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Free functions with symmetry

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

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D. Cushing

J.E. Pascoe

R. Tully-Doyle


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.


Cushing, D., Pascoe, J., & Tully-Doyle, R. (2018). Free functions with symmetry. Mathematische Zeitschrift, 289(3-4), 837-857.

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


Published Journal Article (Advance online version) (471 Kb)

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Copyright Statement
Advance online version © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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