Invariants of symplectic and orthogonal groups acting on $\text{GL}(n,{\mathbb C})$-modules

Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We take a polynomial $\text{GL}(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=\text{SL}(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Then we give explicit examples for computing $H(S(W)^G, t)$. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $Λ(S^2 V)^G$ and $Λ(Λ^2 V)^G$, where $V = {\mathbb C}^n$ denotes the standard $GL(n)$-module.

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