📝 SummarXiv

Abstract

To the best of our knowledge, there is no explicit, constructive description of the generating set for the unit group $A(G)^\times$ of the Burnside ring associated with a finite group $G$. We resolve this long-standing open question, proving that $A(G)^\times$ is generated by the set of \emph{basic degrees} -- canonical Burnside ring elements arising from the $G$-equivariant degree of the identity map on irreducible $G$-representations. In particular, we demonstrate that every unit in $A(G)$ is realized as the equivariant degree of a linear $G$-isomorphism on a suitable orthogonal $G$-representation which, in turn, can be described as the Burnside ring product of a finite number of basic degrees, establishing a concrete link between the multiplicative structure of the Burnside ring and the field of equivariant topology.