Re: the concept of a representation of a group
- From: Igor Khavkine <igor.kh@xxxxxxxxx>
- Date: Tue, 23 Aug 2005 13:52:04 -0400
On 2005-08-23, m1ngleg02@xxxxxxxxxxxxxx <m1ngleg02@xxxxxxxxxxxxxx> wrote:
> I am not sure whether I am understanding it, when one goes from groups
> to representations of groups. A representation f:G->H is a homomorphism
> (f(uv)=f(u)*f(v)) from the group (G) into GL(n) (H). The image of the
> homomorphism f is a subalgebra of H. The kernel of f is a normal
> subgroup of G. Now we can construct the quotient group: G/ker(f) \cong
> image(f) (can we use this quotient group?). The homomorphism is
> one-to-one iff ker(f)=identity of G. So the dimension of im(f) can be
> at most the dimension of G. Do we take the dimension of H as the
> dimension of the representation? I am asking this, as the dimension
> of representations can be higher as the dimension of the group.
I'm not sure what you mean "dimension of the group". The only dimension
that figures here the dimension of the vector space on which the group
acts linearly. That is n in your notation. In general, a group is not a
vector space. And even if it were, the dimension of a representation
would be independent of that of the group.
I don't know what your background is, but it seems that you are trying
to somehow previously gained knowledge of abstract group theory to the
analysis of representations. In my experience, this is not particularly
useful. The simpleminded analysis of group representations has more to
do with linear algebra than anything else.
The central problem of group representation theory is (a) given a
representation of a group to decompose it into irreducible components,
(b) given a group to classify all of its irreducible representations.
For (a) the basic tool is Schur's lemma. Simply stated, it says that
commuting operators share invariant subspaces. It is a generalization of
the well known result from linear algebra about simultaneous
diagonalization of commuting operators. A representation of a group is a
collection of linear operators R_g, one for each element g of the group.
It might be hard to find invariant subspaces common to all R_g. But
Schur's lemma helps by saying that if you can find the invariant
subspaces of a single operator T such that TR_g = R_gT for all g, then
the same invariant subspaces are common to all of the R_g's.
For (b) one has to note that the matrix elements of the representation
operators R_g (for any representation) are functions on the group. These
functions form a vector space (element-wise addition and scalar
multiplication). This vector space also carries a natural representation
of the group (the regular representation). It turns out that the matrix
elements of *irreducible* representations of the group form an
orthonormal basis for this space (with respect to a natural inner
product). Moreover, once the regular representation is split into
irreducible components, each possible irreducible representation can be
found there. I think this is called the Peter-Weyl theorem.
The above outline works for finite and compact topological groups.
Various complications arise for more general cases.
Hope this helps.
Igor
.
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