Induced equators in flag spheres

TL;DR

This paper introduces a combinatorial approach to a strengthened version of Gal's conjecture involving gamma-vectors of flag homology spheres, providing partial evidence and proving a related nonlinear inequality for boundary complexes of flag polytopes.

Contribution

It proposes a new combinatorial method to strengthen Gal's conjecture and proves a nonlinear inequality for flag polytope boundary complexes.

Findings

Partial evidence supporting the conjecture.

Proved a nonlinear inequality for boundary complexes of flag polytopes.

Establishes a new connection between gamma-vectors and h-vectors in flag spheres.

Abstract

We propose a combinatorial approach to the following strengthening of Gal's conjecture: $\gamma(\Delta)\ge \gamma(E)$ coefficientwise, where $\Delta$ is a flag homology sphere and $E\subseteq \Delta$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(\Delta) h_i(\Delta) \ge (d-i+1)h_{i-1}(\Delta) + (i+1) h_{i+1}(\Delta)$ for all $0\le i\le d$.