From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: One question of non-normal subgroup
Date: 21 Dec 1999 13:31:03 GMT
Newsgroups: sci.math
Keywords: components of induced representations
In article <83ne1p$bul$1@nnrp1.deja.com>,
jrbao@my-deja.com writes:
>There are very discussions of non-normal subgroup
>in the group books. That's the reason reason I
>have to post this question. I hope someone
>can help me. I really need the answer.
>
>Suppose non-Abelian group G has a non-normal
>subgroup K (K can be Abelian or non-Abelian).
>Suppose Y is the representation of G induced by
>trivial 1-dimensional representation of K. (As I
>know, Y is called the principal induced
>representation). Normally, Y is a reducible
>representation of G, so we can suppose:
>Y=1+X1+X2+...Xn. Here X1,X2, ... Xn are some
>irreducible representations of G. My question is:
>( I guess) At least one of X1, X2, ... Xn is > or
>= 2 dimension. In other words, I want to know the
>EXISTENCE of Xi, which is > or = 2 dimension. How
>to proof it? Please help me.
This is not too hard if you know a bit of character theory.
Let \chi be the character of the induced representation Y.
Then, using the standard product < , > defined on characters,
< \chi, \chi > is equal to 1 + d_1^2 + ... + d_n^2, where d_i
is the degree of the irreducible constituent X_i of Y.
So, if all of the X_i had dimension 1, then we would have
< \chi, \chi > = degree(\chi) = |G:K|, the index of K in G.
On the other hand, by the Frobenius Reciprocity Theorem,
< \chi, \chi > is equal to
< 1_K, \chi_K > = (sum _ {k in K} \chi (k))/|K|
(\chi is a real values cahracter), which by the (Burnside?)
fixed point theorem is equal to the number of orbits of K
in the permutation representation of G on the cosets of K.
Now this number of orbits is equal to |G:K| if and only if
K is a normal subgroup of G. Since you are assuming that K is
not a normal subgroup, it follows that the X_i cannot all have
dimension 1.
Derek Holt.