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arXiv · 2604.04550

Matroid analogues of Gal's conjecture

Basile Coron·Luis Ferroni·Shiyue Li
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Abstract

Well-known conjectures of Charney--Davis, Gal, and Nevo--Petersen predict increasingly strong positivity phenomena for the $h$-vectors of flag simplicial spheres. In this paper, we formulate and prove matroid analogues of these conjectures in the setting of Chow polynomials of matroids with building sets. We introduce a new class of matroids with building sets, called complete built matroids, encompassing many prominent families of built matroids such as arbitrary matroids with maximal building sets and braid matroids with minimal building sets. For complete built matroids, we prove $\gamma$-positivity as an analogue of Gal's conjecture, via a combinatorial formula for the $\gamma$-coefficients. We further realize the $\gamma$-vector as the $f$-vector of a simplicial complex, as an analogue of the Nevo--Petersen conjecture. As an application, we obtain a new formula for the $\gamma$-polynomial of the Poincar\'e polynomial of $\overline{\mathcal{M}}_{0,n}$, together with new coefficient inequalities. We also study flag built matroids, and prove $\gamma$-positivity of their Chow polynomials, extending several known results. Our proofs crucially use toric geometry and tropical intersection theory. Finally, we construct an infinite family of flag chordal nestohedra whose $h$-polynomials are not real-rooted, invalidating a natural strengthening of our result at this level of generality.

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Basile Coron, Luis Ferroni, Shiyue Li. 2026-04-06. Matroid analogues of Gal's conjecture. https://arxiv.org/abs/2604.04550

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