REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
*** @ already defined as operator
% test symmetry package
% implementation of theory of linear representations
% for small groups
availablegroups();
{z2,k4,d3,d4,d5,d6,c3,c4,c5,c6,s4,a4}
printgroup(D4);
{id,rd4,rot2d4,rot3d4,sd4,srd4,sr2d4,sr3d4}
generators(D4);
{rd4,sd4}
charactertable(D4);
{{d4,{{id},1},{{rd4,rot3d4},1},{{rot2d4},1},{{sd4,sr2d4},1},{{sr3d4,srd4},1}},
{d4,{{id},1},{{rd4,rot3d4},1},{{rot2d4},1},{{sd4,sr2d4},-1},{{sr3d4,srd4},-1}},
{d4,{{id},1},{{rd4,rot3d4},-1},{{rot2d4},1},{{sd4,sr2d4},1},{{sr3d4,srd4},-1}},
{d4,{{id},1},{{rd4,rot3d4},-1},{{rot2d4},1},{{sd4,sr2d4},-1},{{sr3d4,srd4},1}},
{d4,{{id},2},{{rd4,rot3d4},0},{{rot2d4},-2},{{sd4,sr2d4},0},{{sr3d4,srd4},0}}}
characternr(D4,1);
{d4,{{id},1},{{rd4,rot3d4},1},{{rot2d4},1},{{sd4,sr2d4},1},{{sr3d4,srd4},1}}
characternr(D4,2);
{d4,{{id},1},{{rd4,rot3d4},1},{{rot2d4},1},{{sd4,sr2d4},-1},{{sr3d4,srd4},-1}}
characternr(D4,3);
{d4,{{id},1},{{rd4,rot3d4},-1},{{rot2d4},1},{{sd4,sr2d4},1},{{sr3d4,srd4},-1}}
characternr(D4,4);
{d4,{{id},1},{{rd4,rot3d4},-1},{{rot2d4},1},{{sd4,sr2d4},-1},{{sr3d4,srd4},1}}
characternr(D4,5);
{d4,{{id},2},{{rd4,rot3d4},0},{{rot2d4},-2},{{sd4,sr2d4},0},{{sr3d4,srd4},0}}
irreduciblereptable(D4);
{{d4,
id=
[1]
,
rd4=
[1]
,
rot2d4=
[1]
,
rot3d4=
[1]
,
sd4=
[1]
,
srd4=
[1]
,
sr2d4=
[1]
,
sr3d4=
[1]
},
{d4,
id=
[1]
,
rd4=
[1]
,
rot2d4=
[1]
,
rot3d4=
[1]
,
sd4=
[ - 1]
,
srd4=
[ - 1]
,
sr2d4=
[ - 1]
,
sr3d4=
[ - 1]
},
{d4,
id=
[1]
,
rd4=
[ - 1]
,
rot2d4=
[1]
,
rot3d4=
[ - 1]
,
sd4=
[1]
,
srd4=
[ - 1]
,
sr2d4=
[1]
,
sr3d4=
[ - 1]
},
{d4,
id=
[1]
,
rd4=
[ - 1]
,
rot2d4=
[1]
,
rot3d4=
[ - 1]
,
sd4=
[ - 1]
,
srd4=
[1]
,
sr2d4=
[ - 1]
,
sr3d4=
[1]
},
{d4,
id=
[1 0]
[ ]
[0 1]
,
rd4=
[ 0 1]
[ ]
[ - 1 0]
,
rot2d4=
[ - 1 0 ]
[ ]
[ 0 - 1]
,
rot3d4=
[0 - 1]
[ ]
[1 0 ]
,
sd4=
[1 0 ]
[ ]
[0 - 1]
,
srd4=
[0 1]
[ ]
[1 0]
,
sr2d4=
[ - 1 0]
[ ]
[ 0 1]
,
sr3d4=
[ 0 - 1]
[ ]
[ - 1 0 ]
}}
irreduciblerepnr(D4,1);
{d4,
id=
[1]
,
rd4=
[1]
,
rot2d4=
[1]
,
rot3d4=
[1]
,
sd4=
[1]
,
srd4=
[1]
,
sr2d4=
[1]
,
sr3d4=
[1]
}
irreduciblerepnr(D4,2);
{d4,
id=
[1]
,
rd4=
[1]
,
rot2d4=
[1]
,
rot3d4=
[1]
,
sd4=
[ - 1]
,
srd4=
[ - 1]
,
sr2d4=
[ - 1]
,
sr3d4=
[ - 1]
}
irreduciblerepnr(D4,3);
{d4,
id=
[1]
,
rd4=
[ - 1]
,
rot2d4=
[1]
,
rot3d4=
[ - 1]
,
sd4=
[1]
,
srd4=
[ - 1]
,
sr2d4=
[1]
,
sr3d4=
[ - 1]
}
irreduciblerepnr(D4,4);
{d4,
id=
[1]
,
rd4=
[ - 1]
,
rot2d4=
[1]
,
rot3d4=
[ - 1]
,
sd4=
[ - 1]
,
srd4=
[1]
,
sr2d4=
[ - 1]
,
sr3d4=
[1]
}
irreduciblerepnr(D4,5);
{d4,
id=
[1 0]
[ ]
[0 1]
,
rd4=
[ 0 1]
[ ]
[ - 1 0]
,
rot2d4=
[ - 1 0 ]
[ ]
[ 0 - 1]
,
rot3d4=
[0 - 1]
[ ]
[1 0 ]
,
sd4=
[1 0 ]
[ ]
[0 - 1]
,
srd4=
[0 1]
[ ]
[1 0]
,
sr2d4=
[ - 1 0]
[ ]
[ 0 1]
,
sr3d4=
[ 0 - 1]
[ ]
[ - 1 0 ]
}
rr:=mat((1,0,0,0,0),
(0,0,1,0,0),
