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PreprintarXiv licencearXiv

Dot actions and Hessenberg spaces for dihedral groups

  • Jonathan Boretsky
  • Reuven Hodges
  • Nathan Lesnevich
  • Tyrrell McAllister
  • Joseph Pappe
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Abstract

Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the natural group action on them encodes the monodromy action. The resulting representations are of great interest in algebraic combinatorics for their connections to Hecke algebra characters, LLT polynomials, and chromatic symmetric functions. We study the analogous construction for dihedral groups, which in general have no underlying geometry. Using purely algebraic methods, we show that these spline modules are free, construct explicit bases, and compute their Hilbert series, recovering the familiar properties of the geometric setting. We also determine the associated dihedral group representations completely. Unlike what is conjectured for the symmetric group case, these need not be permutation representations.

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Cite

BibTeXRIS
BibTeX
@misc{boretsky2026actions,
  title = {{Dot actions and Hessenberg spaces for dihedral groups}},
  author = {Jonathan Boretsky and Reuven Hodges and Nathan Lesnevich and Tyrrell McAllister and Joseph Pappe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16001},
  eprint = {2610.16001},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16001},
}
RIS
TY  - UNPB
TI  - Dot actions and Hessenberg spaces for dihedral groups
AU  - Jonathan Boretsky
AU  - Reuven Hodges
AU  - Nathan Lesnevich
AU  - Tyrrell McAllister
AU  - Joseph Pappe
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16001
UR  - https://arxiv.org/abs/2610.16001
AB  - Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the natural group action on them encodes the monodromy action. The resulting representations are of great interest in algebraic combinatorics for their connections to Hecke algebra characters, LLT polynomials, and chromatic symmetric functions. We study the analogous construction for dihedral groups, which in general have no underlying geometry. Using purely algebraic methods, we show that these spline modules are free, construct explicit bases, and compute their Hilbert series, recovering the familiar properties of the geometric setting. We also determine the associated dihedral group representations completely. Unlike what is conjectured for the symmetric group case, these need not be permutation representations.
ER  -