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MathGroup Archive 2006

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Re: matrix substitution-> theta1 minimal Pisot quotient group

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67350] Re: matrix substitution-> theta1 minimal Pisot quotient group
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Mon, 19 Jun 2006 00:01:11 -0400 (EDT)
  • References: <e665nv$n43$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Polynomial modular quotiennt groups for the groups associate with
Pisot numbers seems an interesting technical number theory topic.

Theta1 Minimal Pisot quotient  group:
c = Table[x^n, {n, 0, 3}]
b = Table[PolynomialMod[c[[n]]*c[[m]], x^4 - x^3 - 1], {n, 1, 4}, {m, 1, 
4}];
MatrixForm[b]

4by4 Matrix solution found by Artur Jasinski in True Number egroup at yahoo:
(*e ->*)
a[0] = {{1, 0, 0, 0},
{0, 1, 0, 0},
{0, 0, 1, 0},
{0, 0, 0, 1}}
(*i ->*)
a[1] = {{0, 0, 0, 1},
{1, 0, 0, 0},
{0, 1, 0, 0},
{0, 0, 1, 1}}
(*j ->*)
a[2] = {{0, 0, 1, 1},
{0, 0, 0, 1},
{1, 0, 0, 0},
{0, 1, 1, 1}}
(*k ->*)
a[3] = {{0, 1, 1, 1},
{0, 0, 1, 1},
{0, 0, 0, 1},
{1, 1, 1, 1}}
aa = Table[a[n].a[m], {n, 0, 3}, {m, 0, 3}];
MatrixForm[aa]

It seems to check. I have no idea what use this manifold my have in 
physical terms:
in polynomial terms it is nearer to a Galois field based on x^4+1+1
than to a quaternion group.

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