2.1. T-sets and isotropy types. Let T be a group and X be a T-set. If x G T,
then T x denotes the T-orbit of x and Tx denotes the isotropy subgroup of T at
x. Let (Tx) denote the conjugacy class of the subgroup Tx in T. We say that (Tx)
is the isotropy type or orbit type of x. Let 0(X, T) denote the set of isotropy types
for the T-set X. We frequently abbreviate 0(X,T) to O if X and T are implicit
from the context. For x G T, let L(X) denote the isotropy type of x. If r G 0(X1 T),
let XT = {x G X | t(x) = r} be the set of points of isotropy type r. Let X
the fixed point set for the action of T on X. We define the usual partial order for
bifurcation theory on 0(X, T) by "r /x if there exists H G T, K G // such that
2.2. Representations. Suppose that V is a compact Lie group. Let F denote
either the field of real numbers IR or the field of complex numbers C. Let V be a
(finite dimensional) vector space over F. We say that (V, V) is an F-representation
if T acts on V as a group of F-linear transformations. We refer to V as a real or
complex representation space for V according as to whether F = R or F = C.
Let V be a nontrivial (finite-dimensional) real representation space for V. Av-
eraging a positive definite inner product for V over Y using Haar measure we may
suppose that V has a positive definite T-invariant inner product (,) with asso-
ciated norm | |. In particular, we may regard Y as acting on V by orthogonal
Suppose instead that V is a nontrivial (finite-dimensional) complex representa-
tion space for I\ As above, we may assume that V has a positive definite T-invariant
hermitian inner product , . In particular, we may regard (V, Y) as a unitary rep-
resentation. If we let (,) denote the real part of , , then (,) is a T-invariant
inner product on V. Let Jy denote the complex structure on V defined by scalar
multiplication by i. That is, Jy(v) iv, all v G V. If we regard S1 C C as the
group of complex numbers of unit modulus, we may extend the action of T on V
to an action of V x S1 on V. The resulting representation (V, T x S'1) is complex
and both , and (,) will be F x
-invariant. In particular, we have
(4) (v, Jvv) = 0, (tiE V)
In the sequel, we always assume that the T-invariant inner product on V satisfies (4).
We always use the notation S1 for the group of complex numbers of unit modulus
and take the 51-action on V defined by scalar multiplication.
Suppose that (V, T) is a real representation. Let Lr(V, V) denote the linear space
of all T-equivariant E-linear maps from V to V.
Definition 2.2.1. Let (V, T) be a nontrivial irreducible T-representation. We say
(1) (V, T) is absolutely irreducible if £,r(V, V) ^ R.
(2) (V, T) is irreducible of complex type if Lr(V, V) = C.
(3) (V, r) is irreducible of quaternionic type if Lr(V, V) = H (the quaternions).
Remarks 2.2.2. (1) By Frobenius' Theorem [51, 7.7, p.430], every real irreducible
representation (V, V) is either absolutely irreducible, irreducible of complex type
or irreducible of quaternionic type. In this work, we shall not explicitly consider
examples where (V,r) is irreducible of quaternionic type. (See [19, §3] for an
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