Central idempotents of the bifree and left-free double Burnside ring

TL;DR

This paper identifies the primitive central idempotents of specific double Burnside rings and their scalar extensions, providing a detailed algebraic structure analysis relevant to group theory and representation theory.

Contribution

It explicitly determines the blocks and primitive central idempotents of the bifree and left-free double Burnside rings, advancing understanding of their algebraic structure.

Findings

Identified primitive central idempotents of $B^ riangle(G,G)$ and $B^{ rl}(G,G)$

Described the blocks of these double Burnside rings

Analyzed scalar extensions to $ extbf{QQ}$ for these algebras

Abstract

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring $B^\Delta(G,G)$ and the left-free double Burnside ring $B^{\trl}(G,G)$, as well as the primitive central idempotents of the algebras arising from scalar extension to $\QQ$.