The β-construction assigns to each complex representation
φ of the compact Lie group G a unique
element β(φ) in (G). For the details of this
construction the reader is referred to [1] or [5].
The purpose of the present paper is to determine some of the properties of
the element β(φ) in terms of the invariants of the
representation φ. More precisely, we consider the following
question. Let G be a simple, simply-connected compact Lie group and let
f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator
x = β(φ1), φ1 the fundamental representation of S3 , so that if
φ is a representation of
G,f*(φ) = n(φ)x, where
n(φ) is an integer depending on φ and
f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that
ch2, the component of the Chern character in dimension 4 takes
its values in H4(SG,Z)≅Z. Let u be a generator of
H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ)
an integer depending on φ.