REDUCE 3.4.1, 15-Jul-92 ...
1:
(SYMMETRY)
% 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: 2975 ms plus GC time: 102 ms
Quitting