$S$-operations in representation theory

Abstract:

For $G$ a group and ${\text {A} ^G}$ the category of $G$-objects in a category $\text {A}$, a collection of functors, called “$S$-operations,” is introduced under mild restrictions on $\text {A}$. With certain assumptions on $\text {A}$ and with $G$ the symmetric group ${S_k}$, one obtains a unigeneration theorem for the Grothendieck ring formed from the isomorphism classes of objects in ${\text {A} ^{{S_k}}}$. For $\text {A}$ = finite-dimensional vector spaces over $C$, the result says that the representation ring $R({S_k})$ is generated, as a $\lambda$-ring, by the canonical $k$-dimensional permutation representation. When $\text {A}$ = finite sets, the $S$-operations are called “$\beta$-operations,” and the result says that the Burnside ring $B({S_k})$ is generated by the canonical ${S_k}$-set if $\beta$-operations are allowed along with addition and multiplication.