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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 31278, 911]*) (*NotebookOutlinePosition[ 31954, 935]*) (* CellTagsIndexPosition[ 31910, 931]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "The sum of all Young symmetrizers of standard tableaux of ", Cell[BoxData[ \(TraditionalForm\`S\_5\)]] }], "Title"], Cell["Bernd Fiedler, Leipzig, October 2000", "Subtitle"], Cell["\<\ Bernd Fiedler, Alfred-Rosch-Str. 13, D-04249 Leipzig, Germany Bernd.Fiedler.RoschStr.Leipzig@t-online.de\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["< Default Intput Format Type -> InputForm -> Default Output Format Type -> OutputForm Enter the PERMS configuration which is intended to load. ------------------------------------------------------------- (m) Minimal configuration with character tables of S1...S10 (v) Full version with character tables of S1...S17 The evaluation of CHARTAB.M is running. Please wait.\ \>", "Print"] }, Open ]], Cell["First we generate all partitions of 5.", "Text"], Cell[CellGroupData[{ Cell["parts = AllPartitions[5]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[1, 1, 1, 1, 1], Parti[2, 1, 1, 1], Parti[2, 2, 1], Parti[3, 1, \ 1], Parti[3, 2], Parti[4, 1], Parti[5]]\ \>", "\<\ {{1, 1, 1, 1, 1}, {2, 1, 1, 1}, {2, 2, 1}, {3, 1, 1}, {3, 2}, {4, 1}, {5}}\ \>"], "Output"] }, Open ]], Cell["Next we determine all standard tableaux of these partitions.", "Text"], Cell["standardtableaux = StandardTableaux[#]& /@ parts ;", "Input"], Cell["standardtableaux = Flatten[standardtableaux,1,HoldList];", "Input"], Cell[CellGroupData[{ Cell["Length[standardtableaux]", "Input"], Cell[OutputFormData["\<\ 26\ \>", "\<\ 26\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["standardtableaux", "Input"], Cell[OutputFormData["\<\ HoldList[Tableau[TabRow[1], TabRow[2], TabRow[3], TabRow[4], TabRow[5]], Tableau[TabRow[1, 2], TabRow[3], TabRow[4], TabRow[5]], Tableau[TabRow[1, 3], TabRow[2], TabRow[4], TabRow[5]], Tableau[TabRow[1, 4], TabRow[2], TabRow[3], TabRow[5]], Tableau[TabRow[1, 5], TabRow[2], TabRow[3], TabRow[4]], Tableau[TabRow[1, 2], TabRow[3, 4], TabRow[5]], Tableau[TabRow[1, 2], TabRow[3, 5], TabRow[4]], Tableau[TabRow[1, 3], TabRow[2, 4], TabRow[5]], Tableau[TabRow[1, 3], TabRow[2, 5], TabRow[4]], Tableau[TabRow[1, 4], TabRow[2, 5], TabRow[3]], Tableau[TabRow[1, 2, 3], TabRow[4], TabRow[5]], Tableau[TabRow[1, 2, 4], TabRow[3], TabRow[5]], Tableau[TabRow[1, 2, 5], TabRow[3], TabRow[4]], Tableau[TabRow[1, 3, 4], TabRow[2], TabRow[5]], Tableau[TabRow[1, 3, 5], TabRow[2], TabRow[4]], Tableau[TabRow[1, 4, 5], TabRow[2], TabRow[3]], Tableau[TabRow[1, 2, 3], \ TabRow[4, 5]], Tableau[TabRow[1, 2, 4], TabRow[3, 5]], Tableau[TabRow[1, 2, 5], TabRow[3, \ 4]], Tableau[TabRow[1, 3, 4], TabRow[2, 5]], Tableau[TabRow[1, 3, 5], TabRow[2, \ 4]], Tableau[TabRow[1, 2, 3, 4], TabRow[5]], Tableau[TabRow[1, 2, 3, 5], \ TabRow[4]], Tableau[TabRow[1, 2, 4, 5], TabRow[3]], Tableau[TabRow[1, 3, 4, 5], \ TabRow[2]], Tableau[TabRow[1, 2, 3, 4, 5]]]\ \>", "\<\ {{1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 2}, {1, 2}, {1, 3}, {1, 3}, {1, 4}, \ {1, 2, 3}, {2} {3} {2} {2} {2} {3, 4} {3, 5} {2, 4} {2, 5} {2, 5} \ {4} {3} {4} {4} {3} {3} {5} {4} {5} {4} {3} \ {5} {4} {5} {5} {5} {4} {5} {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {1, 2, 3}, {1, 2, \ 4}, {1, 2, 5}, {3} {3} {2} {2} {2} {4, 5} {3, 5} \ {3, 4} {5} {4} {5} {4} {3} {1, 3, 4}, {1, 3, 5}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, \ 5}, {1, 2, 3, 4, 5}} {2, 5} {2, 4} {5} {4} {3} {2}\ \>"], "Output"] }, Open ]], Cell["\<\ We control the number of these standard tableaux be means of the dimensions \ of the corresponding minimal left ideals.