Hilbert series and invariants in exterior algebras

In this paper, we consider the exterior algebra $\mathrm{\Lambda }(W)$ of a polynomial $\mathrm{G}\mathrm{L}(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $\mathrm{\Lambda }(W{)}^G$, where $G$ is one of the classical complex subgroups of $\mathrm{G}\mathrm{L}(n)$, namely $\mathrm{S}\mathrm{L}(n)$, $\mathrm{O}(n)$, $\mathrm{S}\mathrm{O}(n)$, or $\mathrm{S}\mathrm{p}(2d)$ (for $n=2d$). Since $\mathrm{\Lambda }(W{)}^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $\mathrm{\Lambda }(S^3{\mathbb{C}}^3{)}^{\mathrm{S}\mathrm{L}(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.