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. > > >