On Monday, August 20, 2012 10:36:54 AM UTC-7, David Joyner wrote:
>
> On Mon, Aug 20, 2012 at 12:58 PM, bsmile <[email protected] <javascript:>>
> wrote:
> >
> >
> > On Monday, August 20, 2012 11:30:17 AM UTC-5, David Joyner wrote:
> >>
> >> On Mon, Aug 20, 2012 at 11:46 AM, bsmile <[email protected]> wrote:
> >> > Thanks, Volker, This is by far I can get by studying the manual and
> help
> >> > document. But I still cannot get the corresponding irreducible matrix
> >> > representations, and to which group element each representation and
> >> > character correspond to. Can you help me further on it? I was told
> from
> >> > GAP
> >> > forum that SAGE is able to give matrix representation in real
> numbers, I
> >> > hope I can successfully achieve this with SAGE.
> >> >
> >>
> >> Do you mean this
> >> http://www.gap-system.org/Packages/repsn.html ?
> >>
> > Sorry, I don't know which package it is, but the command would be
> something
> > like the following in GAP to get irreducible matrix representations
> >
> > gap> gr := SymmetricGroup(4);;
> > gap> reps := IrreducibleRepresentations(gr);;
> > gap> List(gr,x->[x,x^reps[3]]);
>
> You might want to look at the documentation of that package.
>
> >
> >> >
> >> >
> >> >
> >> >
> >> > On Monday, August 20, 2012 12:17:50 AM UTC-5, Volker Braun wrote:
> >> >>
> >> >> sage: S7 = SymmetricGroup(7)
> >> >> sage: S7.character_table()
> >> >> [ 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1]
> >> >> [ 6 -4 2 0 3 -1 -1 0 -2 0 1 1 1 0 -1]
> >> >> [14 -6 2 -2 2 0 2 -1 0 0 0 -1 -1 1 0]
> >> >> [14 -4 2 0 -1 -1 -1 2 2 0 -1 -1 1 0 0]
> >> >> [15 -5 -1 3 3 1 -1 0 -1 -1 -1 0 0 0 1]
> >> >> [35 -5 -1 -1 -1 1 -1 -1 1 1 1 0 0 -1 0]
> >> >> [21 -1 1 3 -3 -1 1 0 1 -1 1 1 -1 0 0]
> >> >> [21 1 1 -3 -3 1 1 0 -1 -1 -1 1 1 0 0]
> >> >> [20 0 -4 0 2 0 2 2 0 0 0 0 0 0 -1]
> >> >> [35 5 -1 1 -1 -1 -1 -1 -1 1 -1 0 0 1 0]
> >> >> [14 4 2 0 -1 1 -1 2 -2 0 1 -1 -1 0 0]
> >> >> [15 5 -1 -3 3 -1 -1 0 1 -1 1 0 0 0 1]
> >> >> [14 6 2 2 2 0 2 -1 0 0 0 -1 1 -1 0]
> >> >> [ 6 4 2 0 3 1 -1 0 2 0 -1 1 -1 0 -1]
> >> >> [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
> >> >> sage: S7.irreducible_characters()[4]
> >> >> Character of Symmetric group of order 7! as a permutation group
> >> >> sage: list(_)
> >> >> [15, -5, -1, 3, 3, 1, -1, 0, -1, -1, -1, 0, 0, 0, 1]
> >> >>
> >> >>
> >> >> On Monday, August 20, 2012 12:51:04 AM UTC-4, bsmile wrote:
> >> >>>
> >> >>> I need to calculate very simple properties of the symmetric group,
> say
> >> >>> list group elements, their characters and irreducible
> representations.
> >> >>> Would
> >> >>> you please let me know the related commands to achieve these goals?
> >> >>> Thank
> >> >>> you very much!!
>
I don't know much about this aspect of Sage, but you can also try this:
sage: rep = SymmetricGroupRepresentation([3,1])
Now rep is the representation of S_4 (since 4=3+1) corresponding to the
partition [3,1]. We can compute find the matrix associated to each element
of S_4:
sage: rep([1,4,3,2])
[ 1 0 0]
[ 1 -1 1]
[ 0 0 1]
sage: REPS = [(x,rep(x)) for x in S4.list()]
This defines REPS to be a list of pairs, (elt of S_4, corresponding matrix).
Is that the sort of thing you want?
--
John
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