The aim of this work is to obtain an improved theoretical understanding of
codimension one symmetry breaking bifurcations. In this introduction, we present
an overview of our aims and describe our main results. We assume the reader has
some familiarity with the basic concepts and terminology of equivariant bifurcation
theory as described in the text by Golubitsky, Stewart & Schaeffer [43]. Some
familiarity with Field & Richardson [37, 38] would also be helpful.
Throughout, we assume that F is a compact Lie group. Let V be a finite di-
mensional representation space for F over R or C. In the sequel, we write (V,T)
to signify that V is a representation space for F. Unless indicated to the contrary,
we suppose that (V, F) is a non-trivial representation. From the point of view
of generic T-equivariant bifurcation theory, we can assume, using either the center
manifold theorem or Liapunov Schmidt reduction (see for example [43]), that (V, T)
is irreducible over either E or C. We focus on three (exclusive) cases important
for applications: (A) (V, T) is an absolutely irreducible representation; (B) (V, T)
is the complexification of an absolutely irreducible representation; (C) (V, F) is ir-
reducible over C and is neither quaternionic nor reducible as a real representation.
We remark that case (A) is important in the study of static equivariant bifurcation
problems while cases (B,C) are of interest in the study of the equivariant Hopf
bifurcation. We refer to [44] and [24] for background and references.
In the papers by Field & Richardson [35, 36, 37, 38], an analysis was made of
the branching patterns for a large class of absolutely irreducible representations
associated to the Weyl groups W(Bk) and W(Ak) (we refer the reader to these
papers for details about the Weyl groups). One feature of this analysis was that,
on the basis of known information about the invariants and equivariants of the Weyl
groups, it was possible to obtain fairly complete information about the equivariant
bifurcation theory of absolutely irreducible representations of subgroups of the Weyl
groups without having to compute equivariants of degree higher than two or three.
Unfortunately, the study of generic bifurcations for general irreducible represen-
tations (V, r ) cannot be reduced to such simple computations. It is worthwhile
indicating some of the technical and theoretical difficulties that arise. To sim-
plify matters, suppose that (V,r) is absolutely irreducible and F is finite. Let
/ : V x R—V be a smooth (that is, C°°) T-equivariant map, where we take the
trivial action of F on the R-factor and the given action on the F-factor. Since
(V, F) is irreducible and non-trivial, we have /(0, A) = 0 for all A E R. Our interest
lies in the local study of generic bifurcations of the trivial solution x = 0. That is,
in the local study of the zero locus / _ 1 (0) near a bifurcation point. It is no loss
of generality to assume that bifurcations occur at A = 0. If we make the generic
hypothesis that there is a non-degenerate change of stability of the trivial solution
at A = 0, we may normalize and restrict attention to maps / of the form
(1) f(x,\) = \x + g(x,\),
where g(x, A) comprises terms of order at least 2 in x. Of course, without further
hypotheses on / , the problem of describing the local zero set of (1) is still far too
general. Our next step is to impose additional generic restrictions on / that allow us
to reduce to an elementary (though far from trivial) problem in real algebraic geom-
etry. We let P{V)r denote the R-algebra of real valued polynomial T-invariants on
V and Pr(V, V) denote the P(y)r-module of polynomial equivariants from V to V.
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