(*********************************************************************** 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[ 23169, 746]*) (*NotebookOutlinePosition[ 23845, 770]*) (* CellTagsIndexPosition[ 23801, 766]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The symmetry class of the Riemannian curvature tensor", "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["References", "Section"], Cell[TextData[{ "[1] Fulling, S.A., King, R.C., Wybourne, B.G. and Cummins, C.J., Normal \ forms for tensor polynomials: I. The Riemann tensor, Class. Quantum Grav., ", StyleBox["9", FontWeight->"Bold"], ", 1992, 1151 - 1197.\n\n[2] Fiedler, Bernd, A characterization of the \ dependence of the Riemannian metric on the curvature tensor by Young \ symmetrizers, Z. Anal. Anw., ", StyleBox["17", FontWeight->"Bold"], " (1) 1998, 135 -- 157." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The problem", "Section"], Cell[TextData[{ "A result of [1] says that the symmetry class of the Riemannian curvature \ tensor is generated by ", Cell[BoxData[ \(TraditionalForm\`\(y\^\[Star]\)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`y\)]], " is the normalized Young symmetrizer of the standard tableau ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "3"}, {"2", "4"} }], ")"}], TraditionalForm]]], ". A prove of this fact is given in [2], too.\n\nWe verify this result by a \ calculation, which uses the packages PERMS, Ricci and RicciPerms." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The calculation", "Section"], Cell[CellGroupData[{ Cell["< Default Output Format Type -> OutputForm ------------------------------------------------------------ PERMS, Version 2.2, (05.07.2000), for Mathematica 2.x-4.0 Copyright (c) 1994-2000 Bernd Fiedler A.-Rosch-Str. 13, Leipzig, Germany email: Bernd.Fiedler.RoschStr.Leipzig@t-online.de ------------------------------------------------------------ -- For formatted output in a Version 3 or 4 notebook: Cell menu -> 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"], Cell["\<\ Type::shdw: Symbol Type appears in multiple contexts {Perms`, Ricci`}; \ definitions in context Perms` may shadow or be shadowed by other definitions.\ \>", "Message"] }, Open ]], Cell["First we define a differentable Riemannian manifold.", "Text"], Cell[CellGroupData[{ Cell["\<\ DefineBundle[TM,n,g,{i,j,k,l,r,s,t}, MetricType -> Riemannian]\ \>", "Input"], Cell["\<\ Constant n defined. Conjugate[n] = n. Index i associated with TM Index j associated with TM Index k associated with TM Index l associated with TM Index r associated with TM Index s associated with TM Index t associated with TM Tensor g defined. Rank = 2 Symmetries = Symmetric Type = {Real} Bundle = TM Variance = Covariant Tensor Rm defined. Rank = 4 Symmetries = RiemannSymmetries Type = {Real} Bundle = TM Variance = Covariant Tensor Rc defined. Rank = 2 Symmetries = Symmetric Type = {Real} Bundle = TM Variance = Covariant Tensor Sc defined. Rank = 0 Symmetries = NoSymmetries Type = {Real} Bundle = TM Variance = Covariant Bundle TM defined. Metric = g Dimension = n Indices = {i, j, k, l, r, s, t} Bundle Type = Real Metric Type = Riemannian Tangent Bundle = {TM} Connection is torsion free.