# Noncommutative invariant theory of symplectic and orthogonal groups

## TL;DR

This paper introduces a method to compute the Hilbert series of invariant algebras under symplectic and orthogonal groups acting on graded noncommutative algebras, with applications to specific algebra varieties.

## Contribution

It develops a novel computational approach for invariant theory in noncommutative settings involving symplectic and orthogonal groups.

## Key findings

- Computed Hilbert series for invariants of symplectic and orthogonal group actions
- Applied method to free algebras generated by Grassmann and upper triangular matrices
- Identified minimal varieties of exponent 2 in the context of invariants

## Abstract

We present a method for computing the Hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group. We apply our method to compute the Hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated, respectively, by the Grassmann algebra and the algebra of $2\times 2$ upper triangular matrices. These two varieties are remarkable with the property that they are the only minimal varieties of exponent 2.