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 $R$.