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

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 $R$.

References in corpus (1)

• On the Table of Marks of a Direct Product of Finite Groups