Counting points on Hessenberg Varieties over finite fields

TL;DR

This paper provides a new counting formula for points on Hessenberg varieties over finite fields using symmetric functions, linking algebraic geometry with combinatorics and offering new proofs for related polynomials.

Contribution

It introduces a novel counting formula involving Hall-Littlewood polynomials and chromatic quasisymmetric functions for Hessenberg varieties over finite fields.

Findings

Counting formula expressed via modified Hall-Littlewood polynomials and chromatic quasisymmetric functions.

Poincaré polynomials of complex Hessenberg varieties are described using symmetric functions.

New proof of a combinatorial formula for modified Hall-Littlewood polynomials.

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 Poincar\'e 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.