\ \>", "Text"], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ parts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 4, 5, 6, 5, 4, 1]\ \>", "\<\ {1, 4, 5, 6, 5, 4, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 26\ \>", "\<\ 26\ \>"], "Output"] }, Open ]], Cell["\<\ Now we generate the Young symmetrizer for every of the above standard \ tableaux.\ \>", "Text"], Cell["symmetrizers = YoungSymmetrizer[#]& /@ standardtableaux;", "Input"], Cell["Some symmetrizers:", "Text"], Cell[CellGroupData[{ Cell["symmetrizers[[1]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] - Perm[1, 2, 3, 5, 4] - Perm[1, 2, 4, 3, 5] + Perm[1, 2, \ 4, 5, 3] + Perm[1, 2, 5, 3, 4] - Perm[1, 2, 5, 4, 3] - Perm[1, 3, 2, 4, 5] + Perm[1, \ 3, 2, 5, 4] + Perm[1, 3, 4, 2, 5] - Perm[1, 3, 4, 5, 2] - Perm[1, 3, 5, 2, 4] + Perm[1, \ 3, 5, 4, 2] + Perm[1, 4, 2, 3, 5] - Perm[1, 4, 2, 5, 3] - Perm[1, 4, 3, 2, 5] + Perm[1, \ 4, 3, 5, 2] + Perm[1, 4, 5, 2, 3] - Perm[1, 4, 5, 3, 2] - Perm[1, 5, 2, 3, 4] + Perm[1, \ 5, 2, 4, 3] + Perm[1, 5, 3, 2, 4] - Perm[1, 5, 3, 4, 2] - Perm[1, 5, 4, 2, 3] + Perm[1, \ 5, 4, 3, 2] - Perm[2, 1, 3, 4, 5] + Perm[2, 1, 3, 5, 4] + Perm[2, 1, 4, 3, 5] - Perm[2, \ 1, 4, 5, 3] - Perm[2, 1, 5, 3, 4] + Perm[2, 1, 5, 4, 3] + Perm[2, 3, 1, 4, 5] - Perm[2, \ 3, 1, 5, 4] - Perm[2, 3, 4, 1, 5] + Perm[2, 3, 4, 5, 1] + Perm[2, 3, 5, 1, 4] - Perm[2, \ 3, 5, 4, 1] - Perm[2, 4, 1, 3, 5] + Perm[2, 4, 1, 5, 3] + Perm[2, 4, 3, 1, 5] - Perm[2, \ 4, 3, 5, 1] - Perm[2, 4, 5, 1, 3] + Perm[2, 4, 5, 3, 1] + Perm[2, 5, 1, 3, 4] - Perm[2, \ 5, 1, 4, 3] - Perm[2, 5, 3, 1, 4] + Perm[2, 5, 3, 4, 1] + Perm[2, 5, 4, 1, 3] - Perm[2, \ 5, 4, 3, 1] + Perm[3, 1, 2, 4, 5] - Perm[3, 1, 2, 5, 4] - Perm[3, 1, 4, 2, 5] + Perm[3, \ 1, 4, 5, 2] + Perm[3, 1, 5, 2, 4] - Perm[3, 1, 5, 4, 2] - Perm[3, 2, 1, 4, 5] + Perm[3, \ 2, 1, 5, 4] + Perm[3, 2, 4, 1, 5] - Perm[3, 2, 4, 5, 1] - Perm[3, 2, 5, 1, 4] + Perm[3, \ 2, 5, 4, 1] + Perm[3, 4, 1, 2, 5] - Perm[3, 4, 1, 5, 2] - Perm[3, 4, 2, 1, 5] + Perm[3, \ 4, 2, 5, 1] + Perm[3, 4, 5, 1, 2] - Perm[3, 4, 5, 2, 1] - Perm[3, 5, 1, 2, 4] + Perm[3, \ 5, 1, 4, 2] + Perm[3, 5, 2, 1, 4] - Perm[3, 5, 2, 4, 1] - Perm[3, 5, 4, 1, 2] + Perm[3, \ 5, 4, 2, 1] - Perm[4, 1, 2, 3, 5] + Perm[4, 1, 2, 5, 3] + Perm[4, 1, 3, 2, 5] - Perm[4, \ 1, 3, 5, 2] - Perm[4, 1, 5, 2, 3] + Perm[4, 1, 5, 3, 2] + Perm[4, 2, 1, 3, 5] - Perm[4, \ 2, 1, 5, 3] - Perm[4, 2, 3, 1, 5] + Perm[4, 2, 3, 5, 1] + Perm[4, 2, 5, 1, 3] - Perm[4, \ 2, 5, 3, 1] - Perm[4, 3, 1, 2, 5] + Perm[4, 3, 1, 5, 2] + Perm[4, 3, 2, 1, 5] - Perm[4, \ 3, 2, 5, 1] - Perm[4, 3, 5, 1, 2] + Perm[4, 3, 5, 2, 1] + Perm[4, 5, 1, 2, 3] - Perm[4, \ 5, 1, 3, 2] - Perm[4, 5, 2, 1, 3] + Perm[4, 5, 2, 3, 1] + Perm[4, 5, 3, 1, 2] - Perm[4, \ 5, 3, 2, 1] + Perm[5, 1, 2, 3, 4] - Perm[5, 1, 2, 4, 3] - Perm[5, 1, 3, 2, 4] + Perm[5, \ 1, 3, 4, 2] + Perm[5, 1, 4, 2, 3] - Perm[5, 1, 4, 3, 2] - Perm[5, 2, 1, 3, 4] + Perm[5, \ 2, 1, 4, 3] + Perm[5, 2, 3, 1, 4] - Perm[5, 2, 3, 4, 1] - Perm[5, 2, 4, 1, 3] + Perm[5, \ 2, 4, 3, 1] + Perm[5, 3, 1, 2, 4] - Perm[5, 3, 1, 4, 2] - Perm[5, 3, 2, 1, 4] + Perm[5, \ 3, 2, 4, 1] + Perm[5, 3, 4, 1, 2] - Perm[5, 3, 4, 2, 1] - Perm[5, 4, 1, 2, 3] + Perm[5, \ 4, 1, 3, 2] + Perm[5, 4, 2, 1, 3] - Perm[5, 4, 2, 3, 1] - Perm[5, 4, 3, 1, 2] + Perm[5, \ 4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 5 ) - ( 1 2 3 5 4 ) - ( 1 2 4 3 5 ) + ( 1 2 4 5 3 ) + ( 1 2 5 3 4 ) \ - ( 1 2 5 4 3 ) - ( 1 3 2 4 5 ) + ( 1 3 2 5 4 ) + ( 1 3 4 2 5 ) - ( 1 3 4 5 2 ) - ( 1 3 5 2 4 \ ) + ( 1 3 5 4 2 ) + ( 1 4 2 3 5 ) - ( 1 4 2 5 3 ) - ( 1 4 3 2 5 ) + ( 1 4 3 5 2 \ ) + ( 1 4 5 2 3 ) - ( 1 4 5 3 2 ) - ( 1 5 2 3 4 ) + ( 1 5 2 4 3 ) + ( 1 5 3 2 4 \ ) - ( 1 5 3 4 2 ) - ( 1 5 4 2 3 ) + ( 1 5 4 3 2 ) - ( 2 1 