[GAP Forum] Can GAP compute indecomposble modules of group algebra KSn?

[Frank Lübeck]

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: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
\\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\