Re: Induced representations theorem..

James wrote:

Dear all,

Could someone point to me a reference for the proof of the following
theorem :

Suppose G is a group and H is a subgroup of index 2. Let f be an
irreducible representation of H. Let t be an automorphism of H coming
from G - H (H is normal so conjugation of H is an automorphism). t acts
on the irreducible representations of H by t*g(x) = g(t(x)) = g(txt^(-1)).
Then,

Theorem : Ind_H^G (f) is an irreducible representation if t*f is not
isomorphic to f, and it is the direct sum of two irreducible
representations if t*f is isomorphic to f.

At least if G is finite this follows fairly easily from character theory.
Let F be the induced character of h.
Then
chi_F(x) = chi_f(x) + chi_f(txt^{-1}) = chi_f(x) + chi_{t*f}(x).
Recall the formula for the intertwining number: if u,v are characters of G,
I(u,v) = 1/|G| sum_{x in G} chi_u(x) chi_v(x^{-1}).
It follows from this that
I_G(F,F) = 1/2 I_H(f + t*f, f + t*f).
If t*f = f this gives I_G(F,F) = 4/2 = 2, ie F splits into 2 simple parts;
if t*f <> f then I_G(F,F) = 2/2 = 1, ie F is irreducible.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.