Invariants of symplectic and orthogonal groups acting on $GL(n,CC)$-modules

Authors: VESSELIN DRENSKY, ELITZA HRISTOVA

Abstract: Let $GL(n) = GL(n, CC)$ denote the complex general linear group and let $G \subset GL(n)$ be one of the classical complex subgroups $OO(n)$, $SO(n)$, and $Sp(2k)$ (in the case $n = 2k$). We take a finite dimensional polynomial $GL(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=SL(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing $H(S(W)^G, t)$. As an application, we consider the question of regularity of the algebra $S(W)^{OO(n)}$. For $n=2$ and $n=3$ we give a complete list of modules $W$, so that if $S(W)^{OO(n)}$ is regular then $W$ is in this list. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $Λ(S^2 V)^G$ and $Λ(Λ^2 V)^G$, where $V = CC^n$ denotes the standard $GL(n)$-module.

Keywords: Invariant theory, Hilbert series, Schur function

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