On Jan 14, 1:33 pm, "David Joyner" <[EMAIL PROTECTED]> wrote:
> On Jan 14, 2008 9:31 AM, [EMAIL PROTECTED] <[EMAIL PROTECTED]> wrote:
>
>
>
> > Does SAGE currently include any functionality for manipulating Lie
> > algebras? (I only need reductive Lie algebras, because I'm using them
> > to study compact Lie groups.) For instance, I'd like to be able to
> > manipulate irreducibles (encoded by their highest weights), so that I
> > can form a tensor product and decompose it into irreducibles.
>
> GAP has this:http://www.gap-system.org/Manuals/doc/htm/ref/CHAP061.htm
> AFAIK, nothing is wrapped.
>
> On Jan 14, 2008 1:09 PM, [EMAIL PROTECTED] <[EMAIL PROTECTED]> wrote:
>
>
>
> > I just looked at the GAP docs, and I don't quite see how to do what I
> > want (decomposing some random thing into irreducibles, or at least
> > getting the multiplicity of the trivial representation), but I'm sure
> > it's possible.
>
> I don't know how you want to specify "some random thing". As a
> tensor product of a list of irreducibles determined by their
> highest weight?
>
In my case, I start with a single representation (given by the list of
highest weights of its irreducible components, specified with
multiplicities) and I want the multiplicity of the trivial
representation in some tensor power of that. (I might generate that
initial representation by tensoring together some irreducibles, again
indicated by their highest weights.)
With Willem de Graaf's assistance, I have now Magma code that does
exactly this. (One has to be a bit careful, since there are two ways
to compute this. If you actually try to construct the tensor power and
then decompose it, you get hosed. Instead, you only "construct" it as
a list of highest weights, using some combinatorial method for
computing pairwise tensor product multiplicities.)
Ultimately I might need GAP after all, though, because I'm interested
in compact Lie groups which occasionally are not connected. In that
case, I'll need characters of finite groups too.
Anyway, I'm hoping to get together with Mike Hansen at MSRI next week
to discuss how things should look in SAGE. (I'll also be at SAGE Days
7 two weeks later.)
Kiran
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