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coordmaker.mw

The following program will produce the coordinates of the representation of group S4, Conjugacy Class 1, Example 1.

> restart;

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> with(LinearAlgebra):

Warning, the name GramSchmidt has been rebound

> w:=1;

Z is a matrix which will be used to compute the equilibrium stress later.

> z:=matrix([[2,0,0],[0,2,0],[0,0,2]]);

ng1, ng2, and ng3 are each the matricies for cable 1, cable 2, and the strut, respectively.

> ng1:=matrix([[0,1,0],[-1,0,0],[0,0,1]]);

> ng2:=matrix([[0,0,-1],[-1,0,0],[0,1,0]]);

> ng3:=matrix([[1,0,0],[0,-1,0],[0,0,-1]]);

The following lines compute (1-g1)+(1-inverse(g1))+w(1-g2)+w(1-inverse(g2))+t(1-g3)+t(1-inverse(g3)), where g1, g2, and g3 are represented by ng1, ng2, and ng3. (g1, g2, and g3 are used in thw code as the matricies that generate all of S4.) w is the stress ratio between cable 1 and cable 2, and t is the resultant equilibrium stress on the strut.

> a:=matrix(evalm(z-ng1-transpose(ng1)));

> tb:=matrix(evalm(z-ng2-transpose(ng2)));

> wmat:=matrix([[w,0,0],[0,w,0],[0,0,w]]);

> b:=evalm(tb.wmat);

> td:=evalm(z-ng3-transpose(ng3));

> tmat:=matrix([[t,0,0],[0,t,0],[0,0,t]]);

> d:=evalm(td.tmat);

> evalm(a+b+d);

Finding the determinant of the above matrix will produce the equilibrium stress for t.

> det(%);

> solve(%=0,t);

Taking t to be the least negative value of the above choices:

> t:=-1/8*(2+6*w+3*w^2-(4+8*w+8*w^2+12*w^3+9*w^4)^(1/2))/(1+w);

> tmat:=matrix([[t,0,0],[0,t,0],[0,0,t]]);

> d:=evalm(td.tmat);

> evalm(a+b+d);

The kernel of this matrix will then produce the seed for the representation, which will produce each point in the representation.

> kernel(%);

${vector([-9/20-1/20*41^(1/2), 1, 4/5+1/5*41^(1/2)])}$

> tk:=%[1];

> k:=matrix([[tk[1]],[tk[2]],[tk[3]]]);

g1, g2, and g3 are the matricies which will generate S4.

> g1:=matrix([[0,-1,0],[-1,0,0],[0,0,-1]]);

> g2:=matrix([[0,0,1],[1,0,0],[0,1,0]]);

> g3:=matrix([[0,0,-1],[0,1,0],[1,0,0]]);

Multiplying be each of the following will produce the coordinates for each vertex of the representation.

> evalf(transpose(evalm(g1.k)));
evalf(transpose(evalm(g2.k)));
evalf(transpose(evalm(g3.k)));
evalf(transpose(evalm(g1.g2.k)));
evalf(transpose(evalm(g1.g3.k)));
evalf(transpose(evalm(g2.g2.k)));
evalf(transpose(evalm(g2.g3.k)));
evalf(transpose(evalm(g3.g3.k)));
evalf(transpose(evalm(g1.g2.g2.k)));
evalf(transpose(evalm(g1.g2.g3.k)));
evalf(transpose(evalm(g1.g3.g3.k)));
evalf(transpose(evalm(g2.g2.g3.k)));
evalf(transpose(evalm(g2.g3.g3.k)));
evalf(transpose(evalm(g3.g3.g3.k)));
evalf(transpose(evalm(g1.g2.g2.g3.k)));
evalf(transpose(evalm(g1.g2.g3.g3.k)));
evalf(transpose(evalm(g1.g3.g3.g3.k)));
evalf(transpose(evalm(g2.g2.g3.g3.k)));
evalf(transpose(evalm(g2.g3.g3.g3.k)));
evalf(transpose(evalm(g1.g2.g2.g3.g3.k)));
evalf(transpose(evalm(g1.g2.g3.g3.g3.k)));
evalf(transpose(evalm(g2.g2.g3.g3.g3.k)));
evalf(transpose(evalm(g1.g2.g2.g3.g3.g3.k)));