3 4 5 ) + ( 2 1 3 5 4 \ ) + ( 2 1 4 3 5 ) - ( 2 1 4 5 3 ) - ( 2 1 5 3 4 ) + ( 2 1 5 4 3 ) + ( 2 3 1 4 5 \ ) - ( 2 3 1 5 4 ) - ( 2 3 4 1 5 ) + ( 2 3 4 5 1 ) + ( 2 3 5 1 4 ) - ( 2 3 5 4 1 \ ) - ( 2 4 1 3 5 ) + ( 2 4 1 5 3 ) + ( 2 4 3 1 5 ) - ( 2 4 3 5 1 ) - ( 2 4 5 1 3 \ ) + ( 2 4 5 3 1 ) + ( 2 5 1 3 4 ) - ( 2 5 1 4 3 ) - ( 2 5 3 1 4 ) + ( 2 5 3 4 1 \ ) + ( 2 5 4 1 3 ) - ( 2 5 4 3 1 ) + ( 3 1 2 4 5 ) - ( 3 1 2 5 4 ) - ( 3 1 4 2 5 \ ) + ( 3 1 4 5 2 ) + ( 3 1 5 2 4 ) - ( 3 1 5 4 2 ) - ( 3 2 1 4 5 ) + ( 3 2 1 5 4 \ ) + ( 3 2 4 1 5 ) - ( 3 2 4 5 1 ) - ( 3 2 5 1 4 ) + ( 3 2 5 4 1 ) + ( 3 4 1 2 5 \ ) - ( 3 4 1 5 2 ) - ( 3 4 2 1 5 ) + ( 3 4 2 5 1 ) + ( 3 4 5 1 2 ) - ( 3 4 5 2 1 \ ) - ( 3 5 1 2 4 ) + ( 3 5 1 4 2 ) + ( 3 5 2 1 4 ) - ( 3 5 2 4 1 ) - ( 3 5 4 1 2 \ ) + ( 3 5 4 2 1 ) - ( 4 1 2 3 5 ) + ( 4 1 2 5 3 ) + ( 4 1 3 2 5 ) - ( 4 1 3 5 2 \ ) - ( 4 1 5 2 3 ) + ( 4 1 5 3 2 ) + ( 4 2 1 3 5 ) - ( 4 2 1 5 3 ) - ( 4 2 3 1 5 \ ) + ( 4 2 3 5 1 ) + ( 4 2 5 1 3 ) - ( 4 2 5 3 1 ) - ( 4 3 1 2 5 ) + ( 4 3 1 5 2 \ ) + ( 4 3 2 1 5 ) - ( 4 3 2 5 1 ) - ( 4 3 5 1 2 ) + ( 4 3 5 2 1 ) + ( 4 5 1 2 3 \ ) - ( 4 5 1 3 2 ) - ( 4 5 2 1 3 ) + ( 4 5 2 3 1 ) + ( 4 5 3 1 2 ) - ( 4 5 3 2 1 \ ) + ( 5 1 2 3 4 ) - ( 5 1 2 4 3 ) - ( 5 1 3 2 4 ) + ( 5 1 3 4 2 ) + ( 5 1 4 2 3 \ ) - ( 5 1 4 3 2 ) - ( 5 2 1 3 4 ) + ( 5 2 1 4 3 ) + ( 5 2 3 1 4 ) - ( 5 2 3 4 1 \ ) - ( 5 2 4 1 3 ) + ( 5 2 4 3 1 ) + ( 5 3 1 2 4 ) - ( 5 3 1 4 2 ) - ( 5 3 2 1 4 \ ) + ( 5 3 2 4 1 ) + ( 5 3 4 1 2 ) - ( 5 3 4 2 1 ) - ( 5 4 1 2 3 ) + ( 5 4 1 3 2 \ ) + ( 5 4 2 1 3 ) - ( 5 4 2 3 1 ) - ( 5 4 3 1 2 ) + ( 5 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["symmetrizers[[2]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] - Perm[1, 2, 3, 5, 4] - Perm[1, 2, 4, 3, 5] + Perm[1, 2, \ 4, 5, 3] + Perm[1, 2, 5, 3, 4] - Perm[1, 2, 5, 4, 3] + Perm[2, 1, 3, 4, 5] - Perm[2, \ 1, 3, 5, 4] - Perm[2, 1, 4, 3, 5] + Perm[2, 1, 4, 5, 3] + Perm[2, 1, 5, 3, 4] - Perm[2, \ 1, 5, 4, 3] - Perm[3, 1, 2, 4, 5] + Perm[3, 1, 2, 5, 4] + Perm[3, 1, 4, 2, 5] - Perm[3, \ 1, 4, 5, 2] - Perm[3, 1, 5, 2, 4] + Perm[3, 1, 5, 4, 2] - Perm[3, 2, 1, 4, 5] + Perm[3, \ 2, 1, 5, 4] + Perm[3, 2, 4, 1, 5] - Perm[3, 2, 4, 5, 1] - Perm[3, 2, 5, 1, 4] + Perm[3, \ 2, 5, 4, 1] + Perm[4, 1, 2, 3, 5] - Perm[4, 1, 2, 5, 3] - Perm[4, 1, 3, 2, 5] + Perm[4, \ 1, 3, 5, 2] + Perm[4, 1, 5, 2, 3] - Perm[4, 1, 5, 3, 2] + Perm[4, 2, 1, 3, 5] - Perm[4, \ 2, 1, 5, 3] - Perm[4, 2, 3, 1, 5] + Perm[4, 2, 3, 5, 1] + Perm[4, 2, 5, 1, 3] - Perm[4, \ 2, 5, 3, 1] - Perm[5, 1, 2, 3, 4] + Perm[5, 1, 2, 4, 3] + Perm[5, 1, 3, 2, 4] - Perm[5, \ 1, 3, 4, 2] - Perm[5, 1, 4, 2, 3] + Perm[5, 1, 4, 3, 2] - Perm[5, 2, 1, 3, 4] + Perm[5, \ 2, 1, 4, 3] + Perm[5, 2, 3, 1, 4] - Perm[5, 2, 3, 4, 1] - Perm[5, 2, 4, 1, 3] + Perm[5, \ 2, 4, 3, 1]\ \>", "\<\ ( 1 2 3 4 5 ) - ( 1 2 3 5 4 ) - ( 1 2 4 3 5 ) + ( 1 2 4 5 3 ) + ( 1 2 5 3 4 ) \ - ( 1 2 5 4 3 ) + ( 2 1 3 4 5 ) - ( 2 1 3 5 4 ) - ( 2 1 4 3 5 ) + ( 2 1 4 5 3 ) + ( 2 1 5 3 4 \ ) - ( 2 1 5 4 3 ) - ( 3 1 2 4 5 ) + ( 3 1 2 5 4 ) + ( 3 1 4 2 5 ) - ( 3 1 4 5 2 \ ) - ( 3 1 5 2 4 ) + ( 3 1 5 4 2 ) - ( 3 2 1 4 5 ) + ( 3 2 1 5 4 ) + ( 3 2 4 1 5 \ ) - ( 3 2 4 5 1 ) - ( 3 2 5 1 4 ) + ( 3 2 5 4 1 ) + ( 4 1 2 3 5 ) - ( 4 1 2 5 3 \ ) - ( 4 1 3 2 5 ) + ( 4 1 3 5 2 ) + ( 4 1 5 2 3 ) - ( 4 1 5 3 2 ) + ( 4 2 1 3 5 \ ) - ( 4 2 1 5 3 ) - ( 4 2 3 1 5 ) + ( 4 2 3 5 1 ) + ( 4 2 5 1 3 ) - ( 4 2 5 3 1 \ ) - ( 5 1 2 3 4 ) + ( 5 1 2 4 3 ) + ( 5 1 3 2 4 ) - ( 5 1 3 4 2 ) - ( 5 1 4 2 3 \ ) + ( 5 1 4 3 2 ) - ( 5 2 1 3 4 ) + ( 5 2 1 4 3 ) + ( 5 2 3 1 4 ) - ( 5 2 3 4 1 \ ) - ( 5 2 4 1 3 ) + ( 5 2 4 3 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["symmetrizers[[26]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] + Perm[1, 2, 3, 5, 4] + Perm[1, 2, 4, 3, 5] + Perm[1, 