\ \>", "Print"], Cell[OutputFormData["\<\ TM\ \>", "\<\ TM\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Next we define a pure function that yields the coordinates ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " of the Riemannian curvarure tensor, if the names ", StyleBox["i, j, k, l ", FontSlant->"Italic"], "of indizes", StyleBox[" ", FontSlant->"Italic"], "are given", StyleBox[".", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell["riemfunc = Function[{i,j,k,l},Rm[L[i],L[j],L[k],L[l]]]", "Input"], Cell[OutputFormData["\<\ Function[{i, j, k, l}, Rm[L[i], L[j], L[k], L[l]]]\ \>", "\<\ Function[{i, j, k, l}, Rm[L[i], L[j], L[k], L[l]]]\ \>"], "Output"] }, Open ]], Cell["We determine all partitions of 4.", "Text"], Cell[CellGroupData[{ Cell["allparts = AllPartitions[4]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[1, 1, 1, 1], Parti[2, 1, 1], Parti[2, 2], Parti[3, 1], \ Parti[4]]\ \>", "\<\ {{1, 1, 1, 1}, {2, 1, 1}, {2, 2}, {3, 1}, {4}}\ \>"], "Output"] }, Open ]], Cell["We generate all standard tableaux of all these partitions.", "Text"], Cell[CellGroupData[{ Cell["alltabls = StandardTableaux[#]& /@ allparts", "Input"], Cell[OutputFormData["\<\ HoldList[HoldList[Tableau[TabRow[1], TabRow[2], TabRow[3], TabRow[4]]], HoldList[Tableau[TabRow[1, 2], TabRow[3], TabRow[4]], Tableau[TabRow[1, 3], TabRow[2], TabRow[4]], Tableau[TabRow[1, 4], TabRow[2], TabRow[3]]], HoldList[Tableau[TabRow[1, 2], TabRow[3, 4]], Tableau[TabRow[1, 3], TabRow[2, 4]]], HoldList[Tableau[TabRow[1, 2, 3], TabRow[4]], Tableau[TabRow[1, 2, 4], TabRow[3]], Tableau[TabRow[1, 3, 4], TabRow[2]]], \ HoldList[Tableau[TabRow[1, 2, 3, 4]]]]\ \>", "\<\ {{{1}}, {{1, 2}, {1, 3}, {1, 4}}, {{1, 2}, {1, 3}}, {2} {3} {2} {2} {3, 4} {2, 4} {3} {4} {4} {3} {4} {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 2, 3, 4}}} {4} {3} {2}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["alltabls = Flatten[alltabls]", "Input"], Cell[OutputFormData["\<\ HoldList[Tableau[TabRow[1], TabRow[2], TabRow[3], TabRow[4]], Tableau[TabRow[1, 2], TabRow[3], TabRow[4]], Tableau[TabRow[1, 3], TabRow[2], TabRow[4]], Tableau[TabRow[1, 4], TabRow[2], TabRow[3]], Tableau[TabRow[1, 2], TabRow[3, 4]], Tableau[TabRow[1, 3], TabRow[2, 4]], Tableau[TabRow[1, 2, 3], TabRow[4]], Tableau[TabRow[1, 2, 4], TabRow[3]], Tableau[TabRow[1, 3, 4], TabRow[2]], Tableau[TabRow[1, 2, 3, 4]]]\ \>", "\<\ {{1}, {1, 2}, {1, 3}, {1, 4}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 2, 4}, {1, 3, \ 4}, {2} {3} {2} {2} {3, 4} {2, 