Free functions with symmetry

TL;DR

This paper generalizes Wolf's theorem, showing that rings of invariant free polynomials are isomorphic to free polynomial rings and extends this to a norm-preserving isomorphism in free functional calculus.

Contribution

It establishes a general theory linking invariant free polynomials to free polynomial rings and extends the isomorphism to function spaces on the row ball.

Findings

Ring of invariant free polynomials is isomorphic to a free polynomial ring.

Explicit constructions of invariant free polynomials using representation theory.

Generated functions and explicit formulas for bases in the abelian case.

Abstract

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf's theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.