Dear Forum,
I tried to verify, for the A5 group, whether the group of 60 matrices produced
by Repsn is homomorphic to A5.
They appear to be not
One example:
The 60 group elements g1, g2, ..., g60 are sort-listed using List(G).
3D representation matrices Mi are obtained using Repsn: Mi =
gi^IrreducibleAffordingRepresentation(selChar), for some fixed character
selChar.
It is expected that Mi * Mj = Mk, where k is chosen such that gi * gj = gk;
then we are dealing with a homomorphism.
For group A5 take i = 2, j = 3. Then:
g2 = (1,5,4),
g3 = (1,4,5),
g2 * g3 = g1 = () , the identity element. That is g2 and g3 are inverses of
each other.
Now from Repsn we obtain:
M2 =
[ [ -3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^4,
2/5*E(5)-2/5*E(5)^2-2/5*E(5)^3+2/5*E(5)^4,
2/5*E(5)+3/5*E(5)^2+3/5*E(5)^3+2/5*E(5)^4 ],
[ 0, 0, -1 ],
[ -2/5*E(5)+2/5*E(5)^2+2/5*E(5)^3-2/5*E(5)^4,
-2/5*E(5)-3/5*E(5)^2-3/5*E(5)^3-2/5*E(5)^4,
3/5*E(5)+2/5*E(5)^2+2/5*E(5)^3+3/5*E(5)^4 ] ]
M3 = [ [ -E(5)-E(5)^4, E(5)^2+E(5)^3, -2 ], [ E(5)^2+E(5)^3, -1, E(5)^2+E(5)^3
],
[ 1, -E(5)^2-E(5)^3, -E(5)^2-E(5)^3 ] ]
M1 = [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
So M2 * M3 should be equal to M1.
In Gap we find:
M2 * M3 =
[ [ 2/5*E(5)+8/5*E(5)^2+8/5*E(5)^3+2/5*E(5)^4,
-2/5*E(5)+7/5*E(5)^2+7/5*E(5)^3-2/5*E(5)^4,
11/5*E(5)+14/5*E(5)^2+14/5*E(5)^3+11/5*E(5)^4 ],
[ -1, E(5)^2+E(5)^3, E(5)^2+E(5)^3 ],
[ -2/5*E(5)+2/5*E(5)^2+2/5*E(5)^3-2/5*E(5)^4,
-3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^4,
4/5*E(5)+1/5*E(5)^2+1/5*E(5)^3+4/5*E(5)^4 ] ]
which is not equal to M1.
All products Mi * Mj for which neither Mi nor Mj are the identity appear
inconsistent with a homomorphism.
BTW: I know that for A5 "correct" representations have been found. However, I
need a reliable method to generate representations for automatic processing of
many groups.
Any advise is welcome,
Joris
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