characters of the symmetric group

[amps]

I see that there is a function to compute the character table of the
symmetric group, but is there one where you input two partitions and
it outputs the value of the character indexed by the first partition
evaluated at the second? I have been searching for some time and
can't find the answer.

Thanks.

[David Joyner]

I don't know either and would be interested as well.
Do you know how to do this in GAP?

>
> Thanks.
>
> >
>

[Jason Bandlow]

One way is to use symmetric function theory:

sage: s = SFASchur(QQ); p = SFAPower(QQ)
sage: s(p([2,2])).coefficient([3,1])
-1

This says that the value of the irreducible character indexed by the
partition (3,1) is -1 when evaluated on a conjugacy class of size (2,2).

Cheers,
Jason

[David Joyner]

This is cool - thanks!


>
> Cheers,
> Jason
>
>
> >
>

[amps]

Yes, I know about the symmetric functions way, but I thought that
there would be a built in function that does this.

I guess the easiest thing to do would be to define a function that
uses the symmetric functions to compute this. I guess this can be
submitted to sage so such a function exists, but I don't know how
computationally efficient it would be (my guess is that it won't be
very efficient).



On 13 May, 21:32, David Joyner <wdjoy...@gmail.com> wrote:

> On Wed, May 13, 2009 at 10:33 AM, Jason Bandlow <jband...@gmail.com> wrote:
>
> > David Joyner wrote:

> >> On Tue, May 12, 2009 at 6:43 PM, amps <arat...@gmail.com> wrote:
> >>> I see that there is a function to compute the character table of the

> >>>symmetricgroup, but is there one where you input two partitions and

> >>> it outputs the value of the character indexed by the first partition
> >>> evaluated at the second?  I have been searching for some time and
> >>> can't find the answer.
>
> >> I don't know either and would be interested as well.
> >> Do you know how to do this in GAP?
>

> > One way is to usesymmetricfunction theory:

[Franco Saliola]

I posted a patch about a month ago that constructs irreducible matrix
representations of the symmetric group.

http://trac.sagemath.org/sage_trac/ticket/5878

So you could also do it with that functionality (once the patch is
reviewed!). This is not the fastest way to do what you want, though,
since it computes the entire matrix, and you're interested only in its
trace. I think that it would not be too hard to adapted the code to
return only the trace without computing the entire matrix.

Take care,
Franco

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