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PreprintCC BY 4.0arXiv

Permutation Representations on Cohomology of Toric Varieties

  • Tao Gui
  • Chushi Qin
  • Kaizhe Shen
  • Rui Xiong
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Abstract

Let G be a finite group acting properly by lattice automorphisms on a complete simplicial fan Σ. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology H*(XΣ) of the associated toric variety XΣ is isomorphic to a permutation representation of G. We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.

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Cite

BibTeXRIS
BibTeX
@misc{gui2026permutation,
  title = {{Permutation Representations on Cohomology of Toric Varieties}},
  author = {Tao Gui and Chushi Qin and Kaizhe Shen and Rui Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06597},
  eprint = {2609.06597},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06597},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Permutation Representations on Cohomology of Toric Varieties
AU  - Tao Gui
AU  - Chushi Qin
AU  - Kaizhe Shen
AU  - Rui Xiong
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06597
UR  - https://arxiv.org/abs/2609.06597
AB  - Let G be a finite group acting properly by lattice automorphisms on a complete simplicial fan Σ. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology H^*(X_Σ) of the associated toric variety X_Σ is isomorphic to a permutation representation of G. We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.
ER  -