(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 16216, 669]*) (*NotebookOutlinePosition[ 16892, 693]*) (* CellTagsIndexPosition[ 16848, 689]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Symmetry operators connected with algebraic curvature tensors\ \>", "Title"], Cell["Bernd Fiedler, Leipzig, June 2001", "Subtitle"], Cell["\<\ Bernd Fiedler, Eichelbaumstr. 13, D-04249 Leipzig, Germany Bernd.Fiedler.RoschStr.Leipzig@t-online.de\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["The problem", "Section"], Cell[TextData[{ "The calculations in this notebook are a contribution to the paper\n\nB. \ Fiedler, Determination of the structure of algebraic curvature tensors by \ means of Young symmetrizers, Seminaire Lotharingien de Combinatoire ", StyleBox["46", FontWeight->"Bold"], " (2001), Article B46???. Submitted to SLC. ", StyleBox["http://www.mat.univie.ac.at/~slc/", FontFamily->"System", FontWeight->"Bold"], StyleBox[".\n\n", FontFamily->"System"], StyleBox[ "We calculate the following group ring elements (symmetry operators) of \ \[DoubleStruckCapitalK][", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalS]\_4\)]], StyleBox["] and determine some properties of these operators.\n\n", FontFamily->"Times New Roman"], Cell[BoxData[GridBox[{ {"operator", \(notation\ under\ Mathematica\)}, { \(\[Xi]\_1\ := \ \((id\ + \ \((1\ 2)\))\)\[CenterDot]\((id\ + \ \((3\ 4)\)) \)\), "sym"}, { \(\[Xi]\_\(-1\)\ := \ \((id\ - \ \((1\ 2)\))\)\[CenterDot]\((id\ - \ \((3\ 4)\)) \)\), "alt"}, {\(f\ := \ id\ + \ \((1\ 3)\) \((2\ 4)\)\), "swap"}, { RowBox[{\(y\_t\), " ", ":=", " ", RowBox[{ "Young", " ", "symmetrizer", " ", "of", " ", GridBox[{ {"1", "3"}, {"2", "4"} }]}]}], "riemy"}, {\(\(y\_t\%*\)\ := \ Star[y\_t]\), \(\ riemystar\)}, { \(\[Sigma]\_1\ := \ \(y\_t\%*\)\[CenterDot]f\[CenterDot]\[Xi]\_1\), "symop1"}, { \(\[Sigma]\_\(-1\)\ := \ \(y\_t\%*\)\[CenterDot]f\[CenterDot]\[Xi]\_\(-1\)\), "symop2"} }, GridFrame->True, RowLines->True, ColumnLines->True]]], "\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The calculation", "Section"], 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: Character tables of S1...S17, DFT of S10 The evaluation of precomputed data is running. Please wait.\ \>", "Print"] }, Open ]], Cell[TextData[{ "First we determine the group ring element ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], " (", StyleBox["sym", FontWeight->"Bold"], ")." }], "Text"], Cell[CellGroupData[{ Cell["sym1 = Perm[1,2,3,4] + Perm[2,1,3,4]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] + Perm[2, 1, 3, 4]\ \>", "\<\ ( 1 2 3 4 ) + ( 2 1 3 4 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["sym2 = Perm[1,2,3,4] + Perm[1,2,4,3]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3]\ \>", "\<\ ( 1 2 3 4 ) + ( 1 2 4 3 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["sym = PermProd[sym1,sym2]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3] + Perm[2, 1, 3, 4] + Perm[2, 1, 4, 3]\ \>", "\<\ ( 1 2 3 4 ) + ( 1 2 4 3 ) + ( 2 1 3 4 ) + ( 2 1 4 3 