2025/09/05 by Ziad Ghanem, Ghanem, Ziad
Mathematics · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.05432
To the best of our knowledge, there is no explicit, constructive description of the generating set for the unit group A(G)^× of the Burnside ring associated with a finite group G. We resolve this long-standing open question, proving that A(G)^× is generated by the set of 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.