Brauer relations in finite groups

Abstract

If $G$ is a non-cyclic finite group, non-isomorphic $G$-sets $X,Y$ may give rise to isomorphic permutation representations $\mathbb{C}[X]\cong \mathbb{C}[Y]$. Equivalently, the map from the Burnside ring to the rational representation ring of $G$ has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of $p$-groups.