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Existence of Strong Lefschetz algebras with Chow polynomials as Hilbert series

Adam Schweitzer, Lorenzo Vecchi

TL;DR

This work shows that for any finite, bounded, weakly ranked poset $P$, there exists a Gorenstein algebra with the Strong Lefschetz property whose Hilbert–Poincaré series matches the Chow polynomial $H_P(t)$, lending support to a broad generalization of Chow rings beyond geometric lattices. The authors construct the $ ext{FY}$-based $h$-vector $h(P)$ and prove it is an $ ext{SI}$-sequence by realizing it as the $h$-vector of a monomial order ideal via a symmetric chain decomposition, yielding a differential $ ext{O}$-sequence. In the ranked case, the consecutive differences $(h_0, h_1-h_0, frac{h_{ loor{(n-1)/2}}-h_{ loor{(n-1)/2}-1}}{}})$ form a pure $ ext{O}$-sequence, and pureness holds with a counterexample in non-ranked cases. The paper also proves Chow polynomials are log-concave for weak rank at most $6$, while constructing counterexamples for higher ranks, clarifying the extent of regularity these polynomials can satisfy. Overall, the results advance the algebraic/combinatorial understanding of Chow polynomials and provide new models realizing their properties via Lefschetz algebras, supporting broader conjectures about combinatorial Chow rings.

Abstract

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the strong Lefschetz property such that their Hilbert-Poincaré series agrees with the Chow polynomial, providing evidence in support of a conjecture by Ferroni, Matherne and the second author. This allows us to show strong inequalities for the coefficients of Chow polynomials; we prove log-concavity for all posets of weak rank at most six and provide counterexamples to log-concavity for any higher rank. For ranked posets we recover an even stronger condition, showing that the differences between consecutive coefficients constitute a pure O-sequence.

Existence of Strong Lefschetz algebras with Chow polynomials as Hilbert series

TL;DR

This work shows that for any finite, bounded, weakly ranked poset , there exists a Gorenstein algebra with the Strong Lefschetz property whose Hilbert–Poincaré series matches the Chow polynomial , lending support to a broad generalization of Chow rings beyond geometric lattices. The authors construct the -based -vector and prove it is an -sequence by realizing it as the -vector of a monomial order ideal via a symmetric chain decomposition, yielding a differential -sequence. In the ranked case, the consecutive differences form a pure -sequence, and pureness holds with a counterexample in non-ranked cases. The paper also proves Chow polynomials are log-concave for weak rank at most , while constructing counterexamples for higher ranks, clarifying the extent of regularity these polynomials can satisfy. Overall, the results advance the algebraic/combinatorial understanding of Chow polynomials and provide new models realizing their properties via Lefschetz algebras, supporting broader conjectures about combinatorial Chow rings.

Abstract

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the strong Lefschetz property such that their Hilbert-Poincaré series agrees with the Chow polynomial, providing evidence in support of a conjecture by Ferroni, Matherne and the second author. This allows us to show strong inequalities for the coefficients of Chow polynomials; we prove log-concavity for all posets of weak rank at most six and provide counterexamples to log-concavity for any higher rank. For ranked posets we recover an even stronger condition, showing that the differences between consecutive coefficients constitute a pure O-sequence.
Paper Structure (11 sections, 18 theorems, 48 equations, 3 figures)

This paper contains 11 sections, 18 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Properties of sequences
  4. Standard graded algebras
  5. Chow polynomials
  6. Monomial order ideals and (pure) O-sequences
  7. SI-sequences
  8. Symmetric chain decompositions
  9. Proof of Theorem \ref{['thm:main1']}
  10. Proof of Theorem \ref{['thm:main2']}
  11. Log-concavity of the Chow polynomial

Key Result

Theorem 1.1

Let $P$ be a weakly ranked, finite, bounded poset of weak rank $n$. Then $h(P)$ forms a $\operatorname{SI}$-sequence.

Figures (3)

  • Figure 1: Symmetric chain decomposition of the product $C_{r_1}\times S'_i$
  • Figure 2: Counter-example to pureness
  • Figure 3: Counter-example to log-concavity for $\rho(P)\geq 7$

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: ferroni-matherne-vecchi
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 28 more