  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.
