The integral double Burnside ring of the symmetric group $\text{S}_3$

Nora Krauss

The integral double Burnside ring of the symmetric group $\text{S}_3$

Representation Theory 2020-10-16 v2

Authors: Nora Krauss

Abstract

The double Burnside $R$ -algebra $\text{B}_R(G,G)$ of a finite group $G$ with coefficients in a commutative ring $R$ has been introduced by S. Bouc. It is $R$ -linearly generated by finite $(G,G)$ -bisets, modulo a relation identifying disjoint union and sum. Its multiplication is induced by the tensor product. B. Masterson described $\text{B}_{\mathbf{Q}}(\text{S}_3,\text{S}_3)$ as a subalgebra of $\mathbf{Q}^{8\times 8}$ . We give a variant of this description and continue to describe $\text{B}_R(\text{S}_3,\text{S}_3)$ for $R\in\{\mathbf{Z},\mathbf{Z}_{(2)},\mathbf{F}_2,\mathbf{Z}_{(3)},\mathbf{F}_3\}$ via congruences as suborders of certain $R$ -orders respectively via path algebras over $R$ .

Keywords

double burnside ring symmetric group bisets r-orders path algebras finite groups

Cite

@article{arxiv.2002.01493,
  title  = {The integral double Burnside ring of the symmetric group $\text{S}_3$},
  author = {Nora Krauss},
  journal= {arXiv preprint arXiv:2002.01493},
  year   = {2020}
}

The integral double Burnside ring of the symmetric group $\text{S}_3$

Abstract

Keywords

Cite

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