Dear Dmitrii, thank you! Problem solved. Joris ________________________________________ From: Dmitrii Pasechnik <dmitrii.pasech...@cs.ox.ac.uk> Sent: Saturday, March 9, 2019 08:55 To: Joris Vergeest Cc: forum@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 > > > _______________________________________________ > Forum mailing list > Forum@gap-system.org > https://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
