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

TL;DR

This paper develops a method to compute the Hilbert series of invariant algebras under classical groups acting on polynomial modules, extending previous results and providing explicit examples for symplectic and orthogonal groups.

Contribution

It introduces a new approach for calculating Hilbert series of invariants for classical groups acting on polynomial modules, expanding prior work on SL(n) to O(n), SO(n), and Sp(2k).

Findings

Derived explicit formulas for Hilbert series of invariants.

Extended methods to compute invariants of symmetric and exterior powers.

Provided concrete examples illustrating the computation process.

Abstract

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 $\Lambda(S^2 V)^G$ and $\Lambda(\Lambda^2 V)^G$, where $V = {\mathbb C}^n$ denotes the standard $GL(n)$-module.