Commutative and Noncommutative Invariant Theory

TL;DR

This survey explores classical and modern invariant theory, covering key theorems in commutative and noncommutative settings, including algebraic invariants, free algebras, and generic matrices, for a broad mathematical audience.

Contribution

It provides a comprehensive overview of invariant theory topics in both commutative and noncommutative contexts, highlighting recent developments and analogues of classical results.

Findings

Classical invariant theory results like the Endlichkeitssatz and Basissatz are extended to noncommutative algebras.

The Molien formula and Shephard-Todd-Chevalley theorem are adapted for free and relatively free algebras.

Invariant theory related to generic matrices is also discussed.

Abstract

The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only several topics from the classical invariant theory -- the finite generating (the Endlichkeitssatz) and the finite presenting (the Basissatz) of the algebra of invariants, the Molien formula for its Hilbert series and the Shephard-Todd-Chevalley theorem for the invariants of a finite group generated by pseudo-reflections. Then we give analogues of these results for free and relatively free associative and Lie algebras. Finally we deal with the algebra of generic matrices and the invariant theory related with it.