Noncommutative invariants of finite and classical groups

Abstract

We investigate the structure of the invariant subring of the tensor algebra $T(W)$ of a $G$-representation $W$, viewed as a twisted commutative algebra (tca). For a faithful representation $W$ of a finite group $G$ over a field $k$, we show that if char$(k)\mid \#G$, then $T(W{)}^G$ is not finitely generated as a tca. In contrast, for a representation $W$ of a classical group $G_{\mathbb{Z}}$, we prove that the invariant subring $T(W_k{)}^{G_k}$ is finitely generated as a tca when $k$ is algebraically closed of sufficiently large characteristic, provided that $W$ admits a good filtration over $\mathbb{Z}$. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$ for $T({\mathbb{C}}^n)$ as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants.

12 pages; significant expository revisions. To appear in Algebra & Number Theory