Noncommutative Noether's Problem vs Classical Noether's Problem

TL;DR

This paper explores the Noncommutative Noether's Problem, establishing conditions under which invariants of Weyl fields are rational, and provides solutions for specific finite groups, with implications for invariant differential operators.

Contribution

It proves the positive solution of the Noncommutative Noether's Problem for certain finite groups based on the rationality of quotient varieties, and introduces an effective algorithm for Weyl generator computation.

Findings

Positive solutions for pseudo-reflection groups

Solutions for alternating groups (n=3,4,5)

Algorithm for Weyl generators

Abstract

We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and any field k of characteristic zero. In particular, this gives positive solution for all pseudo-reflections groups, for the alternating groups (n = 3,4,5) and for any finite group when n = 3. Alternative proofs are given for the complex field and for all pseudo-reflections groups. In the later case an effective algorithm of finding the Weyl generators is described. We also study birational equivalence for the rings of invariant differential operators on complex affine irreducible varieties.