simplicial complexes under group actions, establishing inequalities and applying results to combinatorial structures like posets and graph colorings.

Contribution

It introduces new inequalities for equivariant flag vectors of complexes satisfying Serre's condition and connects these to combinatorial applications.

Findings

Equivariant flag h-vector satisfies specific inequalities under Serre's condition.

Results apply to P-partitions of double posets.

Insights into weak colorings of mixed graphs.

Abstract

We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector satisfy several inequalities. We apply these results to the study of $P$-partitions of double posets, and weak colorings of mixed graphs.