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Matroid analogues of Gal's conjecture

Basile Coron, Luis Ferroni, Shiyue Li

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. Our proofs rely on toric geometry and make crucial use of tropical intersection theory. We begin by introducing complete building sets, a class encompassing all maximal building sets and other important families such as minimal building sets of braid matroids. For matroids with complete building sets, we analyze the Chow rings of the associated toric varieties, and prove that their Hilbert--Poincaré polynomials are gamma-positive. From this analysis, we derive a combinatorial formula for the coefficients of the gamma-expansion, and use it to explicitly construct a simplicial complex $Γ$, whose f-vector coincides with the gamma-vector. This establishes a matroid analogue of the Nevo--Petersen conjecture. When the building set is maximal, we further prove that $Γ$ is balanced, confirming the strongest such analogue in this case. As an application, we obtain a new combinatorial formula for the gamma-expansion of the Poincaré polynomial of the Deligne--Mumford--Knudsen compactification $\overline{\mathcal{M}}_{0,n}$, and derive several novel numerical inequalities for its coefficients. We also study the toric varieties of matroids with flag building sets, another class containing maximal building sets as well as several other prominent families. We prove that the Hilbert--Poincaré polynomials of these toric varieties are gamma-positive. This result establishes matroid analogues of the Charney--Davis and Gal conjectures, and simultaneously extends several recent gamma-positivity results for Chow polynomials.

Matroid analogues of Gal's conjecture

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. Our proofs rely on toric geometry and make crucial use of tropical intersection theory. We begin by introducing complete building sets, a class encompassing all maximal building sets and other important families such as minimal building sets of braid matroids. For matroids with complete building sets, we analyze the Chow rings of the associated toric varieties, and prove that their Hilbert--Poincaré polynomials are gamma-positive. From this analysis, we derive a combinatorial formula for the coefficients of the gamma-expansion, and use it to explicitly construct a simplicial complex , whose f-vector coincides with the gamma-vector. This establishes a matroid analogue of the Nevo--Petersen conjecture. When the building set is maximal, we further prove that is balanced, confirming the strongest such analogue in this case. As an application, we obtain a new combinatorial formula for the gamma-expansion of the Poincaré polynomial of the Deligne--Mumford--Knudsen compactification , and derive several novel numerical inequalities for its coefficients. We also study the toric varieties of matroids with flag building sets, another class containing maximal building sets as well as several other prominent families. We prove that the Hilbert--Poincaré polynomials of these toric varieties are gamma-positive. This result establishes matroid analogues of the Charney--Davis and Gal conjectures, and simultaneously extends several recent gamma-positivity results for Chow polynomials.

Paper Structure

This paper contains 39 sections, 59 theorems, 187 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Feichtner--Yuzvinsky Chow rings and their Hilbert series
  3. Gamma-positivity for flag simplicial spheres
  4. Main results
  5. Built matroids and nested set complexes
  6. Simplicial complexes
  7. Built matroids
  8. Nested set complexes
  9. Restriction and contraction
  10. Single-element extension
  11. Deletion
  12. Truncation
  13. Complete and flag built matroids
  14. Complete built matroids
  15. Flag built matroids
  16. ...and 24 more sections

Key Result

Theorem 1.6

If $(\mathrm{M},\mathcal{G})$ is a complete built matroid, and $\gamma(\mathrm{M}, \mathcal{G})(t)$ is the $\gamma$-expansion of $\mathrm{H}(\mathrm{M}, \mathcal{G})(t)$, then we have the formula In particular, the polynomial $\mathrm{H}(\mathrm{M}, \mathcal{G})$ is $\gamma$-positive. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Hierarchy of positivity properties for positive, palindromic polynomials.
  • Figure 2: Strengths of conjectures on $\gamma(\Delta)$ for a flag simplicial sphere $\Delta$.
  • Figure 3: Relationships between relevant built matroids.
  • Figure 4: The fan of $(\mathrm{U}_{2, 3}, \mathcal{G}_{\max})$ is a tropical Weil divisor of the fan of $(\mathrm{U}_{3, 3}, \mathcal{G}_{\max})$ in $\mathbb{R}^{[3]}/\mathbb{R}\langle \mathbf{e}_{123} \rangle$ defined by the piecewise linear function $f = - \min_{i \in [3]}(x_i)$.
  • Figure 5: Maximal nested sets of $\mathrm{B}_4$ with $\mathcal{G} = \{1, 2, 3, 4, 12, 34, 1234\}$ with labels in blueish green, and descent labels in pink.

Theorems & Definitions (145)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3: Charney--Davis conjecture
  • Conjecture 1.4: Gal's conjecture
  • Conjecture 1.5: Nevo--Petersen conjecture
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 135 more