Hilbert series and invariants in exterior algebras

TL;DR

This paper computes the Hilbert series of invariant algebras in exterior algebras under classical group actions, providing explicit examples and generators, especially for the case of $ ext{SL}(3)$ acting on $igwedge^3 ext{C}^3$.

Contribution

It introduces methods to determine Hilbert series of invariants in exterior algebras under classical groups and provides explicit computations and generators.

Findings

Computed Hilbert series for various classical group invariants

Identified explicit generators for $ ext{SL}(3)$ invariants in $igwedge^3 ext{C}^3$

Demonstrated the finiteness of invariant algebras in these settings

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.