Gamma-positivity in combinatorics and geometry

TL;DR

This paper surveys the concept of gamma-positivity, a property of symmetric polynomials implying unimodality, highlighting its origins, applications in combinatorics and geometry, and various proof techniques.

Contribution

It provides a comprehensive overview of gamma-positivity, including main results, open problems, and diverse methods used to establish this property across different mathematical contexts.

Findings

Gamma-positivity implies polynomial unimodality.

Multiple combinatorial and geometric applications of gamma-positivity.

Various proof techniques have been developed for gamma-positivity.

Abstract

Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Sch\"utzenberger and Strehl on the Eulerian polynomials; it was revived independently by Br\"and\'en and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.