On Thu, Nov 08, 2007 at 01:15:51PM +0800, 董井成 wrote:
> Can GAP compute indecomposble modules of group algebra KSn? For example
> KS3,where K is a finite field with 2 elements.
Dear Dong, dear Forum,
For small n you can do this as follows:
gap> # create the group
gap> s3 := SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> # and the irreducible representation over GF(2)
gap> reps := IrreducibleRepresentations(s3, GF(2));
[ [ (1,2,3), (1,2) ] -> [ <an immutable 1x1 matrix over GF2>,
<an immutable 1x1 matrix over GF2> ],
[ (1,2,3), (1,2) ] -> [ <an immutable 2x2 matrix over GF2>,
<an immutable 2x2 matrix over GF2> ] ]
gap> # there are 2 of them
gap> Length(reps);
2
gap> # compute image of (2,3) in s3 under second representation
gap> Image(reps[2], (2,3));
<an immutable 2x2 matrix over GF2>
gap> # for looking at the entries do
gap> Display(last);
. 1
1 .
Please, use the GAP help system to find out more about the mentioned commands.
For larger n it would be more complicated, but one could use matrices for
the characteristic 0 Specht modules and find the irreducibles over a finite
field with the 'MeatAxe'.
To get larger dimensional Specht modules into GAP some programs available
from http://www.math.rwth-aachen.de/~Frank.Luebeck/software.html can be
useful. For MeatAxe functionality in GAP see '?the meataxe' in GAPs help
system. (In the small n cases as above the MeatAxe is applied to the regular
representation.)
Best regards,
Frank
--
/// Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64, ///
\\\ 52062 Aachen, Germany \\\
/// E-mail: [EMAIL PROTECTED] ///
\\\ WWW: http://www.math.rwth-aachen.de/~Frank.Luebeck/ \\\
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