Re: Induced representations theorem..
- From: mareg@xxxxxxxxxxxxxxxxxxxxxxxx ()
- Date: Sat, 25 Feb 2006 12:07:35 +0000 (UTC)
In article <7203553.1140797494040.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
James <greenthorn@xxxxxxxxx> writes:
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.
Thank you for your assistance,
For ordinary (complex) representations, you can deduce this easily from the
Frobenius reciprocity theorem. If chi is the character of f, then
< chi^G, chi^G >_G = < chi, (chi^G)_H >_H,
It is easy to see from the definition of the induced representation that
Ind_H^G (f) restricted to H is the direct sum of the representations f
and t*f of H, so the right hand side of the equation is 2 or 1 according to
whether or not f and t*f are isomorphic.
I think it is true representations over all fields, but I would need to think a
little more about that. It is clear however that if f and t*f are not
isomorphic then Ind_H^G (f) is irreducible, because in that case the
restriction to H is the sum of two nonisomorphic H-modules, neither of which
is fixed by elements outside of H.
Derek Holt.
.
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- Induced representations theorem..
- From: James
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