[GAP Forum] Using repsn package on A5
krishna mohan
trebauchet1986 at yahoo.co.in
Wed Oct 13 03:05:39 BST 2010
Thanks Vahid,
Just one more quick question...so is there a way to know, for a particular
group and character, whether repsn is extending a representation or not (i.e
for the command IrreucibleAffordingRepresentation)?
Krishna
________________________________
From: Vahid Dabbaghian <vdabbagh at sfu.ca>
To: krishna mohan <trebauchet1986 at yahoo.co.in>
Cc: gap forum <forum at gap-system.org>
Sent: Tue, 12 October, 2010 9:01:21 AM
Subject: Re: [GAP Forum] Using repsn package on A5
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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