Artifact 021187975e512b63164f25606d6bba8be57db33541b40202ecc3d110574dfcac:
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r36/xlog/SYMMETRY.LOG
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2011-09-02 18:13:33
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— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 9800) [annotate] [blame] [check-ins using] [more...]
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)