2, \ 4, 5, 3] + Perm[1, 2, 5, 3, 4] + Perm[1, 2, 5, 4, 3] + Perm[1, 3, 2, 4, 5] + Perm[1, \ 3, 2, 5, 4] + Perm[1, 3, 4, 2, 5] + Perm[1, 3, 4, 5, 2] + Perm[1, 3, 5, 2, 4] + Perm[1, \ 3, 5, 4, 2] + Perm[1, 4, 2, 3, 5] + Perm[1, 4, 2, 5, 3] + Perm[1, 4, 3, 2, 5] + Perm[1, \ 4, 3, 5, 2] + Perm[1, 4, 5, 2, 3] + Perm[1, 4, 5, 3, 2] + Perm[1, 5, 2, 3, 4] + Perm[1, \ 5, 2, 4, 3] + Perm[1, 5, 3, 2, 4] + Perm[1, 5, 3, 4, 2] + Perm[1, 5, 4, 2, 3] + Perm[1, \ 5, 4, 3, 2] + Perm[2, 1, 3, 4, 5] + Perm[2, 1, 3, 5, 4] + Perm[2, 1, 4, 3, 5] + Perm[2, \ 1, 4, 5, 3] + Perm[2, 1, 5, 3, 4] + Perm[2, 1, 5, 4, 3] + Perm[2, 3, 1, 4, 5] + Perm[2, \ 3, 1, 5, 4] + Perm[2, 3, 4, 1, 5] + Perm[2, 3, 4, 5, 1] + Perm[2, 3, 5, 1, 4] + Perm[2, \ 3, 5, 4, 1] + Perm[2, 4, 1, 3, 5] + Perm[2, 4, 1, 5, 3] + Perm[2, 4, 3, 1, 5] + Perm[2, \ 4, 3, 5, 1] + Perm[2, 4, 5, 1, 3] + Perm[2, 4, 5, 3, 1] + Perm[2, 5, 1, 3, 4] + Perm[2, \ 5, 1, 4, 3] + Perm[2, 5, 3, 1, 4] + Perm[2, 5, 3, 4, 1] + Perm[2, 5, 4, 1, 3] + Perm[2, \ 5, 4, 3, 1] + Perm[3, 1, 2, 4, 5] + Perm[3, 1, 2, 5, 4] + Perm[3, 1, 4, 2, 5] + Perm[3, \ 1, 4, 5, 2] + Perm[3, 1, 5, 2, 4] + Perm[3, 1, 5, 4, 2] + Perm[3, 2, 1, 4, 5] + Perm[3, \ 2, 1, 5, 4] + Perm[3, 2, 4, 1, 5] + Perm[3, 2, 4, 5, 1] + Perm[3, 2, 5, 1, 4] + Perm[3, \ 2, 5, 4, 1] + Perm[3, 4, 1, 2, 5] + Perm[3, 4, 1, 5, 2] + Perm[3, 4, 2, 1, 5] + Perm[3, \ 4, 2, 5, 1] + Perm[3, 4, 5, 1, 2] + Perm[3, 4, 5, 2, 1] + Perm[3, 5, 1, 2, 4] + Perm[3, \ 5, 1, 4, 2] + Perm[3, 5, 2, 1, 4] + Perm[3, 5, 2, 4, 1] + Perm[3, 5, 4, 1, 2] + Perm[3, \ 5, 4, 2, 1] + Perm[4, 1, 2, 3, 5] + Perm[4, 1, 2, 5, 3] + Perm[4, 1, 3, 2, 5] + Perm[4, \ 1, 3, 5, 2] + Perm[4, 1, 5, 2, 3] + Perm[4, 1, 5, 3, 2] + Perm[4, 2, 1, 3, 5] + Perm[4, \ 2, 1, 5, 3] + Perm[4, 2, 3, 1, 5] + Perm[4, 2, 3, 5, 1] + Perm[4, 2, 5, 1, 3] + Perm[4, \ 2, 5, 3, 1] + Perm[4, 3, 1, 2, 5] + Perm[4, 3, 1, 5, 2] + Perm[4, 3, 2, 1, 5] + Perm[4, \ 3, 2, 5, 1] + Perm[4, 3, 5, 1, 2] + Perm[4, 3, 5, 2, 1] + Perm[4, 5, 1, 2, 3] + Perm[4, \ 5, 1, 3, 2] + Perm[4, 5, 2, 1, 3] + Perm[4, 5, 2, 3, 1] + Perm[4, 5, 3, 1, 2] + Perm[4, \ 5, 3, 2, 1] + Perm[5, 1, 2, 3, 4] + Perm[5, 1, 2, 4, 3] + Perm[5, 1, 3, 2, 4] + Perm[5, \ 1, 3, 4, 2] + Perm[5, 1, 4, 2, 3] + Perm[5, 1, 4, 3, 2] + Perm[5, 2, 1, 3, 4] + Perm[5, \ 2, 1, 4, 3] + Perm[5, 2, 3, 1, 4] + Perm[5, 2, 3, 4, 1] + Perm[5, 2, 4, 1, 3] + Perm[5, \ 2, 4, 3, 1] + Perm[5, 3, 1, 2, 4] + Perm[5, 3, 1, 4, 2] + Perm[5, 3, 2, 1, 4] + Perm[5, \ 3, 2, 4, 1] + Perm[5, 3, 4, 1, 2] + Perm[5, 3, 4, 2, 1] + Perm[5, 4, 1, 2, 3] + Perm[5, \ 4, 1, 3, 2] + Perm[5, 4, 2, 1, 3] + Perm[5, 4, 2, 3, 1] + Perm[5, 4, 3, 1, 2] + Perm[5, \ 4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 5 ) + ( 1 2 3 5 4 ) + ( 1 2 4 3 5 ) + ( 1 2 4 5 3 ) + ( 1 2 5 3 4 ) \ + ( 1 2 5 4 3 ) + ( 1 3 2 4 5 ) + ( 1 3 2 5 4 ) + ( 1 3 4 2 5 ) + ( 1 3 4 5 2 ) + ( 1 3 5 2 4 \ ) + ( 1 3 5 4 2 ) + ( 1 4 2 3 5 ) + ( 1 4 2 5 3 ) + ( 1 4 3 2 5 ) + ( 1 4 3 5 2 \ ) + ( 1 4 5 2 3 ) + ( 1 4 5 3 2 ) + ( 1 5 2 3 4 ) + ( 1 5 2 4 3 ) + ( 1 5 3 2 4 \ ) + ( 1 5 3 4 2 ) + ( 1 5 4 2 3 ) + ( 1 5 4 3 2 ) + ( 2 1 3 4 5 ) + ( 2 1 3 5 4 \ ) + ( 2 1 4 3 5 ) + ( 2 1 4 5 3 ) + ( 2 1 5 3 4 ) + ( 2 1 5 4 3 ) + ( 2 3 1 4 5 \ ) + ( 2 3 1 5 4 ) + ( 2 3 4 1 5 ) + ( 2 3 4 5 1 ) + ( 2 3 5 1 4 ) + ( 2 3 5 4 1 \ ) + ( 2 4 1 3 5 ) + ( 2 4 1 5 3 ) + ( 2 4 3 1 5 ) + ( 2 4 3 5 1 ) + ( 2 4 5 1 3 \ ) + ( 2 4 5 3 1 ) + ( 2 5 1 3 4 ) + ( 2 5 1 4 3 ) + ( 2 5 3 1 4 ) + ( 2 5 3 4 1 \ ) + ( 2 5 4 1 3 ) + ( 2 5 4 3 1 ) + ( 3 1 2 4 5 ) + ( 3 1 2 5 4 ) + ( 3 1 4 2 5 \ ) + ( 3 1 4 5 2 ) + ( 3 1 5 2 4 ) + ( 3 1 5 4 2 ) + ( 3 2 1 4 5 ) + ( 3 2 1 5 4 \ ) + ( 3 2 4 1 5 ) + ( 3 2 4 5 1 ) + ( 3 2 5 1 4 ) + ( 3 2 5 4 1 ) + ( 3 4 1 2 5 \ ) + ( 3 4 1 5 2 ) + ( 3 4 2 1 5 ) + ( 3 4 2 5 1 ) + ( 3 4 5 