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 Post subject: Induced RepresentationsPosted: Thu, 17 May 2012 01:03:06 UTC
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It's an easy example (better to start off with something simple) but I'm trying to find
, where 1 is the trivial representation on .

So, I want to use Froebenius Reciprocity. Then
; doing a similar calculation with the character of the sign representation gives 0, so therefore
, where U is an irreducible 2-dimensional representation on and n is the multiplicity.
Now, the dimension of V is equal to ; but the dimension of the trivial representation is 1, and , so dimV = 3 and hence n=1, i.e. .
Of course, to show this explicitly I would have to calculate the restriction of U on , show that it is equal to and then go back to characters, but without doing that bit (im still working through the restriction of U) does it look like a semi-convincing argument?

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 Post subject: Re: Induced RepresentationsPosted: Thu, 17 May 2012 07:59:56 UTC
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peccavi_2006 wrote:
It's an easy example (better to start off with something simple) but I'm trying to find
, where 1 is the trivial representation on .

So, I want to use Froebenius Reciprocity. Then
; doing a similar calculation with the character of the sign representation gives 0, so therefore
, where U is an irreducible 2-dimensional representation on and n is the multiplicity.
Now, the dimension of V is equal to ; but the dimension of the trivial representation is 1, and , so dimV = 3 and hence n=1, i.e. .
Of course, to show this explicitly I would have to calculate the restriction of U on , show that it is equal to and then go back to characters, but without doing that bit (im still working through the restriction of U) does it look like a semi-convincing argument?

Well, you can do it that way. In this case, it is much faster to just work out directly from the formula (where is a right transversal of ) to show , since there is only three conjugacy classes of .

But for more complicated examples, yes, Frobenius reciprocity is a great tool to work out the decomposition.

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