Noncommutative rational functions invariant under the action of a finite solvable group

TL;DR

This paper investigates the structure of invariant skew fields under finite solvable group actions, revealing finite generation and explicit generators, with applications to positivity certificates in noncommutative rational functions.

Contribution

It characterizes invariant skew fields for solvable groups, showing finite generation and explicit generators, and introduces positivity certificates for invariant rational functions.

Findings

Invariant skew fields are always finitely generated.

For abelian or well-behaved solvable groups, invariant skew fields have explicit generators.

Positivity certificates for invariant rational functions are established.

Abstract

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.