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Chow and augmented Chow polynomials as evaluations of Poincaré-extended ab-indices

Christian Stump

TL;DR

This work shows that Chow polynomials and augmented Chow polynomials of matroids, and more generally finite graded posets with $R$-labelings, arise as evaluations of the Poincaré-extended $\mathbf{a}\mathbf{b}$-index, yielding explicit $\gamma$-positive expansions without relying on the Kähler package. The authors establish key evaluations linking the extended index to the Chow polynomials via $${\text{ex}\Psi_P(-x,1,x)}=(1-x)^n\cdot H_P(x)$$ and $${\text{ex}\widetilde{\Psi}_P(-x,1,x)}=(1-x)^n\cdot \underline{H}_P(x)$$, and show these results extend to general finite graded posets with $R$-labelings. They further relate the coarse flag Hilbert-Poincaré series to the extended index, enabling explicit computations for the braid arrangement $\mathcal B_n$, where the Chow and augmented Chow polynomials admit closed forms in terms of descent-isolated sequences and inversion sequences. The findings provide a Hodge-theory-free proof of $\gamma$-positivity, unify combinatorial interpretations across posets and matroids, and suggest real-rootedness questions for various evaluations of the extended index.

Abstract

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labelings, are obtained as evaluations of their Poincaré-extended ab-indices. This implies in particular explicit combinatorial $γ$-positive expansions for both, providing the first proof of the $γ$-positivity not relying on the Kähler package for the Chow ring. We then evaluate this expansion to obtain an explicit closed formula for the braid arrangement.

Chow and augmented Chow polynomials as evaluations of Poincaré-extended ab-indices

TL;DR

This work shows that Chow polynomials and augmented Chow polynomials of matroids, and more generally finite graded posets with -labelings, arise as evaluations of the Poincaré-extended -index, yielding explicit -positive expansions without relying on the Kähler package. The authors establish key evaluations linking the extended index to the Chow polynomials via and , and show these results extend to general finite graded posets with -labelings. They further relate the coarse flag Hilbert-Poincaré series to the extended index, enabling explicit computations for the braid arrangement , where the Chow and augmented Chow polynomials admit closed forms in terms of descent-isolated sequences and inversion sequences. The findings provide a Hodge-theory-free proof of -positivity, unify combinatorial interpretations across posets and matroids, and suggest real-rootedness questions for various evaluations of the extended index.

Abstract

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labelings, are obtained as evaluations of their Poincaré-extended ab-indices. This implies in particular explicit combinatorial -positive expansions for both, providing the first proof of the -positivity not relying on the Kähler package for the Chow ring. We then evaluate this expansion to obtain an explicit closed formula for the braid arrangement.
Paper Structure (7 sections, 10 theorems, 45 equations)

This paper contains 7 sections, 10 theorems, 45 equations.

Table of Contents

  1. Main results
  2. The Poincaré-extended $\mathbf{a}\mathbf{b}$-index and its evaluations
  3. The Chow and the augmented Chow polynomials
  4. The Poincaré-extended $\mathbf{a}\mathbf{b}$-index
  5. Evaluating the Poincaré-extended $\mathbf{a}\mathbf{b}$-index
  6. Evaluating the coarse flag Hilbert-Poincaré series
  7. Chow and augmented Chow polynomials for braid arrangement

Key Result

Theorem 1.1

The Chow polynomial $\underline{\textsf{H}}_M(x)$ has the expansion where the sum ranges over all maximal chains $\mathcal{F}$ in $\mathcal{L}(M)$ such that ${\sf Des}(\mathcal{F})$ is isolated and $1 \notin {\sf Des}(\mathcal{F})$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4: Generalization to finite graded posets admitting $R$-labelings
  • Remark 1.5: Three existing real-rootedness conjectures
  • Definition 2.1: FerroniEtAl
  • Definition 2.2
  • Proposition 2.3: poincareextended
  • Theorem 2.4: poincareextended
  • Corollary 2.5: poincareextended
  • ...and 12 more