  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.
