arXiv subjects

Basile Coron

Publications and source records attributed to Basile Coron.

8 recordsLinked to original sources

arXiv2026

Matroid analogues of Gal's conjecture

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.

math.CO↗

arXiv2026

Structural properties of nested set complexes

We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of matroids. As a first application, we show that the $h$-vector of a nested set complex is strongly flawless and, in particular, top-heavy. We then specialize to the boundary complex of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0, n}$ of rational stable marked curves, which coincides with the complex of trees, establishing new structural decomposition theorems and deriving combinatorial formulas for its face enumeration polynomials.

math.CO↗

arXiv2025

Chow polynomials of rank-uniform labeled posets

We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schr\"oter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincar\'e series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincar\'e series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory.

math.CO↗

arXiv2025

Matroid complexes and Orlik-Solomon algebras

In this article we construct a combinatorial quasi-free differential graded model for the Orlik-Solomon algebra of supersolvable matroids, which generalizes in a matroidal setting the cdga of admissible graphs introduced by M. Kontsevich for the braid arrangements. Our construction draws on well-known concepts from matroid theory, including modularity, single-element extensions, and generalized parallel connections. We also show that this model carries a cooperadic structure in a suitably generalized sense. As an application, we use this model to give a new proof that the Orlik-Solomon algebras of supersolvable matroids are Koszul.

math.CO↗

arXiv2024

An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gr\"obner methods

Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done using a quadratic Gr\"obner basis associated to a new presentation of those rings, which is obtained by iterating the semi-small decomposition of Chow rings of matroids. This Gr\"obner basis can also be applied to compute certain principal ideals in these rings, and ultimately reestablish the known connection between derangement polynomials and the Hilbert series of Chow rings for corank 1 uniform matroids. More broadly, this approach enables us to express the Hilbert series of Chow rings for any uniform matroid as polynomials related to the ascent statistics on particular sets of inversion sequences.

math.CO↗

arXiv2024

Operadic Kazhdan-Lusztig-Stanley theory

We introduce a new type of operad-like structure called a P-operad, which depends on the choice of some collection of posets P, and which is governed by chains in posets of P. We introduce several examples of such structures which are related to classical poset theoretic notions such as poset homology, Cohen--Macaulayness and lexicographic shellability. We then show that P-operads form a satisfactory framework to categorify Kazhdan--Lusztig polynomials of geometric lattices and their kernel. In particular, this leads to a new proof of the positivity of the coefficients of Kazhdan--Lusztig polynomials of geometric lattices.

math.CO↗

arXiv2023

Supersolvability of built lattices and Koszulness of generalized Chow rings

We give an explicit quadratic Grobner basis for generalized Chow rings of supersolvable built lattices, with the help of the operadic structure on geometric lattices introduced in a previous article. This shows that the generalized Chow rings associated to minimal building sets of supersolvable lattices are Koszul. As another consequence, we get that the cohomology algebras of the components of the extended modular operad in genus 0 are Koszul.

math.CO↗

arXiv2022

Matroids, Feynman categories, and Koszul duality

We show that various combinatorial invariants of matroids such as Chow rings and Orlik--Solomon algebras may be assembled into "operad-like" structures. Specifically, one obtains several operads over a certain Feynman category which we introduce and study in detail. In addition, we establish a Koszul-type duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a celebrated result of Getzler. This provides a new interpretation of combinatorial Leray models of Orlik--Solomon algebras.

math.CO↗