Hilbert Series and Invariants in Exterior Algebras

Abstract

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