[GAP Forum] Using repsn package on A5

[Vahid Dabbaghian]

Dear Krishna,

The answer to your question is yes. You may get different representations for different trials even for solvable groups. For solvable groups, the only case that you will get the same answer is when it doesn't need to extend a representation. 

I hope it helps.

Regards
Vahid

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----- Original Message -----
From: "krishna mohan" <trebauchet1986 at yahoo.co.in>
To: "gap forum" <forum at gap-system.org>
Sent: Sunday, October 10, 2010 5:29:01 AM
Subject: [GAP Forum] Using repsn package on A5

Hi....

I do the following:

gap> g:=AllSmallGroups(60)[5];
Group([ (1,2,3,4,5), (1,2,3) ])

gap> StructureDescription(g);
"A5"

gap> LoadPackage("repsn");
true

gap> chi:=Irr(g);;

gap> rep1:=IrreducibleAffordingRepresentation(chi[4]);
[ (1,2)(3,4), (1,4)(2,3), (1,3)(4,5) ] -> 
[ [ [ E(3), -1/2*E(3)^2, -1/2*E(3)+1/2*E(3)^2, 1/2*E(3)^2 ], [ 1, -E(3), E(3)^2, 
-E(3)^2 ], [ E(3), 1/2*E(3), 1/2*E(3)^2, -1/2*E(3) ], 

      [ E(3), -E(3)-1/2*E(3)^2, -3/2*E(3)-1/2*E(3)^2, -1/2*E(3)^2 ] ], 
  [ [ E(3)^2, 1/2*E(3), E(3)-1/2*E(3)^2, -1/2*E(3) ], [ 0, -1/2*E(3), 
-E(3)-1/2*E(3)^2, -1/2*E(3)-E(3)^2 ], 

      [ E(3)^2, -1/2*E(3)^2, 1/2*E(3)-1/2*E(3)^2, 1/2*E(3)^2 ], [ 1, 
-E(3)-1/2*E(3)^2, -1/2, -1/2*E(3)^2 ] ], 

  [ [ 1, -1/2*E(3)+1/2*E(3)^2, 1/2*E(3)+E(3)^2, 1/2*E(3)-1/2*E(3)^2 ], [ 0, -1, 
-1, -E(3) ], [ 0, 0, 0, E(3) ], [ 0, 0, E(3)^2, 0 ] ] ]

gap> rep2:=IrreducibleAffordingRepresentation(chi[4]);
[ (1,2)(3,4), (1,4)(2,3), (1,3)(4,5) ] -> 
[ [ [ 1/2*E(3), -1/2*E(3)-E(3)^2, E(3)^2, 1/2*E(3)^2 ], [ 0, -E(3)^2, E(3)^2, 
E(3) ], [ E(3)+1/2*E(3)^2, -3/2*E(3)^2, E(3)^2, 1/2*E(3)-1/2*E(3)^2 ], 

      [ 1/2*E(3)-1/2*E(3)^2, -1/2*E(3)+1/2*E(3)^2, 0, -1/2*E(3) ] ], 
  [ [ -1/2, 1/2, E(3), -1/2*E(3)-E(3)^2 ], [ 1/2*E(3)+E(3)^2, 1/2*E(3), 1, 
-1/2*E(3)^2 ], [ 1/2*E(3)+E(3)^2, -1/2*E(3)-E(3)^2, 0, -1/2*E(3)^2 ], 

      [ -1/2*E(3)-E(3)^2, -3/2*E(3), E(3), -E(3)-1/2*E(3)^2 ] ], 
  [ [ -1/2*E(3), 1/2*E(3)+E(3)^2, 0, E(3)+1/2*E(3)^2 ], [ 1/2*E(3), -1/2*E(3), 
E(3), -1/2*E(3)^2 ], [ -1/2*E(3)^2, -E(3)+1/2*E(3)^2, E(3), 1/2 ], 

      [ E(3)^2, -1, 1, 0 ] ] ]


As one can see, the representation matrices are different in the two instances. 
I tried this with a couple of other groups, but always got the same 
representation matrices.

On going through the repsn manual, it seemed to me that this must be due to the 
fact that A5 is not solvable.

So, if the group is solvable, then does the program always give the same 
representation matrices?
Or is it that the matrices generated may be different for different trials even 
for a solvable group?

Krishnamohan P


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