Counting points on Hessenberg Varieties over finite fields

TL;DR

Counting points on Hessenberg varieties over ${\mathbb F}_q$ is extended to non-split cases by a universal formula $|\mathscr{H}({\mathsf m},T)|=\langle F_T(x),\omega X_{G({\mathsf m})}(x;q)\rangle$, linking arithmetic counts to symmetric function theory. The approach introduces the invariant flag generating function $F_T(x)$ and expresses it via similarity-class data, specifically through modified Hall-Littlewood polynomials. The modular law enables an explicit decomposition of counts and yields corollaries for the Poincaré polynomials of complex Hessenberg varieties and a new derivation of Brosnan-Chow-type results about the omega-dual chromatic quasisymmetric function. The work also provides a combinatorial route to a formula for modified Hall-Littlewood polynomials via tabloids, strengthening the bridge between arithmetic geometry over finite fields and symmetric-function combinatorics.

Abstract

We give a counting formula in terms of modified Hall-Littlewood polynomials and the chromatic quasisymmetric function for the number of points on an arbitrary Hessenberg variety over a finite field. As a consequence, we express the Poincaré polynomials of complex Hessenberg varieties in terms of a Hall scalar product involving the symmetric functions above. We use these results to give a new proof of a combinatorial formula for the modified Hall-Littlewood polynomials.

Figures (3)

• Figure 1: Left: A Hessenberg function ${\mathsf m}$ with the corresponding Dyck path; Right: The indifference graph $G({\mathsf m})$ and the irreducible components ${\mathsf m}_1,{\mathsf m}_2,{\mathsf m}_3$ of ${\mathsf m}$.
• Figure 2: Tabloid and value of an element.
• Figure 3: Overview of possible positions for $i$ and $k$.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Definition 3.1