On the essential algebra of the shifted Burnside biset functor

TL;DR

This paper characterizes the essential algebra of the shifted Burnside biset functor for specific classes of groups, providing explicit algebraic descriptions and bases in certain cases.

Contribution

It offers a detailed description of the essential algebra for the shifted Burnside biset functor in two cases, including isomorphisms and basis constructions.

Findings

For finite abelian groups, the algebra is a quotient of a shifted star algebra.

For coprime order groups, the algebra is a semidirect product involving outer automorphisms.

Identifies conditions under which a basis can be explicitly described.

Abstract

We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set.