[GAP Forum] Repsn matrices appear to be not homomorphic to group (for A5)

[Joris Vergeest]

Dear Dmitrii, thank you! Problem solved.   Joris

________________________________________
From: Dmitrii Pasechnik <dmitrii.pasechnik at cs.ox.ac.uk>
Sent: Saturday, March 9, 2019 08:55
To: Joris Vergeest
Cc: forum at gap-system.org
Subject: Re: [GAP Forum] Repsn matrices appear to be not homomorphic to group (for A5)

Dear Joris,
On Fri, Mar 08, 2019 at 09:28:10PM +0000, Joris Vergeest wrote:
>
> 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.

Different calls to IrreducibleAffordingRepresentation(selChar) might
produce different representations, I suppose this is exactly the
problem you see here.
Store it in a variable, e.g.

 rep:=IrreducibleAffordingRepresentation(selChar);
then compute Mi's as follows:

Mi:=gi^rep;

After this change everything should be working right.

Hope this helps,
Dmitrii

>
> 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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