Given a tuple $({𝒳}_1,...,{𝒳}_k)$ of irreducible characters of $G{L}_n\left(F_q\right)$ we define a star-shaped quiver $\Gamma$ together with a dimension vector $v$. Assume that $({𝒳}_1,...,{𝒳}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product ${𝒳}_1\otimes \cdots \otimes {𝒳}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...

Let $p$ be a prime number. This paper introduces the Roquette category ${ℛ}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In ${ℛ}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of ${ℛ}_p$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors from ${ℛ}_p$ to...