[Paper Review] Gamma-positivity in combinatorics and geometry

Christos A. Athanasiadis|arXiv (Cornell University)|Nov 16, 2017

Advanced Combinatorial Mathematics102 references79 citations

TL;DR

A survey of gamma-positivity, its definitions, and its appearances across combinatorics and geometry, including Eulerian polynomials, posets, Coxeter groups, and geometric h-polynomials, with methods and open problems.

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, Schützenberger and Strehl on the Eulerian polynomials; it was revived independently by Brändén 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.

Motivation & Objective

• Motivate gamma-positivity as a tool that implies symmetry and unimodality for symmetric polynomials.
• Survey main gamma-positive instances across combinatorics and geometry, highlighting how gamma-positivity arises in diverse contexts.
• Summarize methods used to prove gamma-positivity and outline key interpretations of gamma-coefficients.
• Connect combinatorial gamma-positivity to geometric objects like flag triangulations and h-polynomials, including conjectures and generalizations.

Proposed method

• Present explicit gamma-expansions for classical and variant polynomials (e.g., Eulerian A_n(x), binomial Eulerian orms, poset Eulerian A_P(x), Coxeter W(x)).
• Provide combinatorial interpretations of gamma-coefficients via Asc/Des sets, excedances, and related statistics.
• Use real-rootedness and symmetry as underlying principles to deduce gamma-positivity (and unimodality).
• Discuss symmetric function and representation-theoretic approaches that yield gamma-positivity results.
• Explore geometric combinatorics methods (flag triangulations, h-polynomials, local h-polynomials) and valley hopping techniques to prove positivity.
• Offer q-analogues and equivariant/generalized frameworks to extend gamma-positivity beyond the classical setting.

Experimental results

Research questions

• RQ1 In which combinatorial and geometric contexts do symmetric polynomials exhibit gamma-positivity?
• RQ2 What are the combinatorial interpretations of the gamma-coefficients in various Eulerian-type polynomials?
• RQ3 Can gamma-positivity be established for broader families (posets, Coxeter groups, derangements, involutions) and via diverse methods?
• RQ4 How does gamma-positivity relate to geometric objects like flag simplicial spheres and their h-polynomials?
• RQ5 What are the open problems and potential generalizations (e.g., nonsymmetric, equivariant, q-analogs) in gamma-positivity?

Key findings

• Eulerian polynomials A_n(x) are gamma-positive, with gamma-coefficients counting combinatorial structures such as up-down permutations and related statistics.
• Polynomials arising from posets, including A_P(x) for graded posets, are gamma-positive, with Brändean giving combinatorial proofs in broad settings.
• Weyl/Coxeter Eulerian polynomials W(x) for finite Coxeter groups are gamma-positive, with interpretations in classical types and extensions to affine/crystallographic cases.
• Derangement polynomials d_n(x) are gamma-positive, with interpretations in excedances and related permutation statistics, and connections to local h-polynomials.
• Binomial Eulerian polynomials ormed as combinations of A_n(x) and d_k(x) are gamma-positive, with explicit gamma-expansions and q-analogues.
• The geometry-focused section discusses Gal conjecture and h-polynomials of flag triangulations as a framework for gamma-positivity in spheres and simplices.