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Chow polynomials of totally nonnegative matrices and posets

Petter Brändén, Lorenzo Vecchi

TL;DR

The paper proves that Chow polynomials of lower-triangular totally nonnegative ($ ext{TN}$) matrices with unit diagonal are real-rooted, and deduces real-rootedness for Chow polynomials of $ ext{TN}$-posets and many important poset families (e.g., projective/affine geometries, Dowling and lattice of flats of paving matroids). It develops a systematic incidence-algebra framework for Chow functions, augmented Chow functions, and their duality, and introduces interlacing concepts including $ ext{I}_n$-interlacing to preserve real-rootedness through matrix-driven recursions. The authors further connect gamma-Chow polynomials to determinants and non-intersecting paths, extend results to Toeplitz, binomial, and Sheffer posets, and tie Chow polynomials to symmetric/supersymmetric functions, yielding gamma-positivity and Schur-positivity consequences. Collectively, the work provides a unified, expandable toolkit for real-rootedness questions in Chow theory, with broad implications for the combinatorics of matroids, lattice theory, and related algebraic structures.

Abstract

Huh-Stevens and Ferroni-Schröter independently conjectured that Hilbert-Poincaré series of Chow rings of geometric lattices have only real zeros. Ferroni, Matherne and the second author extended this conjecture to Chow polynomials of Cohen-Macaulay poset. In this paper we address the above conjectures by providing new defining relations and properties of Chow functions of posets and matrices. These are used, in conjunction with new techniques on interlacing sequences of polynomials, to prove that Chow polynomials of totally nonnegative matrices have only real zeros, which, in turn, proves the above conjectures for a class of posets that contains projective and affine geometries, face lattices of cubical polytopes, partition lattices and Dowling lattices, perfect matroid designs, and lattices of flats of paving matroids. We also study Chow polynomials of Toeplitz matrices in greater detail, and show how these are related the combinatorics of binomial and Sheffer posets, as well as to a family of generalized Eulerian polynomials with coefficients in the ring of symmetric polynomials that have been studied by e.g. Stanley, Brenti, Stembridge and Shareshian-Wachs.

Chow polynomials of totally nonnegative matrices and posets

TL;DR

The paper proves that Chow polynomials of lower-triangular totally nonnegative () matrices with unit diagonal are real-rooted, and deduces real-rootedness for Chow polynomials of -posets and many important poset families (e.g., projective/affine geometries, Dowling and lattice of flats of paving matroids). It develops a systematic incidence-algebra framework for Chow functions, augmented Chow functions, and their duality, and introduces interlacing concepts including -interlacing to preserve real-rootedness through matrix-driven recursions. The authors further connect gamma-Chow polynomials to determinants and non-intersecting paths, extend results to Toeplitz, binomial, and Sheffer posets, and tie Chow polynomials to symmetric/supersymmetric functions, yielding gamma-positivity and Schur-positivity consequences. Collectively, the work provides a unified, expandable toolkit for real-rootedness questions in Chow theory, with broad implications for the combinatorics of matroids, lattice theory, and related algebraic structures.

Abstract

Huh-Stevens and Ferroni-Schröter independently conjectured that Hilbert-Poincaré series of Chow rings of geometric lattices have only real zeros. Ferroni, Matherne and the second author extended this conjecture to Chow polynomials of Cohen-Macaulay poset. In this paper we address the above conjectures by providing new defining relations and properties of Chow functions of posets and matrices. These are used, in conjunction with new techniques on interlacing sequences of polynomials, to prove that Chow polynomials of totally nonnegative matrices have only real zeros, which, in turn, proves the above conjectures for a class of posets that contains projective and affine geometries, face lattices of cubical polytopes, partition lattices and Dowling lattices, perfect matroid designs, and lattices of flats of paving matroids. We also study Chow polynomials of Toeplitz matrices in greater detail, and show how these are related the combinatorics of binomial and Sheffer posets, as well as to a family of generalized Eulerian polynomials with coefficients in the ring of symmetric polynomials that have been studied by e.g. Stanley, Brenti, Stembridge and Shareshian-Wachs.

Paper Structure

This paper contains 17 sections, 56 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The incidence algebra and Chow functions
  3. Augmented Chow functions
  4. Duality for Chow functions
  5. Chow polynomial of matrices and weak-rank uniform posets
  6. Totally nonnegative matrices and their Chow-polynomials
  7. Totally nonnegative and resolvable matrices
  8. Interlacing zeros and In-interlacing sequences of polynomials
  9. Chow-deranged maps and Chow-Eulerian transformations
  10. gamma-Chow polynomials of matrices
  11. Coefficients of gamma-Chow polynomials of matrices
  12. Interlacing gamma-Chow polynomials
  13. Chow polynomials of TN-posets and dual TN-posets
  14. Chow polynomials of lattices of flats of paving matroids
  15. Toeplitz matrices
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given a $P$-kernel $\kappa$, there exists a unique element $g\in I(P)$ such that and $\kappa = g^{-1}\mathcal{I} (g)$. Conversely, if $g \in I (P)$ satisfies condition for kls, then $\kappa = g^{-1}\mathcal{I} (g)$ is a $P$-kernel.

Figures (1)

  • Figure 1: The directed graph $\Gamma_N$ with its weights.

Theorems & Definitions (102)

  • Theorem 2.1: brenti-kls, proudfoot-kls
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • ...and 92 more