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Peer-reviewedCC BY 4.0Algebraic Combinatorics

Counting points on Hessenberg varieties over finite fields

  • Alex Abreu1
  • Antonio Nigro1
  • Samrith Ram2
  1. 1Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Rio de Janeiro (Brazil)
  2. 2Indraprastha Institute of Information Technology Delhi, New Delhi (India)
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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.

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Cite

BibTeXRIS
BibTeX
@article{abreu2026counting,
  title = {{Counting points on Hessenberg varieties over finite fields}},
  author = {Alex Abreu and Antonio Nigro and Samrith Ram},
  journal = {Algebraic Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.5802/alco.519},
  url = {https://alco.centre-mersenne.org/articles/10.5802/alco.519/},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Counting points on Hessenberg varieties over finite fields
AU  - Alex Abreu
AU  - Antonio Nigro
AU  - Samrith Ram
PY  - 2026
DA  - 2026/10/27
JO  - Algebraic Combinatorics
DO  - 10.5802/alco.519
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.519/
AB  - 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.
SN  - 2589-5486
ER  -