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Next we determine the group ring element ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_\(-1\)\)]], " (", StyleBox["alt", FontWeight->"Bold"], ")." }], "Text"], Cell[CellGroupData[{ Cell["alt1 = Perm[1,2,3,4] - Perm[2,1,3,4]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[2, 1, 3, 4]\ \>", "\<\ ( 1 2 3 4 ) - ( 2 1 3 4 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["alt2 = Perm[1,2,3,4] - Perm[1,2,4,3]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3]\ \>", "\<\ ( 1 2 3 4 ) - ( 1 2 4 3 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["alt = PermProd[alt1,alt2]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] - Perm[2, 1, 3, 4] + Perm[2, 1, 4, 3]\ \>", "\<\ ( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 2 1 3 4 ) + ( 2 1 4 3 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The element f (", StyleBox["swap", FontWeight->"Bold"], ") is defined by hand." }], "Text"], Cell[CellGroupData[{ Cell["swap = Perm[1,2,3,4] + Perm[3,4,1,2]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] + Perm[3, 4, 1, 2]\ \>", "\<\ ( 1 2 3 4 ) + ( 3 4 1 2 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we generate the Young symmetrizer ", Cell[BoxData[ \(TraditionalForm\`y\_t\)]], " ", StyleBox["(riemy", FontWeight->"Bold"], ") of the tableau t := ", Cell[BoxData[ FormBox[GridBox[{ {"1", "3"}, {"2", "4"} }], TraditionalForm]]], " ." }], "Text"], Cell[CellGroupData[{ Cell["riemtabl = DefTableau[{1,3},{2,4}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 3], TabRow[2, 4]]\ \>", "\<\ {1, 3} {2, 4}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["riemy = YoungSymmetrizer[riemtabl]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] - Perm[1, 4, 2, 3] + Perm[1, 4, 3, 2] - \ Perm[2, 1, 3, 4] + Perm[2, 1, 4, 3] - Perm[2, 3, 1, 4] + Perm[2, 3, 4, 1] + Perm[3, 2, 1, 4] - \ Perm[3, 2, 4, 1] + Perm[3, 4, 1, 2] - Perm[3, 4, 2, 1] + Perm[4, 1, 2, 3] - \ Perm[4, 1, 3, 2] - Perm[4, 3, 1, 2] + Perm[4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 1 4 2 3 ) + ( 1 4 3 2 ) - ( 2 1 3 4 ) + ( 2 1 4 \ 3 ) - ( 2 3 1 4 ) + ( 2 3 4 1 ) + ( 3 2 1 4 ) - ( 3 2 4 1 ) + ( 3 4 1 2 ) - ( 3 4 \ 2 1 ) + ( 4 1 2 3 ) - ( 4 1 3 2 ) - ( 4 3 1 2 ) + ( 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Finally, we determine ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], " (", StyleBox["riemystar", FontWeight->"Bold"], "):" }], "Text"], Cell[CellGroupData[{ Cell["riemystar = Star[riemy]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] - Perm[1, 3, 4, 2] + Perm[1, 4, 3, 2] - \ Perm[2, 1, 3, 4] + Perm[2, 1, 4, 3] + Perm[2, 3, 4, 1] - Perm[2, 4, 3, 1] - Perm[3, 1, 2, 4] + \ Perm[3, 2, 1, 4] + Perm[3, 4, 1, 2] - Perm[3, 4, 2, 1] + Perm[4, 1, 2, 3] - \ Perm[4, 2, 1, 3] - Perm[4, 3, 1, 2] + Perm[4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 1 3 4 2 ) + ( 1 4 3 2 ) - ( 2 1 3 4 ) + ( 2 1 4 \ 3 ) + ( 2 3 4 1 ) - ( 2 4 3 1 ) - ( 3 1 2 4 ) + ( 3 2 1 4 ) + ( 3 4 1 2 ) - ( 3 4 \ 2 1 ) + ( 4 1 2 3 ) - ( 4 2 1 3 ) - ( 4 3 1 2 ) + ( 