On the structure theory of graded Burnside rings

Abstract

Let $G$ denote a finite group and let $S$ be a finite $G$-set. It is well known that the Burnside ring $\Omega(G)$ of $G$ has its elements as the formal differences of isomorphism classes of finite G-sets. In \cite{Nw}, the category $(G, S, \Omega(G))$-gr, which consists of $\Omega(G)$-modules graded by $S$ as objects and the degree preserving $\Omega(G)$-linear maps as  morphisms, was introduced. Using this category as a springboard, some interesting results in the structure theory of graded Burnside rings are brandished.