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MathGroup Archive 2006

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Re: too many special linear matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68766] Re: too many special linear matrices
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Fri, 18 Aug 2006 03:12:12 -0400 (EDT)
  • References: <200608160736.DAA06175@smc.vnet.net> <ec1aiv$or3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Carl K. Woll wrote:

>
>
>p = 3;
>r = Reduce[Mod[o l - m n, p] == 1 && 0 <= l < p &&
>0 <= m < p && 0 <= n < p && 0 <= o < p,
>{l, m, n, o}, Integers]
>
>and the list of matrices:
>
>s = {{l,m},{n,o}} /. {ToRules[r]}
>
>Carl Woll
>Wolfram Research
>
>  
>
>>    
>>
>
>  
>
Carl K. Woll,
It always helps if you separate the Mathemaica code for cut and pasting 
to Mathematica!
Here's my effort to get a 24 element (SL[2,3] group) mass Klein-Gordon 
quantum equation.
It appears that the result doesn't have the right angular content for an 
SU(5) like isomorphic group.
If you use p=2 in this it will go much faster to test if it works for 
your version of Mathematica.
Oh, sorry , your code doesn't work at p=2, I just tried it. and got {}.
I got it to work for p=3,5. I haven't tried p=7.
Roger Bagula

Mathematica code:
(*SL[2, Prime] : from Carl K.Woll *)
p = 3;
r = Reduce[Mod[o l - m n, p] == 1 && 0 <= l < p &&
0 <= m < p && 0 <= n < p && 0 <= o < p,
{l, m, n, o}, Integers];
s0 = {{l, m}, {n, o}} /. {ToRules[r]};
(* Cayley multiplication table*)
MatrixForm[Table[Mod[s0[[n]].s0[[m]], p], {n, 1, Length[s]}, {m,
      1, Length[s]}]]
(*  exponential Wave Function  definitions space partition functions*)
phi[n_] := Exp[x[n]*s0[[n]]]
(* Jacobian from Paul Abbott*)
 jacobian = D[Table[phi[n], {n, 1, Length[
  s]}], {Table[x[n], {n, 1, Length[s]}]}];
a = Tr[ Transpose[jacobian] . jacobian, List] // Simplify
(* metric as a vector*)
guv = a.a
(* hyperplane full quantum number wave function of 24 group *)
PhiX = Exp[Sum[q[n]*x[n]*s0[[n]], {n, 1, Length[s]}]]
(* Klein - Gordon Mass gap*)
dd = Sum[D[PhiX, {x[n], 2}], {n, 1, Length[s]}]
Aut = FullSimplify[dd/PhiX]
FullSimplify[Det[Aut]]


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