a1 Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. e-mail: email@example.com
a2 Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada N2L 3G1. e-mail: firstname.lastname@example.org
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)
† Partially supported by the Göran Gustafsson Foundation
‡ Supported by NSERC Discovery Grant OGP0105392