Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

A converse to the Grace–Walsh–Szegő theorem

PETTER BRÄNDÉNa1 and DAVID G. WAGNERa2

a1 Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. e-mail: pbranden@math.kth.se

a2 Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada N2L 3G1. e-mail: dgwagner@math.waterloo.ca

Abstract

We prove that the symmetrizer of a permutation group preserves stability if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace–Walsh–Szegő Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the Grace-like polynomials, introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

(Received September 25 2008)

(Revised January 13 2009)

Footnotes

† Partially supported by the Göran Gustafsson Foundation

‡ Supported by NSERC Discovery Grant OGP0105392