1 2 ) + ( 3 4 5 2 1 \ ) + ( 3 5 1 2 4 ) + ( 3 5 1 4 2 ) + ( 3 5 2 1 4 ) + ( 3 5 2 4 1 ) + ( 3 5 4 1 2 \ ) + ( 3 5 4 2 1 ) + ( 4 1 2 3 5 ) + ( 4 1 2 5 3 ) + ( 4 1 3 2 5 ) + ( 4 1 3 5 2 \ ) + ( 4 1 5 2 3 ) + ( 4 1 5 3 2 ) + ( 4 2 1 3 5 ) + ( 4 2 1 5 3 ) + ( 4 2 3 1 5 \ ) + ( 4 2 3 5 1 ) + ( 4 2 5 1 3 ) + ( 4 2 5 3 1 ) + ( 4 3 1 2 5 ) + ( 4 3 1 5 2 \ ) + ( 4 3 2 1 5 ) + ( 4 3 2 5 1 ) + ( 4 3 5 1 2 ) + ( 4 3 5 2 1 ) + ( 4 5 1 2 3 \ ) + ( 4 5 1 3 2 ) + ( 4 5 2 1 3 ) + ( 4 5 2 3 1 ) + ( 4 5 3 1 2 ) + ( 4 5 3 2 1 \ ) + ( 5 1 2 3 4 ) + ( 5 1 2 4 3 ) + ( 5 1 3 2 4 ) + ( 5 1 3 4 2 ) + ( 5 1 4 2 3 \ ) + ( 5 1 4 3 2 ) + ( 5 2 1 3 4 ) + ( 5 2 1 4 3 ) + ( 5 2 3 1 4 ) + ( 5 2 3 4 1 \ ) + ( 5 2 4 1 3 ) + ( 5 2 4 3 1 ) + ( 5 3 1 2 4 ) + ( 5 3 1 4 2 ) + ( 5 3 2 1 4 \ ) + ( 5 3 2 4 1 ) + ( 5 3 4 1 2 ) + ( 5 3 4 2 1 ) + ( 5 4 1 2 3 ) + ( 5 4 1 3 2 \ ) + ( 5 4 2 1 3 ) + ( 5 4 2 3 1 ) + ( 5 4 3 1 2 ) + ( 5 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell["\<\ Now we calculate the factors which transform the above symmetrizers into \ idempotents.\ \>", "Text"], Cell[CellGroupData[{ Cell["factors = YoungFactor[MakePartition[#]]& /@ standardtableaux", "Input"], Cell[OutputFormData["\<\ HoldList[1/120, 1/30, 1/30, 1/30, 1/30, 1/24, 1/24, 1/24, 1/24, 1/24, 1/20, \ 1/20, 1/20, 1/20, 1/20, 1/20, 1/24, 1/24, 1/24, 1/24, 1/24, 1/30, 1/30, 1/30, 1/30, 1/120]\ \>", "\<\ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \ 1 1 1 1 {---, --, --, --, --, --, --, --, --, --, --, --, --, --, --, --, --, --, --, \ --, --, --, --, 120 30 30 30 30 24 24 24 24 24 20 20 20 20 20 20 24 24 24 \ 24 24 30 30 1 1 1 --, --, ---} 30 30 120\ \>"], "Output"] }, Open ]], Cell["The sum of all normalized Young symmetrizers reads", "Text"], Cell[CellGroupData[{ Cell["symsum = Expand[(List @@ factors).(List @@ symmetrizers)]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] + Perm[1, 3, 5, 2, 4]/24 - Perm[1, 3, 5, 4, 2]/24 - Perm[1, 4, 2, 5, 3]/24 - Perm[1, 4, 3, 5, 2]/24 + Perm[1, 4, 5, 2, 3]/12 - Perm[1, 5, 2, 4, 3]/24 - Perm[1, 5, 3, 2, 4]/24 + Perm[1, 5, 4, 3, 2]/12 + Perm[2, 3, 5, 1, 4]/24 - Perm[2, 3, 5, 4, 1]/24 - Perm[2, 4, 1, 5, 3]/24 - Perm[2, 4, 3, 5, 1]/24 + Perm[2, 4, 5, 1, 3]/12 - Perm[2, 5, 1, 4, 3]/24 - Perm[2, 5, 3, 1, 4]/24 + Perm[2, 5, 4, 3, 1]/12 - Perm[3, 1, 5, 2, 4]/24 + Perm[3, 1, 5, 4, 2]/24 - Perm[3, 2, 5, 1, 4]/24 + Perm[3, 2, 5, 4, 1]/24 - Perm[3, 4, 1, 5, 2]/24 - Perm[3, 4, 2, 5, 1]/24 + Perm[3, 4, 5, 1, 2]/24 + Perm[3, 4, 5, 2, 1]/24 + Perm[3, 5, 1, 2, 4]/24 - Perm[3, 5, 1, 4, 2]/12 + Perm[3, 5, 2, 1, 4]/24 - Perm[3, 5, 2, 4, 1]/12 + Perm[3, 5, 4, 1, 2]/24 + Perm[3, 5, 4, 2, 1]/24 + Perm[4, 1, 2, 5, 3]/24 + Perm[4, 1, 3, 5, 2]/24 - Perm[4, 1, 5, 2, 3]/12 + Perm[4, 2, 1, 5, 3]/24 + Perm[4, 2, 3, 5, 1]/24 - Perm[4, 2, 5, 1, 3]/12 + Perm[4, 3, 1, 5, 2]/24 + Perm[4, 3, 2, 5, 1]/24 - Perm[4, 3, 5, 1, 2]/24 - Perm[4, 3, 5, 2, 1]/24 + Perm[4, 5, 1, 2, 3]/24 - Perm[4, 5, 1, 3, 2]/24 + Perm[4, 5, 2, 1, 3]/24 - Perm[4, 5, 2, 3, 1]/24 + Perm[5, 1, 2, 4, 3]/24 + Perm[5, 1, 3, 2, 4]/24 - Perm[5, 1, 4, 3, 2]/12 + Perm[5, 2, 1, 4, 3]/24 + Perm[5, 2, 3, 1, 4]/24 - Perm[5, 2, 4, 3, 1]/12 - Perm[5, 3, 1, 2, 4]/24 + Perm[5, 3, 1, 4, 2]/12 - Perm[5, 3, 2, 1, 4]/24 + Perm[5, 3, 2, 4, 1]/12 - Perm[5, 3, 4, 1, 2]/24 - Perm[5, 3, 4, 2, 1]/24 - Perm[5, 4, 1, 2, 3]/24 + Perm[5, 4, 1, 3, 2]/24 - Perm[5, 4, 2, 1, 3]/24 + Perm[5, 4, 2, 3, 1]/24\ \>", "\<\ ( 1 3 5 2 4 ) ( 1 3 5 4 2 ) ( 1 4 2 5 3 ) ( 1 4 3 5 2 ) \ ( 1 4 5 2 3 ) ( 1 2 3 4 5 ) + ------------- - ------------- - ------------- - ------------- \ + ------------- - 24 24 24 24 \ 12 ( 1 5 2 4 3 ) ( 1 5 3 2 4 ) ( 1 5 4 3 2 ) ( 2 3 5 1 4 ) ( 2 3 5 4 1 \ ) ------------- - ------------- + ------------- + ------------- - \ ------------- - 24 24 12 24 24 ( 2 4 1 5 3 ) ( 2 4 3 5 1 ) ( 2 4 5 1 3 ) ( 2 5 1 4 3 ) ( 2 5 3 1 4 \ ) ------------- - ------------- + ------------- - ------------- - \ ------------- + 24 24 12 24 24 ( 2 5 4 3 1 ) ( 3 1 5 2 4 ) ( 3 1 5 4 2 ) ( 3 2 5 1 4 ) ( 3 2 5 4 1 \ ) ------------- - ------------- + ------------- - ------------- + \ ------------- - 12 24 24 24 24 ( 3 4 1 5 2 ) ( 3 4 2 5 1 ) ( 3 4 5 1 2 ) ( 3 4 5 2 1 ) ( 3 5 1 2 4 \ ) ------------- - ------------- + ------------- + ------------- + \ ------------- - 24 24 24 24 24 ( 3 5 1 4 2 ) ( 3 5 2 1 4 ) ( 3 5 2 4 1 ) ( 3 5 4 1 2 ) ( 3 5 4 2 1 \ ) ------------- + ------------- - ------------- + ------------- + \ ------------- + 12 24 12 24 24 ( 4 1 2 5 3 ) ( 4 1 3 5 2 ) ( 4 1 5 2 3 ) ( 4 2 1 5 3 ) ( 4 2 3 5 1 \ ) ------------- + ------------- - ------------- + ------------- + \ ------------- - 24 24 12 24 24 ( 4 2 5 1 3 ) ( 4 3 1 5 2 ) ( 4 3 2 5 1 ) ( 4 3 5 1 2 ) ( 4 3 5 2 1 \ ) ------------- + ------------- + ------------- - ------------- - \ ------------- + 12 24 24 24 24 ( 4 5 1 2 3 ) ( 4 5 1 3 2 ) ( 4 5 2 1 3 ) ( 4 5 2 3 1 ) ( 5 1 2 4 3 \ ) ------------- - ------------- + ------------- - ------------- + \ ------------- + 24 24 24 24 24 ( 5 1 3 2 4 ) ( 5 1 4 3 2 ) ( 5 2 1 4 3 ) ( 5 2 3 1 4 ) ( 5 2 4 3 1 \ ) ------------- - ------------- + ------------- + ------------- - \ ------------- - 24 12 24 24 12 ( 5 3 1 2 4 ) ( 5 3 1 4 2 ) ( 5 3 2 1 4 ) ( 5 3 2 4 1 ) ( 5 3 4 1 2 \ ) ------------- + ------------- - ------------- + ------------- - \ ------------- - 24 12 24 12 24 ( 5 3 4 2 1 ) ( 5 4 1 2 3 ) ( 5 4 1 3 2 ) ( 5 4 2 1 3 ) ( 5 4 2 3 1 \ ) ------------- - ------------- + ------------- - ------------- + \ ------------- 24 24 24 24 24\ \>"], "Output"] }, Open ]], Cell["\<\ sumsym is unequalt to (1 2 3 4 5) since the above symmetrizers are not \ pairwise orthogonal. For example:\ \>", "Text"], Cell[CellGroupData[{ Cell["tabl1 = DefTableau[{1,2,3},{4,5}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 2, 3], TabRow[4, 5]]\ \>", "\<\ {1, 2, 3} {4, 5}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tabl2 = DefTableau[{1,3,5},{2,4}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 3, 5], TabRow[2, 4]]\ \>", "\<\ {1, 3, 5} {2, 4}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y1 = YoungSymmetrizer[tabl1]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] + Perm[1, 2, 3, 5, 4] + Perm[1, 3, 2, 4, 5] + Perm[1, 3, \ 2, 5, 4] - Perm[1, 4, 2, 5, 3] - Perm[1, 4, 3, 5, 2] - Perm[1, 5, 2, 4, 3] - Perm[1, \ 5, 3, 4, 2] + Perm[2, 1, 3, 4, 5] + Perm[2, 1, 3, 5, 4] + Perm[2, 3, 1, 4, 5] + Perm[2, \ 3, 1, 5, 4] - Perm[2, 4, 1, 5, 3] - Perm[2, 4, 3, 5, 1] - Perm[2, 5, 1, 4, 3] - Perm[2, \ 5, 3, 4, 1] + Perm[3, 1, 2, 4, 5] + Perm[3, 1, 2, 5, 4] + Perm[3, 2, 1, 4, 5] + Perm[3, \ 2, 1, 5, 4] - Perm[3, 4, 1, 5, 2] - Perm[3, 4, 2, 5, 1] - Perm[3, 5, 1, 4, 2] - Perm[3, \ 5, 2, 4, 1] - Perm[4, 1, 2, 3, 5] - Perm[4, 1, 3, 2, 5] - Perm[4, 2, 1, 3, 5] - Perm[4, \ 2, 3, 1, 5] - Perm[4, 3, 1, 2, 5] - Perm[4, 3, 2, 1, 5] + Perm[4, 5, 1, 2, 3] + Perm[4, \ 5, 1, 3, 2] + Perm[4, 5, 2, 1, 3] + Perm[4, 5, 2, 3, 1] + Perm[4, 5, 3, 1, 2] + Perm[4, \ 5, 3, 2, 1] - Perm[5, 1, 2, 3, 4] - Perm[5, 1, 3, 2, 4] - Perm[5, 2, 1, 3, 4] - Perm[5, \ 2, 3, 1, 4] - Perm[5, 3, 1, 2, 4] - Perm[5, 3, 2, 1, 4] + Perm[5, 4, 1, 2, 3] + Perm[5, \ 4, 1, 3, 2] + Perm[5, 4, 2, 1, 3] + Perm[5, 4, 2, 3, 1] + Perm[5, 4, 3, 1, 2] + Perm[5, \ 4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 5 ) + ( 1 2 3 5 4 ) + ( 1 3 2 4 5 ) + ( 1 3 2 5 4 ) - ( 1 4 2 5 3 ) \ - ( 1 4 3 5 2 ) - ( 1 5 2 4 3 ) - ( 1 5 3 4 2 ) + ( 2 1 3 4 5 ) + ( 2 1 3 5 4 ) + ( 2 3 1 4 5 \ ) + ( 2 3 1 5 4 ) - ( 2 4 1 5 3 ) - ( 2 4 3 5 1 ) - ( 2 5 1 4 3 ) - ( 2 5 3 4 1 \ ) + ( 3 1 2 4 