Chow and augmented Chow polynomials as evaluations of Poincar\'e-extended ab-indices

TL;DR

This paper demonstrates that Chow and augmented Chow polynomials of certain posets are evaluations of their Poincaré-extended ab-indices, leading to explicit combinatorial gamma-positive expansions and a closed formula for the braid arrangement.

Contribution

It establishes a novel connection between Chow polynomials and Poincaré-extended ab-indices, providing the first combinatorial proof of gamma-positivity without Kähler geometry.

Findings

Explicit gamma-positive expansions for Chow polynomials

A closed formula for the braid arrangement

First combinatorial proof of gamma-positivity for these polynomials

Abstract

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labelings, are obtained as evaluations of their Poincar\'e-extended ab-indices. This implies in particular explicit combinatorial $\gamma$-positive expansions for both, providing the first proof of the $\gamma$-positivity not relying on the K\"ahler package for the Chow ring. We then evaluate this expansion to obtain an explicit closed formula for the braid arrangement.