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

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.

24 pages, 3 figures. arXiv admin note: text overlap with arXiv:2112.09109