Grothendieck Rings of Module Categories over Drinfeld Doubles

Abstract

Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\mathcal{R}\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\mathbb C\otimes_{\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer--Picard action on indecomposable ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\oplus\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.