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Operations on G-sets and the Dress map

  • Thomas Brazelton
  • Ben Spitz
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Abstract

We study versions of binomial and multichoose coefficients for finite G-sets, extending a result of Siebeneicher from the 1970s concerning the duality between these operations. If the group G comes from a finite Galois extension of fields, the Dress map provides a bridge from G-sets to quadratic forms over the base field. We show that this Dress map is compatible with binomial coefficients of finite G-sets and a particular operation on quadratic forms we define. This allows us to extend the definition of Brugallé and Wickelgren's quadratically enriched binomial coefficients to arbitrary Grothendieck--Witt classes. This allows for a new proof that quadratically enriched binomial coefficients depend only on the trace form of the inputted étale algebra, as well as an elegant reproof of a result of Chen and Wickelgren about binomial coefficients over finite fields.

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Cite

BibTeXRIS
BibTeX
@misc{brazelton2026operations,
  title = {{Operations on $G$-sets and the Dress map}},
  author = {Thomas Brazelton and Ben Spitz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23758},
  eprint = {2610.23758},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23758},
}
RIS
TY  - UNPB
TI  - Operations on G-sets and the Dress map
AU  - Thomas Brazelton
AU  - Ben Spitz
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23758
UR  - https://arxiv.org/abs/2610.23758
AB  - We study versions of binomial and multichoose coefficients for finite G-sets, extending a result of Siebeneicher from the 1970s concerning the duality between these operations. If the group G comes from a finite Galois extension of fields, the Dress map provides a bridge from G-sets to quadratic forms over the base field. We show that this Dress map is compatible with binomial coefficients of finite G-sets and a particular operation on quadratic forms we define. This allows us to extend the definition of Brugallé and Wickelgren's quadratically enriched binomial coefficients to arbitrary Grothendieck--Witt classes. This allows for a new proof that quadratically enriched binomial coefficients depend only on the trace form of the inputted étale algebra, as well as an elegant reproof of a result of Chen and Wickelgren about binomial coefficients over finite fields.
ER  -