Explicit Generators for the Unit Group of the Burnside ring

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.