Invariant rings and representations of the symmetric groups

In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{\Gamma}$ where $\Gamma$ is a product of general linear groups over a field $K$ of characteristic zero, and $U$ is a finite dimensional rational representation of $\Gamma$. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where $U=\text{End}(W^{\oplus k})$ and $\Gamma = \text{GL}(W)$ and on the case where $U=\text{End}(V\otimes W)$ and $\Gamma = \text{GL}(V)\times \text{GL}(W)$, though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.