Hello,
Please take a look at the theorem of equation 3.33 http://i.imgur.com/mrWey.jpg
I don't even understand the statement. What does it mean that a function "belongs" to an representation. Also, is the statement correct, or should there be some coefficients in front of each f_i?
I have uploaded this definition, which might be useful http://i.imgur.com/ic80p.jpg
Thanks in advance, Tony Bruguier
The best I can tell here is what it is saying. Given a finite groupOriginally Posted by tonybruguier
one can form the group algebra
which is the set
with pointwise scalar multplication/addition and the 'convolution' product
where
. Then, if
denotes the set of all equivalency classes of irreps of
(one can always put an inner product on a space for which any representation
is unitary--so we can a priori assume that
is a pre-Hilbert space and that the image of
lives inside the unitary group
). If we then choose a basis
(assume it's unitary) then for
where
by
(it turns out that
is itself an irrep in the class
). We can then consider that
functions
for
where
is the complex number sitting in the
entry of
. Thus for every
we have created
forms an (orthogonal) basis for
That's how I interpret it, but a lot of the notation is foreign to me...so take that with a grain of salt.
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