Noncommutative invariants of finite and classical groups

TL;DR

This paper studies the structure and finite generation properties of invariant subrings of tensor algebras under group actions, revealing differences between finite and classical groups and introducing a categorical Gelfand--Kirillov dimension.

Contribution

It demonstrates non-finite generation of invariants for finite groups in certain characteristics and establishes finite generation for classical groups under specific conditions, also defining a new categorical dimension.

Findings

Invariant subring not finitely generated for finite groups in characteristic dividing group order.

Invariant subring finitely generated for classical groups over large characteristic fields.

Categorical Gelfand--Kirillov dimension computed as binomial coefficient.

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 $\binom{n+1}{2}$ 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.