Free functions with symmetry

Cushing, D.; Pascoe, J.E.; Tully-Doyle, R.

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.

Citation

Cushing, D., Pascoe, J., & Tully-Doyle, R. (2018). Free functions with symmetry. Mathematische Zeitschrift, 289(3-4), 837-857. https://doi.org/10.1007/s00209-017-1977-x

• Journal Article Type: Article
• Acceptance Date: Sep 20, 2017
• Online Publication Date: Nov 2, 2017
• Publication Date: Aug 1, 2018
• Deposit Date: Nov 1, 2017
• Publicly Available Date: Nov 8, 2017
• Journal: Mathematische Zeitschrift
• Print ISSN: 0025-5874
• Electronic ISSN: 1432-1823
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 289
• Issue: 3-4
• Pages: 837-857
• DOI: https://doi.org/10.1007/s00209-017-1977-x