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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Graduated, and done with maths forever