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Grothendieck Rings of Module Categories over Drinfeld Doubles

  • Dmitri Nikshych
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Abstract

Let k be an algebraically closed field of characteristic zero and G a finite group. We realize the based Grothendieck ring RG of ℛep(D(G))-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of RG on the Grothendieck group of VecG-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that ℂ⊗ℤRG is semisimple exactly when G is cyclic. We study the Brauer--Picard action on indecomposable VecG-module categories and show that it is transitive exactly when G is abelian of square-free exponent. For abelian G, we determine the possible abstract group types of Lagrangian subgroups of G⊕Ĝ, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.

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Cite

BibTeXRIS
BibTeX
@misc{nikshych2026grothendieck,
  title = {{Grothendieck Rings of Module Categories over Drinfeld Doubles}},
  author = {Dmitri Nikshych},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27020},
  eprint = {2609.27020},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27020},
}
RIS
TY  - UNPB
TI  - Grothendieck Rings of Module Categories over Drinfeld Doubles
AU  - Dmitri Nikshych
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.27020
UR  - https://arxiv.org/abs/2609.27020
AB  - Let k be an algebraically closed field of characteristic zero and G a finite group. We realize the based Grothendieck ring R_G of ℛep(D(G))-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of R_G on the Grothendieck group of Vec_G-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that ℂ⊗_ℤR_G is semisimple exactly when G is cyclic. We study the Brauer--Picard action on indecomposable Vec_G-module categories and show that it is transitive exactly when G is abelian of square-free exponent. For abelian G, we determine the possible abstract group types of Lagrangian subgroups of G⊕Ĝ, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.
ER  -