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we calculate ", Cell[BoxData[ \(\[Sigma]\_1\ = \ \(y\_t\%*\)\[CenterDot]f\[CenterDot]\[Xi]\_1\)]], " (", StyleBox["symop1", FontWeight->"Bold"], ") and show that ", Cell[BoxData[ \(\[Sigma]\_1\)]], " \[NotEqual] 0." }], "Text"], Cell[CellGroupData[{ Cell["symop1 = PermProd[riemystar,PermProd[swap,sym]]", "Input"], Cell[OutputFormData["\<\ -4*Perm[1, 3, 2, 4] - 4*Perm[1, 3, 4, 2] + 4*Perm[1, 4, 2, 3] + 4*Perm[1, 4, \ 3, 2] + 4*Perm[2, 3, 1, 4] + 4*Perm[2, 3, 4, 1] - 4*Perm[2, 4, 1, 3] - 4*Perm[2, 4, \ 3, 1] - 4*Perm[3, 1, 2, 4] - 4*Perm[3, 1, 4, 2] + 4*Perm[3, 2, 1, 4] + 4*Perm[3, 2, \ 4, 1] + 4*Perm[4, 1, 2, 3] + 4*Perm[4, 1, 3, 2] - 4*Perm[4, 2, 1, 3] - 4*Perm[4, 2, \ 3, 1]\ \>", "\<\ -4 ( 1 3 2 4 ) - 4 ( 1 3 4 2 ) + 4 ( 1 4 2 3 ) + 4 ( 1 4 3 2 ) + 4 ( 2 3 1 4 \ ) + 4 ( 2 3 4 1 ) - 4 ( 2 4 1 3 ) - 4 ( 2 4 3 1 ) - 4 ( 3 1 2 4 ) - 4 ( 3 1 4 2 \ ) + 4 ( 3 2 1 4 ) + 4 ( 3 2 4 1 ) + 4 ( 4 1 2 3 ) + 4 ( 4 1 3 2 ) - 4 ( 4 2 1 3 \ ) - 4 ( 4 2 3 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[symop1]", "Input"], Cell[OutputFormData["\<\ 16\ \>", "\<\ 16\ \>"], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(\[Sigma]\_1\)]], " is nilpotent:" }], "Text"], Cell[CellGroupData[{ Cell["PermProd[symop1,symop1]", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Further we calculate ", Cell[BoxData[ \(\[Sigma]\_\(-1\)\ = \ \(y\_t\%*\)\[CenterDot]f\[CenterDot]\[Xi]\_\(-1\)\)]], " (", StyleBox["symop2", FontWeight->"Bold"], ") and show that ", Cell[BoxData[ \(\[Sigma]\_\(-1\)\)]], " \[NotEqual] 0." }], "Text"], Cell[CellGroupData[{ Cell["symop2 = PermProd[riemystar,PermProd[swap,alt]]", "Input"], Cell[OutputFormData["\<\ 8*Perm[1, 2, 3, 4] - 8*Perm[1, 2, 4, 3] + 4*Perm[1, 3, 2, 4] - 4*Perm[1, 3, \ 4, 2] - 4*Perm[1, 4, 2, 3] + 4*Perm[1, 4, 3, 2] - 8*Perm[2, 1, 3, 4] + 8*Perm[2, 1, \ 4, 3] - 4*Perm[2, 3, 1, 4] + 4*Perm[2, 3, 4, 1] + 4*Perm[2, 4, 1, 3] - 4*Perm[2, 4, \ 3, 1] - 4*Perm[3, 1, 2, 4] + 4*Perm[3, 1, 4, 2] + 4*Perm[3, 2, 1, 4] - 4*Perm[3, 2, \ 4, 1] + 8*Perm[3, 4, 1, 2] - 8*Perm[3, 4, 2, 1] + 4*Perm[4, 1, 2, 3] - 4*Perm[4, 1, \ 3, 2] - 4*Perm[4, 2, 1, 3] + 4*Perm[4, 2, 3, 1] - 8*Perm[4, 3, 1, 2] + 8*Perm[4, 3, \ 2, 1]\ \>", "\<\ 8 ( 1 2 3 4 ) - 8 ( 1 2 4 3 ) + 4 ( 1 3 2 4 ) - 4 ( 1 3 4 2 ) - 4 ( 1 4 2 3 ) \ + 4 ( 1 4 3 2 ) - 8 ( 2 1 3 4 ) + 8 ( 2 1 4 3 ) - 4 ( 2 3 1 4 ) + 4 ( 2 3 4 1 ) + 4 ( 2 4 1 3 \ ) - 4 ( 2 4 3 1 ) - 4 ( 3 1 2 4 ) + 4 ( 3 1 4 2 ) + 4 ( 3 2 1 4 ) - 4 ( 3 2 4 1 \ ) + 8 ( 3 4 1 2 ) - 8 ( 3 4 2 1 ) + 4 ( 4 1 2 3 ) - 4 ( 4 1 3 2 ) - 4 ( 4 2 1 3 \ ) + 4 ( 4 2 3 1 ) - 8 ( 4 3 1 2 ) + 8 ( 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[symop2]", "Input"], Cell[OutputFormData["\<\ 24\ \>", "\<\ 24\ \>"], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(\[Sigma]\_\(-1\)\)]], " is essentially idempotent" }], "Text"], Cell[CellGroupData[{ Cell["symop2quad = PermProd[symop2,symop2]", "Input"], Cell[OutputFormData["\<\ 768*Perm[1, 2, 3, 4] - 768*Perm[1, 2, 4, 3] + 384*Perm[1, 3, 2, 4] - \ 384*Perm[1, 3, 4, 2] - 384*Perm[1, 4, 2, 3] + 