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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[ 17348, 657]*) (*NotebookOutlinePosition[ 18024, 681]*) (* CellTagsIndexPosition[ 17980, 677]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[ "Symmetrization of S\[CircleTimes]S and A\[CircleTimes]A by means of the \ Young symmetrizer of the Riemann tensor"], "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["Let ", FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ RowBox[{\(y\_t\), " ", StyleBox["be", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["the", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["Young", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["symmetrizer", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["of", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["the", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["Young", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["tableau", FontFamily->"Times New Roman"], " ", "t"}], " ", ":=", " ", GridBox[{ {"1", "3"}, {"2", "4"} }]}]]], " and e := ", Cell[BoxData[ \(TraditionalForm\`1/12\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], " an idempotent formed from ", Cell[BoxData[ \(TraditionalForm\`y\_t\)]], ". Consider tensors S \[CircleTimes] S and A \[CircleTimes] A, where S, A \ \[Element] ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalT]\_2\)]], "V are symmetric or alternating tensors of order 2 over a \ finite-dimensional \[DoubleStruckCapitalK]-vector space V, \ \[DoubleStruckCapitalK] = \[DoubleStruckCapitalR], \[DoubleStruckCapitalC]. \ In this notebook we calculate e(S \[CircleTimes] S) and e(A \[CircleTimes] A) \ and some simple contractions of these tensors.\n\nIn these calculations we \ use RICPERMS which is a ", "coupling of ", "the ", StyleBox["Mathematica", FontSlant->"Italic"], " packages Ricci and PERMS. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The calculation", "Section"], Cell[CellGroupData[{ Cell["< Default Output Format Type -> OutputForm ------------------------------------------------------------ PERMS, Version 2.2, (26.03.2001), for Mathematica 2.x-4.0 Copyright (c) 1994-2001 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: Character tables of S1...S17, DFT of S10 The evaluation of precomputed data 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["\<\ In the first step we define a Riemannian manifold M with metric g and a set \ {i,j,k,l,r,s,t} of index names.\ \>", "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["We define a symmetric tensor S of oder 2.", "Text"], Cell[CellGroupData[{ Cell["DefineTensor[S,2,Symmetries -> Symmetric]", "Input"], Cell["\<\ Tensor S defined. Rank = 2 Symmetries = Symmetric Type = {Real} Bundle = {TM} Variance = Covariant\ \>", "Print"], Cell[OutputFormData["\<\ S\ \>", "\<\ S\ \>"], "Output"] }, Open ]], Cell["We test wether S is symmetric.", "Text"], Cell[CellGroupData[{ Cell["S[L[j],L[i]]", "Input"], Cell[OutputFormData["\<\ S[L[i], L[j]]\ \>", "\<\ S i j\ \>"], "Output"] }, Open ]], Cell["And we define an alternating tensor A of order 2.", "Text"], Cell[CellGroupData[{ Cell["DefineTensor[A,2,Symmetries -> Alternating]", "Input"], Cell["\<\ Tensor A defined. Rank = 2 Symmetries = Alternating Type = {Real} Bundle = {TM} Variance = Covariant\ \>", "Print"], Cell[OutputFormData["\<\ A\ \>", "\<\ A\ \>"], "Output"] }, Open ]], Cell["We verify that A is skew-symmetric.", "Text"], Cell[CellGroupData[{ Cell["A[L[j],L[i]]", "Input"], Cell[OutputFormData["\<\ -A[L[i], L[j]]\ \>", "\<\ -A i j\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we define functions\nsymprod: (a,b,c,d) \[RightTeeArrow] ", Cell[BoxData[ \(TraditionalForm\`S\_\(a\ b\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`S\_\(c\ d\)\)]], "\naltprod: (a,b,c,d) \[RightTeeArrow] ", Cell[BoxData[ \(TraditionalForm\`A\_\(a\ b\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`A\_\(c\ d\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["symprod = Function[{a,b,c,d},S[L[a],L[b]] S[L[c],L[d]]]", "Input"], Cell[OutputFormData["\<\ Function[{a, b, c, d}, S[L[a], L[b]]*S[L[c], L[d]]]\ \>", "\<\ Function[{a, b, c, d}, S[L[a], L[b]] S[L[c], L[d]]]\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["altprod = Function[{a,b,c,d},A[L[a],L[b]] A[L[c],L[d]]]", "Input"], Cell[OutputFormData["\<\ Function[{a, b, c, d}, A[L[a], L[b]]*A[L[c], L[d]]]\ \>", "\<\ Function[{a, b, c, d}, A[L[a], L[b]] A[L[c], L[d]]]\ \>"], "Output"] }, Open ]], Cell[TextData[{ "We generate the Young symmetrizer ", Cell[BoxData[ \(TraditionalForm\`y\_t\)]], " of the Young tableau t = ", Cell[BoxData[ FormBox[GridBox[{ {"1", "3"}, {"2", "4"} }], TraditionalForm]]], " ." }], "Text"], Cell[CellGroupData[{ Cell["riemtableau = DefTableau[{1,3},{2,4}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 3], TabRow[2, 4]]\ \>", "\<\ {1, 3} {2, 4}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["ys = YoungSymmetrizer[riemtableau]", "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 form the idempotent e = ", Cell[BoxData[ \(TraditionalForm\`1\/12\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(y\_t\%*\)\)]], " ." }], "Text"], Cell[CellGroupData[{ Cell["yf = YoungFactor[Parti[2,2]]", "Input"], Cell[OutputFormData["\<\ 1/12\ \>", "\<\ 1 -- 12\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["ee = Star[ys]* yf //Expand", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4]/12 - Perm[1, 2, 4, 3]/12 - Perm[1, 3, 4, 2]/12 + Perm[1, 4, \ 3, 2]/12 - Perm[2, 1, 3, 4]/12 + Perm[2, 1, 4, 3]/12 + Perm[2, 3, 4, 1]/12 - Perm[2, \ 4, 3, 1]/12 - Perm[3, 1, 2, 4]/12 + Perm[3, 2, 1, 4]/12 + Perm[3, 4, 1, 2]/12 - Perm[3, \ 4, 2, 1]/12 + Perm[4, 1, 2, 3]/12 - Perm[4, 2, 1, 3]/12 - Perm[4, 3, 1, 2]/12 + Perm[4, \ 3, 2, 1]/12\ \>", "\<\ ( 1 2 3 4 ) ( 1 2 4 3 ) ( 1 3 4 2 ) ( 1 4 3 2 ) ( 2 1 3 4 ) ( 2 1 4 \ 3 ) ----------- - ----------- - ----------- + ----------- - ----------- + \ ----------- + 12 12 12 12 12 12 ( 2 3 4 1 ) ( 2 4 3 1 ) ( 3 1 2 4 ) ( 3 2 1 4 ) ( 3 4 1 2 ) ( 3 4 \ 2 1 ) ----------- - ----------- - ----------- + ----------- + ----------- - \ ----------- + 12 12 12 12 12 \ 12 ( 4 1 2 3 ) ( 4 2 1 3 ) ( 4 3 1 2 ) ( 4 3 2 1 ) ----------- - ----------- - ----------- + ----------- 12 12 12 12\ \>"], "Output"] }, Open ]], Cell[TextData["Now we calculate e(S \[CircleTimes] S)."], "Text"], Cell[CellGroupData[{ Cell["s2 = Symmetrize[ee,symprod,{i,j,k,l}]", "Input"], Cell[OutputFormData["\<\ (S[L[i], L[l]]*S[L[j], L[k]])/3 - 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A A + - A A 3 i l j k 3 i k j l 3 i j k l\ \>"], "Output"] }, Open ]], Cell[TextData[ "Again we verify that e(A \[CircleTimes] A) is invariant under a second \ application of e."], "Text"], Cell[CellGroupData[{ Cell["altexpr = Function[{i,j,k,l},Evaluate[a2]]", "Input"], Cell[OutputFormData["\<\ Function[{i, j, k, l}, -(A[L[i], L[l]]*A[L[j], L[k]])/3 + (A[L[i], \ L[k]]*A[L[j], L[l]])/3 + (2*A[L[i], L[j]]*A[L[k], L[l]])/3]\ \>", "\<\ 1 1 2 Function[{i, j, k, l}, -(-) A A + - A A + - A A ] 3 i l j k 3 i k j l 3 i j k l\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Symmetrize[ee,altexpr,{i,j,k,l}] //Expand", "Input"], Cell[OutputFormData["\<\ -(A[L[i], L[l]]*A[L[j], L[k]])/3 + (A[L[i], L[k]]*A[L[j], L[l]])/3 + (2*A[L[i], L[j]]*A[L[k], L[l]])/3\ \>", "\<\ 1 1 2 -(-) A A + - A A + - A A 3 i l j k 3 i k j l 3 i j k l\ \>"], "Output"] }, Open ]], Cell[TextData[ "Finally, we calculate the traces of e(S \[CircleTimes] S) and e(A \ \[CircleTimes] A):"], "Text"], Cell[CellGroupData[{ Cell["traces = g[U[i],U[l]] s2 //TensorSimplify", "Input"], Cell[OutputFormData["\<\ (S[L[i], U[i]]*S[L[j], L[k]])/3 - 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