4} {4} {3} {2} {3} {4} {4} {3} {4} {1, 2, 3, 4}}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we generate all Young symmetrizers of all these standard tableaux. \ These Young symmetrizers ", StyleBox["y ", FontSlant->"Italic"], "are not normalized, i.e. they are essentially idempotent elements." }], "Text"], Cell[CellGroupData[{ Cell["allys = YoungSymmetrizer[#]& /@ alltabls", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] - Perm[1, 3, 2, 4] + Perm[1, 3, 4, 2] + 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[2, 4, 1, 3] \ + Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] - Perm[3, 1, 4, 2] - 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, 2, 1, 3] - Perm[4, 2, 3, 1] - Perm[4, 3, 1, 2] \ + Perm[4, 3, 2, 1], Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] + Perm[2, 1, 3, 4] - \ Perm[2, 1, 4, 3] - Perm[3, 1, 2, 4] + Perm[3, 1, 4, 2] - Perm[3, 2, 1, 4] \ + Perm[3, 2, 4, 1] + Perm[4, 1, 2, 3] - Perm[4, 1, 3, 2] + Perm[4, 2, 1, 3] \ - Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] - Perm[1, 4, 3, 2] - Perm[2, 1, 3, 4] - \ Perm[2, 3, 1, 4] + Perm[2, 4, 1, 3] + Perm[2, 4, 3, 1] + Perm[3, 2, 1, 4] \ - Perm[3, 4, 1, 2] + Perm[4, 1, 3, 2] - Perm[4, 2, 1, 3] - Perm[4, 2, 3, 1] \ + Perm[4, 3, 1, 2], Perm[1, 2, 3, 4] - Perm[1, 3, 2, 4] - Perm[2, 1, 3, 4] + \ Perm[2, 3, 1, 4] + Perm[2, 3, 4, 1] - Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] \ - Perm[3, 2, 1, 4] - Perm[3, 2, 4, 1] + Perm[3, 4, 2, 1] + Perm[4, 2, 3, 1] \ - Perm[4, 3, 2, 1], 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], 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], Perm[1, 2, 3, 4] + Perm[1, 3, 2, 4] + Perm[2, 1, 3, 4] + \ Perm[2, 3, 1, 4] + Perm[3, 1, 2, 4] + Perm[3, 2, 1, 4] - Perm[4, 1, 2, 3] \ - Perm[4, 1, 3, 2] - Perm[4, 2, 1, 3] - Perm[4, 2, 3, 1] - Perm[4, 3, 1, 2] \ - Perm[4, 3, 2, 1], Perm[1, 2, 3, 4] + Perm[1, 4, 3, 2] + Perm[2, 1, 3, 4] + \ Perm[2, 4, 3, 1] - Perm[3, 1, 2, 4] - Perm[3, 1, 4, 2] - Perm[3, 2, 1, 4] \ - Perm[3, 2, 4, 1] - Perm[3, 4, 1, 2] - Perm[3, 4, 2, 1] + Perm[4, 1, 3, 2] \ + Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3] - Perm[2, 1, 3, 4] - \ Perm[2, 1, 4, 3] - Perm[2, 3, 1, 4] - Perm[2, 3, 4, 1] - Perm[2, 4, 1, 3] \ - Perm[2, 4, 3, 1] + Perm[3, 2, 1, 4] + Perm[3, 2, 4, 1] + Perm[4, 2, 1, 3] \ + Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3] + Perm[1, 3, 2, 4] + \ Perm[1, 3, 4, 2] + 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[2, 4, 1, 3] \ + Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] + Perm[3, 1, 4, 2] + 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, 2, 1, 3] + Perm[4, 2, 3, 1] + Perm[4, 3, 1, 2] \ + Perm[4, 3, 2, 1]]\ \>", "\<\ {( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 1 3 2 4 ) + ( 1 3 4 2 ) + ( 1 4 2 3 ) - ( 1 4 3 2 ) - ( 2 1 3 4 ) + ( 2 1 4 3 ) + ( 2 3 1 4 ) - ( 2 3 4 1 ) - ( 2 4 1 3 ) + ( 2 4 3 1 ) + ( 3 1 2 4 ) - ( 