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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