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,
, 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?
"It's never crowded along the extra mile"
Graduated, and done with maths forever