3 1 4 2 ) - ( 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 2 1 3 ) - ( 4 2 3 1 ) - ( 4 3 1 2 ) + ( 4 3 2 1 ), ( 1 2 3 4 ) - ( 1 2 4 3 ) + ( 2 1 3 4 ) - ( 2 1 4 3 ) - ( 3 1 2 4 ) + ( 3 1 4 2 ) - ( 3 2 1 4 ) + ( 3 2 4 1 ) + ( 4 1 2 3 ) - ( 4 1 3 2 ) + ( 4 2 1 3 ) - ( 4 2 3 1 ), ( 1 2 3 4 ) - ( 1 4 3 2 ) - ( 2 1 3 4 ) - ( 2 3 1 4 ) + ( 2 4 1 3 ) + ( 2 4 3 1 ) + ( 3 2 1 4 ) - ( 3 4 1 2 ) + ( 4 1 3 2 ) - ( 4 2 1 3 ) - ( 4 2 3 1 ) + ( 4 3 1 2 ), ( 1 2 3 4 ) - ( 1 3 2 4 ) - ( 2 1 3 4 ) + ( 2 3 1 4 ) + ( 2 3 4 1 ) - ( 2 4 3 1 ) + ( 3 1 2 4 ) - ( 3 2 1 4 ) - ( 3 2 4 1 ) + ( 3 4 2 1 ) + ( 4 2 3 1 ) - ( 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 ), ( 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 ), ( 1 2 3 4 ) + ( 1 3 2 4 ) + ( 2 1 3 4 ) + ( 2 3 1 4 ) + ( 3 1 2 4 ) + ( 3 2 1 4 ) - ( 4 1 2 3 ) - ( 4 1 3 2 ) - ( 4 2 1 3 ) - ( 4 2 3 1 ) - ( 4 3 1 2 ) - ( 4 3 2 1 ), ( 1 2 3 4 ) + ( 1 4 3 2 ) + ( 2 1 3 4 ) + ( 2 4 3 1 ) - ( 3 1 2 4 ) - ( 3 1 4 2 ) - ( 3 2 1 4 ) - ( 3 2 4 1 ) - ( 3 4 1 2 ) - ( 3 4 2 1 ) + ( 4 1 3 2 ) + ( 4 2 3 1 ), ( 1 2 3 4 ) + ( 1 2 4 3 ) - ( 2 1 3 4 ) - ( 2 1 4 3 ) - ( 2 3 1 4 ) - ( 2 3 4 1 ) - ( 2 4 1 3 ) - ( 2 4 3 1 ) + ( 3 2 1 4 ) + ( 3 2 4 1 ) + ( 4 2 1 3 ) + ( 4 2 3 1 ), ( 1 2 3 4 ) + ( 1 2 4 3 ) + ( 1 3 2 4 ) + ( 1 3 4 2 ) + ( 1 4 2 3 ) + ( 1 4 3 2 ) + ( 2 1 3 4 ) + ( 2 1 4 3 ) + ( 2 3 1 4 ) + ( 2 3 4 1 ) + ( 2 4 1 3 ) + ( 2 4 3 1 ) + ( 3 1 2 4 ) + ( 3 1 4 2 ) + ( 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 2 1 3 ) + ( 4 2 3 1 ) + ( 4 3 1 2 ) + ( 4 3 2 1 )}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "We form ", Cell[BoxData[ \(TraditionalForm\`\(y\^\[Star]\)\)]], " for all Young symmetrizers ", StyleBox["y", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["allyy = Star[#]& /@ allys", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"allyy\" is similar to existing \ symbol \"allys\".\ \>", "Message"], Cell[OutputFormData["\<\ HoldList[Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] - Perm[1, 3, 2, 4] + Perm[1, 3, 4, 2] + 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[2, 4, 1, 3] \ + Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] - Perm[3, 1, 4, 2] - 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, 2, 1, 3] - Perm[4, 2, 3, 1] - Perm[4, 3, 1, 2] \ + Perm[4, 3, 2, 1], Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] + Perm[2, 1, 3, 4] - \ Perm[2, 1, 4, 3] - Perm[2, 3, 1, 4] + Perm[2, 3, 4, 1] + Perm[2, 4, 1, 3] \ - Perm[2, 4, 3, 1] - Perm[3, 2, 1, 4] + Perm[3, 2, 4, 1] + Perm[4, 2, 1, 3] \ - Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] - Perm[1, 4, 3, 2] - Perm[2, 1, 3, 4] + \ Perm[2, 4, 3, 1] - Perm[3, 1, 2, 4] + Perm[3, 1, 4, 2] + Perm[3, 2, 1, 4] \ - Perm[3, 2, 4, 1] - Perm[3, 4, 1, 2] + Perm[3, 4, 2, 1] + Perm[4, 1, 3, 2] \ - Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] - Perm[1, 3, 2, 4] - Perm[2, 1, 3, 4] + \ Perm[2, 3, 1, 4] + Perm[3, 1, 2, 4] - Perm[3, 2, 1, 4] + Perm[4, 1, 2, 3] \ - Perm[4, 1, 3, 2] - Perm[4, 2, 1, 3] + Perm[4, 2, 3, 1] + Perm[4, 3, 1, 2] \ - Perm[4, 3, 2, 1], 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], 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], Perm[1, 2, 3, 4] + Perm[1, 3, 2, 4] + Perm[2, 1, 3, 4] + \ Perm[2, 3, 1, 4] - Perm[2, 3, 4, 1] - Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] \ + Perm[3, 2, 1, 4] - Perm[3, 2, 4, 1] - Perm[3, 4, 2, 1] - Perm[4, 2, 3, 1] \ - Perm[4, 3, 2, 1], Perm[1, 2, 3, 4] + Perm[1, 4, 3, 2] + Perm[2, 1, 3, 4] - \ Perm[2, 3, 1, 4] - Perm[2, 4, 1, 3] + Perm[2, 4, 3, 1] - Perm[3, 2, 1, 4] \ - Perm[3, 4, 1, 2] + Perm[4, 1, 3, 2] - Perm[4, 2, 1, 3] + Perm[4, 2, 3, 1] \ - Perm[4, 3, 1, 2], Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3] - Perm[2, 1, 3, 4] - \ Perm[2, 1, 4, 3] - Perm[3, 1, 2, 4] - Perm[3, 1, 4, 2] + Perm[3, 2, 1, 4] \ + Perm[3, 2, 4, 1] - Perm[4, 1, 2, 3] - Perm[4, 1, 3, 2] + Perm[4, 2, 1, 3] \ + Perm[4, 2, 3, 1], Perm[1, 2, 3, 4] + Perm[1, 2, 4, 3] + Perm[1, 3, 2, 4] + \ Perm[1, 3, 4, 2] + 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[2, 4, 1, 3] \ + Perm[2, 4, 3, 1] + Perm[3, 1, 2, 4] + Perm[3, 1, 4, 2] + 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, 2, 1, 3] + Perm[4, 2, 3, 1] + Perm[4, 3, 1, 2] \ + Perm[4, 3, 2, 1]]\ \>", "\<\ {( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 1 3 2 4 ) + ( 1 3 4 2 ) + ( 1 4 2 3 ) - ( 1 4 3 2 ) - ( 2 1 3 4 ) + ( 2 1 4 3 ) + ( 2 3 1 4 ) - ( 2 3 4 1 ) - ( 2 4 1 3 ) + ( 2 4 3 1 ) + ( 3 1 2 4 ) - ( 3 1 4 2 ) - ( 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 2 1 3 ) - ( 4 2 3 1 ) - ( 4 3 1 2 ) + ( 4 3 2 1 ), ( 1 2 3 4 ) - ( 1 2 4 3 ) + ( 2 1 3 4 ) - ( 2 1 4 3 ) - ( 2 3 1 4 ) + ( 2 3 4 1 ) + ( 2 4 1 3 ) - ( 2 4 3 1 ) - ( 3 2 1 4 ) + ( 3 2 4 1 ) + ( 4 2 1 3 ) - ( 4 2 3 1 ), ( 1 2 3 4 ) - ( 1 4 3 2 ) - ( 2 1 3 4 ) + ( 2 4 3 1 ) - ( 3 1 2 4 ) + ( 3 1 4 2 ) + ( 3 2 1 4 ) - ( 3 2 4 1 ) - ( 3 4 1 2 ) + ( 3 4 2 1 ) + ( 4 1 3 2 ) - ( 4 2 3 1 ), ( 1 2 3 4 ) - ( 1 3 2 4 ) - ( 2 1 3 4 ) + ( 2 3 1 4 ) + ( 3 1 2 4 ) - ( 3 2 1 4 ) + ( 4 1 2 3 ) - ( 4 1 3 2 ) - ( 4 2 1 3 ) + ( 4 2 3 1 ) + ( 4 3 1 2 ) - ( 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 ), ( 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 ), ( 1 2 3 4 ) + ( 1 3 2 4 ) + ( 2 1 3 4 ) + ( 2 3 1 4 ) - ( 2 3 4 1 ) - ( 2 4 3 1 ) + ( 3 1 2 4 ) + ( 3 2 1 4 ) - ( 3 2 4 1 ) - ( 3 4 2 1 ) - ( 4 2 3 1 ) - ( 4 3 2 1 ), ( 1 2 3 4 ) + ( 1 4 3 2 ) + ( 2 1 3 4 ) - ( 2 3 1 4 ) - ( 2 4 1 3 ) + ( 2 4 3 1 ) - ( 3 2 1 4 ) - ( 3 4 1 2 ) + ( 4 1 3 2 ) - ( 4 2 1 3 ) + ( 4 2 3 1 ) - ( 4 3 1 2 ), ( 1 2 3 4 ) + ( 1 2 4 3 ) - ( 2 1 3 4 ) - ( 2 1 4 3 ) - ( 3 1 2 4 ) - ( 3 1 4 2 ) + ( 3 2 1 4 ) + ( 3 2 4 1 ) - ( 4 1 2 3 ) - ( 4 1 3 2 ) + ( 4 2 1 3 ) + ( 4 2 3 1 ), ( 1 2 3 4 ) + ( 1 2 4 3 ) + ( 1 3 2 4 ) + ( 1 3 4 2 ) + ( 1 4 2 3 ) + ( 1 