5 ) + ( 3 1 2 5 4 ) + ( 3 2 1 4 5 ) + ( 3 2 1 5 4 ) - ( 3 4 1 5 2 \ ) - ( 3 4 2 5 1 ) - ( 3 5 1 4 2 ) - ( 3 5 2 4 1 ) - ( 4 1 2 3 5 ) - ( 4 1 3 2 5 \ ) - ( 4 2 1 3 5 ) - ( 4 2 3 1 5 ) - ( 4 3 1 2 5 ) - ( 4 3 2 1 5 ) + ( 4 5 1 2 3 \ ) + ( 4 5 1 3 2 ) + ( 4 5 2 1 3 ) + ( 4 5 2 3 1 ) + ( 4 5 3 1 2 ) + ( 4 5 3 2 1 \ ) - ( 5 1 2 3 4 ) - ( 5 1 3 2 4 ) - ( 5 2 1 3 4 ) - ( 5 2 3 1 4 ) - ( 5 3 1 2 4 \ ) - ( 5 3 2 1 4 ) + ( 5 4 1 2 3 ) + ( 5 4 1 3 2 ) + ( 5 4 2 1 3 ) + ( 5 4 2 3 1 \ ) + ( 5 4 3 1 2 ) + ( 5 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y2 = YoungSymmetrizer[tabl2]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] - Perm[1, 2, 4, 3, 5] - Perm[1, 2, 4, 5, 3] + Perm[1, 2, \ 5, 4, 3] - Perm[1, 4, 2, 3, 5] - Perm[1, 4, 2, 5, 3] + Perm[1, 4, 3, 2, 5] + Perm[1, \ 4, 5, 2, 3] - Perm[2, 1, 3, 4, 5] + Perm[2, 1, 4, 3, 5] + Perm[2, 1, 4, 5, 3] - Perm[2, \ 1, 5, 4, 3] - Perm[2, 3, 1, 4, 5] + Perm[2, 3, 4, 1, 5] + Perm[2, 3, 4, 5, 1] - Perm[2, \ 3, 5, 4, 1] - Perm[2, 5, 1, 4, 3] - Perm[2, 5, 3, 4, 1] + Perm[2, 5, 4, 1, 3] + Perm[2, \ 5, 4, 3, 1] + Perm[3, 2, 1, 4, 5] - Perm[3, 2, 4, 1, 5] - Perm[3, 2, 4, 5, 1] + Perm[3, \ 2, 5, 4, 1] + Perm[3, 4, 1, 2, 5] - Perm[3, 4, 2, 1, 5] - Perm[3, 4, 2, 5, 1] + Perm[3, \ 4, 5, 2, 1] + Perm[4, 1, 2, 3, 5] + Perm[4, 1, 2, 5, 3] - Perm[4, 1, 3, 2, 5] - Perm[4, \ 1, 5, 2, 3] - Perm[4, 3, 1, 2, 5] + Perm[4, 3, 2, 1, 5] + Perm[4, 3, 2, 5, 1] - Perm[4, \ 3, 5, 2, 1] - Perm[4, 5, 1, 2, 3] + Perm[4, 5, 2, 1, 3] + Perm[4, 5, 2, 3, 1] - Perm[4, \ 5, 3, 2, 1] + Perm[5, 2, 1, 4, 3] + Perm[5, 2, 3, 4, 1] - Perm[5, 2, 4, 1, 3] - Perm[5, \ 2, 4, 3, 1] + Perm[5, 4, 1, 2, 3] - Perm[5, 4, 2, 1, 3] - Perm[5, 4, 2, 3, 1] + Perm[5, \ 4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 5 ) - ( 1 2 4 3 5 ) - ( 1 2 4 5 3 ) + ( 1 2 5 4 3 ) - ( 1 4 2 3 5 ) \ - ( 1 4 2 5 3 ) + ( 1 4 3 2 5 ) + ( 1 4 5 2 3 ) - ( 2 1 3 4 5 ) + ( 2 1 4 3 5 ) + ( 2 1 4 5 3 \ ) - ( 2 1 5 4 3 ) - ( 2 3 1 4 5 ) + ( 2 3 4 1 5 ) + ( 2 3 4 5 1 ) - ( 2 3 5 4 1 \ ) - ( 2 5 1 4 3 ) - ( 2 5 3 4 1 ) + ( 2 5 4 1 3 ) + ( 2 5 4 3 1 ) + ( 3 2 1 4 5 \ ) - ( 3 2 4 1 5 ) - ( 3 2 4 5 1 ) + ( 3 2 5 4 1 ) + ( 3 4 1 2 5 ) - ( 3 4 2 1 5 \ ) - ( 3 4 2 5 1 ) + ( 3 4 5 2 1 ) + ( 4 1 2 3 5 ) + ( 4 1 2 5 3 ) - ( 4 1 3 2 5 \ ) - ( 4 1 5 2 3 ) - ( 4 3 1 2 5 ) + ( 4 3 2 1 5 ) + ( 4 3 2 5 1 ) - ( 4 3 5 2 1 \ ) - ( 4 5 1 2 3 ) + ( 4 5 2 1 3 ) + ( 4 5 2 3 1 ) - ( 4 5 3 2 1 ) + ( 5 2 1 4 3 \ ) + ( 5 2 3 4 1 ) - ( 5 2 4 1 3 ) - ( 5 2 4 3 1 ) + ( 5 4 1 2 3 ) - ( 5 4 2 1 3 \ ) - ( 5 4 2 3 1 ) + ( 5 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["PermProd[y2,y1]", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["PermProd[y1,y2]", "Input"], Cell[OutputFormData["\<\ -24*Perm[1, 4, 2, 5, 3] - 24*Perm[1, 4, 3, 5, 2] + 24*Perm[1, 4, 5, 2, 3] + 24*Perm[1, 4, 5, 3, 2] - 24*Perm[1, 5, 2, 4, 3] - 24*Perm[1, 5, 3, 4, 2] + 24*Perm[1, 5, 4, 2, 3] + 24*Perm[1, 5, 4, 3, 2] - 24*Perm[2, 4, 1, 5, 3] - 24*Perm[2, 4, 3, 5, 1] + 24*Perm[2, 4, 5, 1, 3] + 24*Perm[2, 4, 5, 3, 1] - 24*Perm[2, 5, 1, 4, 3] - 24*Perm[2, 5, 3, 4, 1] + 24*Perm[2, 5, 4, 1, 3] + 24*Perm[2, 5, 4, 3, 1] - 24*Perm[3, 4, 1, 5, 2] - 24*Perm[3, 4, 2, 5, 1] + 24*Perm[3, 4, 5, 1, 2] + 24*Perm[3, 4, 5, 2, 1] - 24*Perm[3, 5, 1, 4, 2] - 24*Perm[3, 5, 2, 4, 1] + 24*Perm[3, 5, 4, 1, 2] + 24*Perm[3, 5, 4, 2, 1] + 24*Perm[4, 1, 2, 5, 3] + 24*Perm[4, 1, 3, 5, 2] - 24*Perm[4, 1, 5, 2, 3] - 24*Perm[4, 1, 5, 3, 2] + 24*Perm[4, 2, 1, 5, 3] + 24*Perm[4, 2, 3, 5, 1] - 24*Perm[4, 2, 5, 1, 3] - 24*Perm[4, 2, 5, 3, 1] + 24*Perm[4, 3, 1, 5, 2] + 24*Perm[4, 3, 2, 5, 1] - 24*Perm[4, 3, 5, 1, 2] - 24*Perm[4, 3, 5, 2, 1] + 24*Perm[5, 1, 2, 4, 3] + 24*Perm[5, 1, 3, 4, 2] - 24*Perm[5, 1, 4, 2, 3] - 24*Perm[5, 1, 4, 3, 2] + 24*Perm[5, 2, 1, 4, 3] + 