Algebraic Combinatorics· 2026Q1
Counting points on Hessenberg varieties over finite fields
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- 2026year
Short summary
A new formula counts points on arbitrary Hessenberg varieties over finite fields using modified Hall–Littlewood polynomials and the chromatic quasisymmetric function.
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Key points
- A counting formula for points on Hessenberg varieties over finite fields is derived.
- The formula utilizes modified Hall–Littlewood polynomials and the chromatic quasisymmetric function.
- Poincaré polynomials of complex Hessenberg varieties are expressed using a Hall scalar product.
- A new combinatorial proof for modified Hall–Littlewood polynomials is presented.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Algebraic Combinatorics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics