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

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.

References in corpus (6)

• Noncommutative symmetric functions
• The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)
• Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables
• Unramified cohomology of degree 3 and Noether's problem
• Regular and positive noncommutative rational functions
• Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables