James Pascoe

Invariant free polynomials

Given a group G which acts on d-dimensional space, it is natural to study the ring of polynomials which are invariant under the action of G, the so-called ring of invariants. Two key classical results in invariant theory are the fact that the ring of invariants is finitely generated, and the Chevalley-Shepard-Todd theorem which characterizes their structure. Similarly, one can study the noncommutative ring of invariant free polynomials, which in turn have their own rich theory, starting in 1936, when 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, which corresponds to a free Chevalley-Shepard-Todd type theorem. In recent work with David Cushing and Ryan Tully-Doyle, we showed 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, and in fact that this isomorphism is isometric when we treat the ring of invariant free polynomials appropriately as a function space. In this talk, I will describe the classical situation and our recent work in terms of several concrete examples: 1. Even functions in two variables (that is, functions which satisfy f(X, Y ) = f (−X, −Y ),) 2. Cyclically symmetric functions in three variables (functions satisfying f(X,Y,Z) = f(Y,Z,X),) 3. Symmetric functions in two variables (functions satisfying f(X,Y)=f(Y,X).)