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.

11 pages