Rationality of Hilbert series in noncommutative invariant theory

TL;DR

This paper extends the understanding of Hilbert series rationality to noncommutative invariant algebras, especially those related to polynomial identities, beyond the classical commutative and finitely generated cases.

Contribution

It proves the rationality of Hilbert series for noncommutative, non-finitely generated invariant algebras arising from linear group actions on tensor algebra factors.

Findings

Rational Hilbert series established for specific noncommutative invariants

Application to algebras related to polynomial identities

Extension of classical rationality results to broader algebra classes

Abstract

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.