384*Perm[1, 4, 3, 2] - 768*Perm[2, 1, 3, 4] + \ 768*Perm[2, 1, 4, 3] - 384*Perm[2, 3, 1, 4] + 384*Perm[2, 3, 4, 1] + 384*Perm[2, 4, 1, 3] - \ 384*Perm[2, 4, 3, 1] - 384*Perm[3, 1, 2, 4] + 384*Perm[3, 1, 4, 2] + 384*Perm[3, 2, 1, 4] - \ 384*Perm[3, 2, 4, 1] + 768*Perm[3, 4, 1, 2] - 768*Perm[3, 4, 2, 1] + 384*Perm[4, 1, 2, 3] - \ 384*Perm[4, 1, 3, 2] - 384*Perm[4, 2, 1, 3] + 384*Perm[4, 2, 3, 1] - 768*Perm[4, 3, 1, 2] + \ 768*Perm[4, 3, 2, 1]\ \>", "\<\ 768 ( 1 2 3 4 ) - 768 ( 1 2 4 3 ) + 384 ( 1 3 2 4 ) - 384 ( 1 3 4 2 ) - 384 ( \ 1 4 2 3 ) + 384 ( 1 4 3 2 ) - 768 ( 2 1 3 4 ) + 768 ( 2 1 4 3 ) - 384 ( 2 3 1 4 ) + 384 \ ( 2 3 4 1 ) + 384 ( 2 4 1 3 ) - 384 ( 2 4 3 1 ) - 384 ( 3 1 2 4 ) + 384 ( 3 1 4 2 ) + 384 \ ( 3 2 1 4 ) - 384 ( 3 2 4 1 ) + 768 ( 3 4 1 2 ) - 768 ( 3 4 2 1 ) + 384 ( 4 1 2 3 ) - 384 \ ( 4 1 3 2 ) - 384 ( 4 2 1 3 ) + 384 ( 4 2 3 1 ) - 768 ( 4 3 1 2 ) + 768 ( 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["symop2quad - 96 symop2 //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Finally, we verify some relations:\n\n", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], "\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]], " = 12 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]] }], "Text"], Cell[CellGroupData[{ Cell["PermProd[riemystar,symop1] - 12 symop1 //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]], "\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`\(\(y\_t\%*\)\ \)\)]], "= 0" }], "Text"], Cell[CellGroupData[{ Cell["PermProd[symop1,riemystar]", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], "\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\(-1\)\)]], " = 12 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\(-1\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["PermProd[riemystar,symop2] - 12 symop2 //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\(-1\)\)]], "\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`\(\(y\_t\%*\)\ \)\)]], " =96 ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["PermProd[symop2,riemystar] - 96 riemystar //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]], " \[CenterDot] ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\(-1\)\)]], " = 0" }], "Text"], Cell[CellGroupData[{ Cell["PermProd[symop1,symop2]", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\(-1\)\)]], " \[CenterDot] ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]], " = 96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_1\)]] }], "Text"], Cell[CellGroupData[{ Cell["PermProd[symop2,symop1] - 96 symop1 //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], " \[CenterDot] ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], " = 12 ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["PermProd[riemystar,riemystar] - 12 riemystar //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{762, 614}, WindowMargins->{{0, Automatic}, {Automatic, 5}} ] (*********************************************************************** Cached data follows. 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