4 3 2 ) + ( 2 1 3 4 ) + ( 2 1 4 3 ) + ( 2 3 1 4 ) + ( 2 3 4 1 ) + ( 2 4 1 3 ) + ( 2 4 3 1 ) + ( 3 1 2 4 ) + ( 3 1 4 2 ) + ( 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 2 1 3 ) + ( 4 2 3 1 ) + ( 4 3 1 2 ) + ( 4 3 2 1 )}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we symmetrize the Riemannian curvature tensor by means of all these ", Cell[BoxData[ \(TraditionalForm\`\(y\^\[Star]\)\)]], ". Only two results are unequal to zero." }], "Text"], Cell[CellGroupData[{ Cell["\<\ symmetrizedriem = List @@ (Symmetrize[#,riemfunc,{i,j,k,l}]& /@ allyy)\ \>", "Input"], Cell[OutputFormData["\<\ {8*Rm[L[i], L[j], L[k], L[l]] - 8*Rm[L[i], L[k], L[j], L[l]] + 8*Rm[L[i], L[l], L[j], L[k]], 0, 0, 0, 0, 8*Rm[L[i], L[j], L[k], L[l]] + 4*Rm[L[i], L[k], L[j], L[l]] - 4*Rm[L[i], L[l], L[j], L[k]], 0, 0, 0, 0}\ \>", "\<\ {8 Rm - 8 Rm + 8 Rm , 0, 0, 0, 0, i j k l i k j l i l j k 8 Rm + 4 Rm - 4 Rm , 0, 0, 0, 0} i j k l i k j l i l j k\ \>"], "Output"] }, Open ]], Cell["\<\ If we apply the First Bianchi identity, we see that only one result is \ unequal to zero.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ symmetrizedriem = symmetrizedriem /. BianchiRules[L[k],L[j],L[l]] //Expand\ \>", "Input"], Cell[OutputFormData["\<\ {0, 0, 0, 0, 0, 12*Rm[L[i], L[j], L[k], L[l]], 0, 0, 0, 0}\ \>", "\<\ {0, 0, 0, 0, 0, 12 Rm , 0, 0, 0, 0} i j k l\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The tableau of the symmetrizer ", StyleBox["y ", FontSlant->"Italic"], "that yields 12 ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], ", is the 6th tableau ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "3"}, {"2", "4"} }], ")"}], TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Complement[Range[Length[symmetrizedriem]], Flatten[Position[symmetrizedriem,0]] ]\ \>", "Input"], Cell[OutputFormData["\<\ {6}\ \>", "\<\ {6}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["alltabls[[6]]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 3], TabRow[2, 4]]\ \>", "\<\ {1, 3} {2, 4}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The above symmetrizer ", StyleBox["y", FontSlant->"Italic"], " can be normalized by a multiplication by ", Cell[BoxData[ \(TraditionalForm\`1\/12\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["YoungFactor[Parti[2,2]]", "Input"], Cell[OutputFormData["\<\ 1/12\ \>", "\<\ 1 -- 12\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Thus the normalized symmetrizer ", StyleBox["e", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`1\/12\)]], " ", StyleBox["y", FontSlant->"Italic"], " fulfils ", StyleBox["e R = R", FontSlant->"Italic"], "." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{685, 594}, WindowMargins->{{0, Automatic}, {Automatic, 5}} ] (*********************************************************************** Cached data follows. 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