On Feb 18, 2:09 am, javier <vengor...@gmail.com> wrote:
> Hi Dima,
>
> On Feb 18, 6:26 am, Dima Pasechnik <dimp...@gmail.com> wrote:
>
> > I am curious to know, how you are doing this. IMHO for this you need
> > to know
> > each irreducible representation explicitly --- but then you can just
> > stack up the right
> > number of copies of each irreducible.
>
> > Or you rather mean a weaker decomposition, into direct sums of
> > "homogeneous components", where the latter are isomorphic to direct
> > sums of copies of the same irreducible?
>
> The latest, indeed. The elements
> e_i = n_i / n * \sum_{g\in G] X_i(g)g
> provide a family of orthogonal idempotents (where X_i is the i-th
> irreducible character and n_i = X_i(1) the size of the corresponding
> irrep). These idempotents provide a decomposition
> A = \oplus e_i A e_i
> which is not the complete AW decomposition but might suffice for my
> needs. Ultimately what I want to do is test some property in PSU(3,4)
> (order 62400) that requires computing the determinant of an operator
> in the group algebra. Since the matrix is too big to fit in memory, my
> idea is to restrict the operator to each of the homogeneous components
> and then multiply the resulting determinants, so replacing computing
> one 62400x62400 determinant by around 20 determinants of 3000x3000
> matrices.
>exact determinants of size 3000x3000 still look quite challenging. For U(3,4) one knows all the irreducible representations explicitly, (most of them are exlicitly given here: http://brauer.maths.qmul.ac.uk/Atlas/v3/clas/U34/ and the remaining few are probably not so hard to get --- I guess Atlas (of Finite Groups Reps) people will be grateful if you contribute the missing ones) and they determine the value of the determinant you are interested in. So it might be much better to work directly with the irreducibles. Dmitrii > From the above decomposition I can now how many copies of each irrep > are there, just have to divide the dimension of the block by the > dimension of the irrep, thought that doesn't provide an explicit > description of the basis. > > Cheers > J -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
