Equivariant log-concavity of graph matchings

TL;DR

This paper proves that the permutation representation of graph automorphisms on matchings exhibits strong equivariant log-concavity, linking combinatorial and algebraic geometric methods.

Contribution

It introduces a new equivariant log-concavity property for graph matchings and connects combinatorial maps with the hard Lefschetz theorem.

Findings

Matchings form a strongly equivariantly log-concave permutation representation.

The proof constructs equivariant injections inspired by Kratthenthaler's combinatorial map.

The approach reduces the problem to the hard Lefschetz theorem.

Abstract

For any graph, we show that the graded permutation representation of the graph automorphism group given by matchings is strongly equivariantly log-concave. The proof gives a family of equivariant injections inspired by a combinatorial map of Kratthenthaler and reduces to the hard Lefschetz theorem.