Date: Tue, 01 Feb 1994 12:07:18 +0000
From: [Permission pending]
Subject: Re: Finding irreducible representations
To: rusin@math.niu.edu (Dave Rusin)
Hi Dave,
Thanks a lot for your reply. I had to dig for my prehistoric
notes to understand what you mean and now I can see that this is exactly
what I wanted..
You are right: my group is one of the crystallographic groups
and I have already a reducible representation - Wigner D(L,m,m') matrices.
Now, I am going to use your block-diagonalization scheme, i.e find the conjugacy
classes sums, compute their images V_i (subspaces in (2L+1)-dimensional space)
and find the new basis vectors for every subspace V_i.
these new bases will give me linear combinations of spherical harmonics which
belong to irreducible representations of my point group
(which is the final goal..)
>>Does anyone know about an algorithm finding all irreducible
>>representations contained in a given representation ? Computing
>>characters of a point group ? Transforming a set of matrices to
>>a block-diagonal form ?
>
> (...)
>
>As for block-diagonalization, recall that a basis for the center of
>the group algebra is the set of conjugacy-class sums. From these you
>can calculate a collection of minimal central idempotents in the
>group ring. Thus, each irreducible representation of the group
>corresponds to a certain sum of conjugacy class sums, the correspondence
>being that a given representation takes its sum to the identity matrix
>and the others to zero. Take such a sum, compute the matrix that
>corresponds to it, and look at its image V_i. These subspaces V_i
>are disjoint and span the original vector space V; each is preserved
>by the matrices in your set; hence in this new basis, the matrices
>will be in block-diagonal form. (Moreover, this is the finest such
>decomposition possible unless one of the irreducible constituents
>of the original representation shows up more than once.)
>
>dave
>
>PS - I take it your group is one of the crystallographic groups.
>If it is of low dimension, then surely its representations can be
>found even in the applied literature. For a general group you might
>want to turn to one of the group-theory-specific packages such as
>Cayley, GAP, or Magma.
>
there is still some coding to be done : I'll send you the result if you're
interested.
[sig deleted -- djr]