(0,0,0,1,0),
(0,0,0,0,1),
(0,1,0,0,0));
[1 0 0 0 0]
[ ]
[0 0 1 0 0]
[ ]
rr := [0 0 0 1 0]
[ ]
[0 0 0 0 1]
[ ]
[0 1 0 0 0]
sp:=mat((1,0,0,0,0),
(0,0,1,0,0),
(0,1,0,0,0),
(0,0,0,0,1),
(0,0,0,1,0));
[1 0 0 0 0]
[ ]
[0 0 1 0 0]
[ ]
sp := [0 1 0 0 0]
[ ]
[0 0 0 0 1]
[ ]
[0 0 0 1 0]
rep:={D4,rD4=rr,sD4=sp};
rep := {d4,
rd4=
[1 0 0 0 0]
[ ]
[0 0 1 0 0]
[ ]
[0 0 0 1 0]
[ ]
[0 0 0 0 1]
[ ]
[0 1 0 0 0]
,
sd4=
[1 0 0 0 0]
[ ]
[0 0 1 0 0]
[ ]
[0 1 0 0 0]
[ ]
[0 0 0 0 1]
[ ]
[0 0 0 1 0]
}
canonicaldecomposition(rep);
teta=2*teta1 + teta4 + teta5
character(rep);
{d4,{{id},5},{{rd4,rot3d4},1},{{rot2d4},1},{{sd4,sr2d4},1},{{sr3d4,srd4},3}}
symmetrybasis(rep,1);
[1 0 ]
[ ]
[ 1 ]
[0 ---]
[ 2 ]
[ ]
[ 1 ]
[0 ---]
[ 2 ]
[ ]
[ 1 ]
[0 ---]
[ 2 ]
[ ]
[ 1 ]
[0 ---]
[ 2 ]
symmetrybasis(rep,2);
symmetrybasis(rep,3);
symmetrybasis(rep,4);
[ 0 ]
[ ]
[ 1 ]
[ --- ]
[ 2 ]
[ ]
[ - 1 ]
[------]
[ 2 ]
[ ]
[ 1 ]
[ --- ]
[ 2 ]
[ ]
[ - 1 ]
[------]
[ 2 ]
symmetrybasis(rep,5);
[ 0 0 ]
[ ]
[ 1 - 1 ]
[ --- ------]
[ 2 2 ]
[ ]
[ 1 1 ]
[ --- --- ]
[ 2 2 ]
[ ]
[ - 1 1 ]
[------ --- ]
[ 2 2 ]
[ ]
[ - 1 - 1 ]
[------ ------]
[ 2 2 ]
symmetrybasispart(rep,5);
[ 0 ]
[ ]
[ 1 ]
[ --- ]
[ 2 ]
[ ]
[ 1 ]
[ --- ]
[ 2 ]
[ ]
[ - 1 ]
[------]
[ 2 ]
[ ]
[ - 1 ]
[------]
[ 2 ]
allsymmetrybases(rep);
[1 0 0 0 0 ]
[ ]
[ 1 1 1 - 1 ]
[0 --- --- --- ------]
[ 2 2 2 2 ]
[ ]
[ 1 - 1 1 1 ]
[0 --- ------ --- --- ]
[ 2 2 2 2 ]
[ ]
[ 1 1 - 1 1 ]
[0 --- --- ------ --- ]
[ 2 2 2 2 ]
[ ]
[ 1 - 1 - 1 - 1 ]
[0 --- ------ ------ ------]
[ 2 2 2 2 ]
% Ritz matrix from Stiefel, Faessler p. 200
m:=mat((eps,a,a,a,a),
(a ,d,b,g,b),
(a ,b,d,b,g),
(a ,g,b,d,b),
(a ,b,g,b,d));
[eps a a a a]
[ ]
[ a d b g b]
[ ]
m := [ a b d b g]
[ ]
[ a g b d b]
[ ]
[ a b g b d]
diagonalize(m,rep);
[eps 2*a 0 0 0 ]
[ ]
[2*a 2*b + d + g 0 0 0 ]
[ ]
[ 0 0 - 2*b + d + g 0 0 ]
[ ]
[ 0 0 0 d - g 0 ]
[ ]
[ 0 0 0 0 d - g]
% eigenvalues are obvious. Eigenvectors may be obtained with
% the coordinate transformation matrix given by allsymmetrybases.
r1:=mat((0,1,0),
(0,0,1),
(1,0,0));
[0 1 0]
[ ]
r1 := [0 0 1]
[ ]
[1 0 0]
repC3:={C3,rC3=r1};
repc3 := {c3,rc3=
[0 1 0]
[ ]
[0 0 1]
[ ]
[1 0 0]
}
mC3:=mat((a,b,c),
(c,a,b),
(b,c,a));
[a b c]
[ ]
mc3 := [c a b]
[ ]
[b c a]
diagonalize(mC3,repC3);
[a + b + c 0 0 ]
[ ]
[ 2*a - b - c sqrt(3)*b - sqrt(3)*c ]
[ 0 ------------- -----------------------]
[ 2 2 ]
[ ]
[ - sqrt(3)*b + sqrt(3)*c 2*a - b - c ]
[ 0 -------------------------- ------------- ]
[ 2 2 ]
% note difference between real and complex case
on complex;
diagonalize(mC3,repC3);
mat((a + b + c,0,0),
i*sqrt(3)*b - i*sqrt(3)*c + 2*a - b - c
(0,-----------------------------------------,0),
2
- i*sqrt(3)*b + i*sqrt(3)*c + 2*a - b - c
(0,0,--------------------------------------------))
2
off complex;
end;
(TIME: symmetry 2870 2870)