24*Perm[5, 2, 3, 4, 1] - 24*Perm[5, 2, 4, 1, 3] - 24*Perm[5, 2, 4, 3, 1] + 24*Perm[5, 3, 1, 4, 2] + 24*Perm[5, 3, 2, 4, 1] - 24*Perm[5, 3, 4, 1, 2] - 24*Perm[5, 3, 4, 2, 1]\ \>", "\<\ -24 ( 1 4 2 5 3 ) - 24 ( 1 4 3 5 2 ) + 24 ( 1 4 5 2 3 ) + 24 ( 1 4 5 3 2 ) - \ 24 ( 1 5 2 4 3 ) - 24 ( 1 5 3 4 2 ) + 24 ( 1 5 4 2 3 ) + 24 ( 1 5 4 3 2 ) - 24 ( 2 4 1 5 3 ) - \ 24 ( 2 4 3 5 1 ) + 24 ( 2 4 5 1 3 ) + 24 ( 2 4 5 3 1 ) - 24 ( 2 5 1 4 3 ) - \ 24 ( 2 5 3 4 1 ) + 24 ( 2 5 4 1 3 ) + 24 ( 2 5 4 3 1 ) - 24 ( 3 4 1 5 2 ) - \ 24 ( 3 4 2 5 1 ) + 24 ( 3 4 5 1 2 ) + 24 ( 3 4 5 2 1 ) - 24 ( 3 5 1 4 2 ) - \ 24 ( 3 5 2 4 1 ) + 24 ( 3 5 4 1 2 ) + 24 ( 3 5 4 2 1 ) + 24 ( 4 1 2 5 3 ) + \ 24 ( 4 1 3 5 2 ) - 24 ( 4 1 5 2 3 ) - 24 ( 4 1 5 3 2 ) + 24 ( 4 2 1 5 3 ) + \ 24 ( 4 2 3 5 1 ) - 24 ( 4 2 5 1 3 ) - 24 ( 4 2 5 3 1 ) + 24 ( 4 3 1 5 2 ) + \ 24 ( 4 3 2 5 1 ) - 24 ( 4 3 5 1 2 ) - 24 ( 4 3 5 2 1 ) + 24 ( 5 1 2 4 3 ) + \ 24 ( 5 1 3 4 2 ) - 24 ( 5 1 4 2 3 ) - 24 ( 5 1 4 3 2 ) + 24 ( 5 2 1 4 3 ) + \ 24 ( 5 2 3 4 1 ) - 24 ( 5 2 4 1 3 ) - 24 ( 5 2 4 3 1 ) + 24 ( 5 3 1 4 2 ) + \ 24 ( 5 3 2 4 1 ) - 24 ( 5 3 4 1 2 ) - 24 ( 5 3 4 2 1 )\ \>"], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{730, 436}, WindowMargins->{{0, Automatic}, {Automatic, 5}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 142, 4, 150, "Title"], Cell[1876, 57, 56, 0, 64, "Subtitle"], Cell[1935, 59, 135, 3, 71, "Subsubtitle"], Cell[CellGroupData[{ Cell[2095, 66, 31, 0, 30, "Input"], Cell[2129, 68, 794, 18, 297, "Print"] }, Open ]], Cell[2938, 89, 54, 0, 33, "Text"], Cell[CellGroupData[{ Cell[3017, 93, 41, 0, 30, "Input"], Cell[3061, 95, 250, 6, 27, "Output"] }, Open ]], Cell[3326, 104, 76, 0, 33, "Text"], Cell[3405, 106, 67, 0, 30, "Input"], Cell[3475, 108, 73, 0, 30, "Input"], Cell[CellGroupData[{ Cell[3573, 112, 41, 0, 30, "Input"], Cell[3617, 114, 58, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3712, 123, 33, 0, 30, "Input"], Cell[3748, 125, 2049, 53, 207, "Output"] }, Open ]], Cell[5812, 181, 143, 3, 33, "Text"], Cell[CellGroupData[{ Cell[5980, 188, 44, 0, 30, "Input"], Cell[6027, 190, 104, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6168, 199, 29, 0, 30, "Input"], Cell[6200, 201, 58, 4, 27, "Output"] }, Open ]], Cell[6273, 208, 105, 3, 33, "Text"], Cell[6381, 213, 73, 0, 30, "Input"], Cell[6457, 215, 34, 0, 33, "Text"], Cell[CellGroupData[{ Cell[6516, 219, 34, 0, 30, "Input"], Cell[6553, 221, 4916, 132, 487, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11506, 358, 34, 0, 30, "Input"], Cell[11543, 360, 1992, 54, 207, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13572, 419, 35, 0, 30, "Input"], Cell[13610, 421, 4916, 132, 487, "Output"] }, Open ]], Cell[18541, 556, 111, 3, 33, "Text"], Cell[CellGroupData[{ Cell[18677, 563, 77, 0, 30, "Input"], Cell[18757, 565, 554, 15, 87, "Output"] }, Open ]], Cell[19326, 583, 66, 0, 33, "Text"], Cell[CellGroupData[{ Cell[19417, 587, 74, 0, 30, "Input"], Cell[19494, 589, 4587, 95, 487, "Output"] }, Open ]], Cell[24096, 687, 130, 3, 33, "Text"], Cell[CellGroupData[{ Cell[24251, 694, 50, 0, 30, "Input"], Cell[24304, 696, 109, 6, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24450, 707, 50, 0, 30, "Input"], Cell[24503, 709, 109, 6, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24649, 720, 45, 0, 30, "Input"], Cell[24697, 722, 1992, 54, 207, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[26726, 781, 45, 0, 30, "Input"], Cell[26774, 783, 1992, 54, 207, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[28803, 842, 32, 0, 30, "Input"], Cell[28838, 844, 56, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[28931, 853, 32, 0, 30, "Input"], Cell[28966, 855, 2284, 52, 247, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)