TY - JOUR

T1 - On the zero set of G-equivariant maps

AU - Buono, P. L.

AU - Helmer, M.

AU - Lamb, J. S.W.

PY - 2009/11

Y1 - 2009/11

N2 - Let G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: V W. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W. We define an index s() for isotropy subgroups of G which is the difference of the dimension of the fixed point subspace of in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup satisfying s() > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup of dimension s(). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G-reversible equivariant vector fields.

AB - Let G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: V W. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W. We define an index s() for isotropy subgroups of G which is the difference of the dimension of the fixed point subspace of in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup satisfying s() > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup of dimension s(). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G-reversible equivariant vector fields.

UR - http://www.scopus.com/inward/record.url?scp=70450237194&partnerID=8YFLogxK

U2 - 10.1017/S0305004109990120

DO - 10.1017/S0305004109990120

M3 - Journal article

AN - SCOPUS:70450237194

VL - 147

SP - 735

EP - 755

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -