(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 37477, 1513]*) (*NotebookOutlinePosition[ 38153, 1537]*) (* CellTagsIndexPosition[ 38109, 1533]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ The left ideal that describes the symmetry of the 1. covariant derivative 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["< 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[TextData[{ "The first covariant derivative", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\ ; \ m\)\)]], " of the Riemannian curvatore tensor ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " has the following index commutation symmetry\n\n(1) ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\ ; \ m\)\)]], " = - ", Cell[BoxData[ \(TraditionalForm\`R\_\(j\ i\ k\ l\ ; \ m\)\)]], " = - ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ l\ k\ ; \ m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`R\_\(k\ l\ i\ j\ ; \ m\)\)]], "\n\nand satisfies the first Bianchi identity\n\n(2) ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\ ; \ m\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(\(R\_\(i\ k\ l\ j\ ; \ m\)\ + \)\ \)\)]], Cell[BoxData[ \(TraditionalForm\`R\_\(i\ l\ j\ k\ ; \ m\)\)]], " = 0\n\nand the second Bianchi identity\n\n(3) ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\ ; \ m\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(\(R\_\(i\ j\ l\ m\ ; \ k\)\ + \)\ \)\)]], Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ m\ k\ ; \ l\)\)]], " = 0.\n\nCondition (1) yields that the group ring elements ", Cell[BoxData[ \(TraditionalForm\`\((\[Del]R)\)\_b\)]], " of ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\ ; m\)\)]], " lie in the left ideal ", Cell[BoxData[ \(TraditionalForm\`I\_1\)]], " = \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "]\[CenterDot]\[Chi] of the following symmetrizer \[Chi]:" }], "Text"], Cell["The generators of the symmetry group:", "Text"], Cell[CellGroupData[{ Cell["\<\ gens = HoldList[Perm[2,1,3,4,5],Perm[1,2,4,3,5],Perm[3,4,1,2,5]]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[2, 1, 3, 4, 5], Perm[1, 2, 4, 3, 5], Perm[3, 4, 1, 2, 5]]\ \>", "\<\ {( 2 1 3 4 5 ), ( 1 2 4 3 5 ), ( 3 4 1 2 5 )}\ \>"], "Output"] }, Open ]], Cell["The symmetry group:", "Text"], Cell[CellGroupData[{ Cell["symgrp = GeneratedGroup[gens]", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[1, 2, 3, 4, 5], Perm[2, 1, 3, 4, 5], Perm[1, 2, 4, 3, 5], Perm[3, 4, 1, 2, 5], Perm[2, 1, 4, 3, 5], Perm[3, 4, 2, 1, 5], Perm[4, 3, \ 1, 2, 5], Perm[4, 3, 2, 1, 5]]\ \>", "\<\ {( 1 2 3 4 5 ), ( 2 1 3 4 5 ), ( 1 2 4 3 5 ), ( 3 4 1 2 5 ), ( 2 1 4 3 5 ), ( 3 4 2 1 5 ), ( 4 3 1 2 5 ), ( 4 3 2 1 5 )}\ \>"], "Output"] }, Open ]], Cell["\<\ Obviously, these permutations lead to the following signs of the curvature \ tensor:\ \>", "Text"], Cell[CellGroupData[{ Cell["signs = {1, -1, -1, 1, 1, -1, -1, 1}", "Input"], Cell[OutputFormData["\<\ {1, -1, -1, 1, 1, -1, -1, 1}\ \>", "\<\ {1, -1, -1, 1, 1, -1, -1, 1}\ \>"], "Output"] }, Open ]], Cell["Thus we obtain the symmetrizer", "Text"], Cell[CellGroupData[{ Cell["chi = signs.(List @@ symgrp)", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] - Perm[1, 2, 4, 3, 5] - Perm[2, 1, 3, 4, 5] + Perm[2, 1, \ 4, 3, 5] + Perm[3, 4, 1, 2, 5] - Perm[3, 4, 2, 1, 5] - Perm[4, 3, 1, 2, 5] + Perm[4, \ 3, 2, 1, 5]\ \>", "\<\ ( 1 2 3 4 5 ) - ( 1 2 4 3 5 ) - ( 2 1 3 4 5 ) + ( 2 1 4 3 5 ) + ( 3 4 1 2 5 ) \ - ( 3 4 2 1 5 ) - ( 4 3 1 2 5 ) + ( 4 3 2 1 5 )\ \>"], "Output"] }, Open ]], Cell[TextData[ "\[Chi] is essentially idempotent, i.e. \[Chi]\[CenterDot]\[Chi] = 8 \ \[Chi]:"], "Text"], Cell[CellGroupData[{ Cell["PermProd[chi,chi]", "Input"], Cell[OutputFormData["\<\ 8*Perm[1, 2, 3, 4, 5] - 8*Perm[1, 2, 4, 3, 5] - 8*Perm[2, 1, 3, 4, 5] + 8*Perm[2, 1, 4, 3, 5] + 8*Perm[3, 4, 1, 2, 5] - 8*Perm[3, 4, 2, 1, 5] - 8*Perm[4, 3, 1, 2, 5] + 8*Perm[4, 3, 2, 1, 5]\ \>", "\<\ 8 ( 1 2 3 4 5 ) - 8 ( 1 2 4 3 5 ) - 8 ( 2 1 3 4 5 ) + 8 ( 2 1 4 3 5 ) + 8 ( 3 4 1 2 5 ) - 8 ( 3 4 2 1 5 ) - 8 ( 4 3 1 2 5 ) + 8 ( 4 3 2 1 5 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Because of (2) , the covariant derivative \[Del]", StyleBox["R", FontSlant->"Italic"], " of the curvature tensor fulfils ", StyleBox["a", FontSlant->"Italic"], "(\[Del]", StyleBox["R", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "= 0 with" }], "Text"], Cell[CellGroupData[{ Cell["a = Perm[1,2,3,4,5] + Perm[1,3,4,2,5] + Perm[1,4,2,3,5]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] + Perm[1, 3, 4, 2, 5] + Perm[1, 4, 2, 3, 5]\ \>", "\<\ ( 1 2 3 4 5 ) + ( 1 3 4 2 5 ) + ( 1 4 2 3 5 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Consequently, all ", Cell[BoxData[ \(TraditionalForm\`\((\[Del]R)\)\_b\)]], " of \[Del]", StyleBox["R ", FontSlant->"Italic"], "lie in the left annihilator ideal ", Cell[BoxData[ \(TraditionalForm\`I\_2\)]], " := ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA]\_l\)]], "(", StyleBox["a", FontSlant->"Italic"], ") = {", StyleBox["x", FontSlant->"Italic"], " \[Element] \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "] | ", StyleBox["x\[CenterDot]a", FontSlant->"Italic"], " = 0 } of ", StyleBox["a", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\^*\)\)]], "." }], "Text"], Cell[TextData[{ "Because of (3) , the covariant derivative \[Del]", StyleBox["R", FontSlant->"Italic"], " of the curvature tensor fulfils ", StyleBox["b", FontSlant->"Italic"], "(\[Del]", StyleBox["R", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "= 0 with" }], "Text"], Cell[CellGroupData[{ Cell["b = Perm[1,2,3,4,5] + Perm[1,2,4,5,3] + Perm[1,2,5,3,4]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5] + Perm[1, 2, 4, 5, 3] + Perm[1, 2, 5, 3, 4]\ \>", "\<\ ( 1 2 3 4 5 ) + ( 1 2 4 5 3 ) + ( 1 2 5 3 4 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Consequently, all ", Cell[BoxData[ \(TraditionalForm\`\((\[Del]R)\)\_b\)]], " of \[Del]", StyleBox["R ", FontSlant->"Italic"], "lie in the left annihilator ideal ", Cell[BoxData[ \(TraditionalForm\`I\_3\)]], " := ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA]\_l\)]], "(", StyleBox["a", FontSlant->"Italic"], ") = {", StyleBox["x", FontSlant->"Italic"], " \[Element] \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "] | ", StyleBox["x\[CenterDot]b", FontSlant->"Italic"], " = 0 } of ", StyleBox["b", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\(b\^*\)\)]], "." }], "Text"], Cell[TextData[{ "Now we search for a generating idempotent ", StyleBox["e", FontSlant->"Italic"], " of the left ideal ", StyleBox["I", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`I\_1\)]], "\[Intersection] ", Cell[BoxData[ \(TraditionalForm\`I\_2\)]], " \[Intersection] ", Cell[BoxData[ \(TraditionalForm\`I\_3\)]], "." }], "Text"], Cell[TextData[{ "The idempotent ", StyleBox["f ", FontSlant->"Italic"], "with f \[Tilde] \[Chi] reads" }], "Text"], Cell[CellGroupData[{ Cell["f = 1/8 chi //Expand", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5]/8 - Perm[1, 2, 4, 3, 5]/8 - Perm[2, 1, 3, 4, 5]/8 + Perm[2, 1, 4, 3, 5]/8 + Perm[3, 4, 1, 2, 5]/8 - Perm[3, 4, 2, 1, 5]/8 - Perm[4, 3, 1, 2, 5]/8 + Perm[4, 3, 2, 1, 5]/8\ \>", "\<\ ( 1 2 3 4 5 ) ( 1 2 4 3 5 ) ( 2 1 3 4 5 ) ( 2 1 4 3 5 ) ( 3 4 1 2 5 ) ------------- - ------------- - ------------- + ------------- + ------------- \ - 8 8 8 8 8 ( 3 4 2 1 5 ) ( 4 3 1 2 5 ) ( 4 3 2 1 5 ) ------------- - ------------- + ------------- 8 8 8\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The right annihilator ideal ", StyleBox["J", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA]\_r\)]], "(", StyleBox["I", FontSlant->"Italic"], ") if ", StyleBox["I", FontSlant->"Italic"], " fulfils ", StyleBox["J", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`J\_1\)]], " + ", Cell[BoxData[ \(TraditionalForm\`J\_2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`J\_3\)]], " where ", Cell[BoxData[ \(TraditionalForm\`J\_1\)]], " = (1 - ", StyleBox["f", FontSlant->"Italic"], ")\[CenterDot]\[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "], ", Cell[BoxData[ \(TraditionalForm\`J\_2\)]], " = ", StyleBox["a", FontSlant->"Italic"], "\[CenterDot]\[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "]. and ", Cell[BoxData[ \(TraditionalForm\`J\_3\)]], " = ", StyleBox["b", FontSlant->"Italic"], "\[CenterDot]\[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_5\)]], "].We have" }], "Text"], Cell[CellGroupData[{ Cell["h = Perm[1,2,3,4,5] - f", "Input"], Cell[OutputFormData["\<\ (7*Perm[1, 2, 3, 4, 5])/8 + Perm[1, 2, 4, 3, 5]/8 + Perm[2, 1, 3, 4, 5]/8 - Perm[2, 1, 4, 3, 5]/8 - Perm[3, 4, 1, 2, 5]/8 + Perm[3, 4, 2, 1, 5]/8 + Perm[4, 3, 1, 2, 5]/8 - Perm[4, 3, 2, 1, 5]/8\ \>", "\<\ 7 ( 1 2 3 4 5 ) ( 1 2 4 3 5 ) ( 2 1 3 4 5 ) ( 2 1 4 3 5 ) ( 3 4 1 2 5 \ ) --------------- + ------------- + ------------- - ------------- - \ ------------- + 8 8 8 8 8 ( 3 4 2 1 5 ) ( 4 3 1 2 5 ) ( 4 3 2 1 5 ) ------------- + ------------- - ------------- 8 8 8\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we determine the discrete Fourier transforms A, B and H of ", StyleBox["a, b ", FontSlant->"Italic"], "and ", StyleBox["h", FontSlant->"Italic"], ". They have the following block matrices:" }], "Text"], Cell[CellGroupData[{ Cell["(A1 = FourierTransform[Parti[5],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A2 = FourierTransform[Parti[4,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3, 0, 0, 0}, {0, 1, 1, 1}, {0, 1, 1, 1}, {0, 1, 1, 1}}\ \>", "\<\ 3 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A3 = FourierTransform[Parti[3,2],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 1, -1, 1, -1}, {1, 1, -1, 1, -1}, {0, 0, 0, 0, 0}, {1, 1, -1, 1, -1}, {0, 0, 0, 0, 0}}\ \>", "\<\ 1 1 -1 1 -1 1 1 -1 1 -1 0 0 0 0 0 1 1 -1 1 -1 0 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A4 = FourierTransform[Parti[3,1,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 1, 0, 1, 0, 0}, {1, 1, 0, 1, 0, 0}, {0, 0, 1, 0, -1, 1}, {1, 1, 0, 1, 0, \ 0}, {0, 0, -1, 0, 1, -1}, {0, 0, 1, 0, -1, 1}}\ \>", "\<\ 1 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 -1 1 1 1 0 1 0 0 0 0 -1 0 1 -1 0 0 1 0 -1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A5 = FourierTransform[Parti[2,2,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0, 0, 0, 0, 0}, {0, 1, 0, -1, 1}, {0, -1, 0, 1, -1}, {0, -1, 0, 1, -1}, {0, 1, 0, -1, 1}}\ \>", "\<\ 0 0 0 0 0 0 1 0 -1 1 0 -1 0 1 -1 0 -1 0 1 -1 0 1 0 -1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A6 = FourierTransform[Parti[2,1,1,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, -1, 1, 0}, {-1, 1, -1, 0}, {1, -1, 1, 0}, {0, 0, 0, 3}}\ \>", "\<\ 1 -1 1 0 -1 1 -1 0 1 -1 1 0 0 0 0 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A7 = FourierTransform[Parti[1,1,1,1,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B1 = FourierTransform[Parti[5],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B2 = FourierTransform[Parti[4,1],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 1, 1, 0}, {1, 1, 1, 0}, {1, 1, 1, 0}, {0, 0, 0, 3}}\ \>", "\<\ 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B3 = FourierTransform[Parti[3,2],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 1, 1, -1, -1}, {1, 1, 1, -1, -1}, {1, 1, 1, -1, -1}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}\ \>", "\<\ 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B4 = FourierTransform[Parti[3,1,1],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, -1, 1, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {1, -1, 1, 0, 0, 0}, {0, 0, 0, 1, \ 1, 1}, {0, 0, 0, 1, 1, 1}, {0, 0, 0, 1, 1, 1}}\ \>", "\<\ 1 -1 1 0 0 0 -1 1 -1 0 0 0 1 -1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B5 = FourierTransform[Parti[2,2,1],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0, 0, 0, 0, 0}, {0, 0, -1, 1, -1}, {0, 0, 1, -1, 1}, {0, 0, -1, 1, -1}, {0, 0, 1, -1, 1}}\ \>", "\<\ 0 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 1 0 0 -1 1 -1 0 0 1 -1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B6 = FourierTransform[Parti[2,1,1,1],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3, 0, 0, 0}, {0, 1, -1, 1}, {0, -1, 1, -1}, {0, 1, -1, 1}}\ \>", "\<\ 3 0 0 0 0 1 -1 1 0 -1 1 -1 0 1 -1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(B7 = FourierTransform[Parti[1,1,1,1,1],b]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H1 = FourierTransform[Parti[5],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1}}\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H2 = FourierTransform[Parti[4,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H3 = FourierTransform[Parti[3,2],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0, -3/4}, {0, 1, 0, 0, -1/4}, {0, 0, 1, 0, -1/2}, {0, 0, 0, 1, \ -1/2}, {0, 0, 0, 0, 0}}\ \>", "\<\ 3 -(-) 1 0 0 0 4 1 -(-) 0 1 0 0 4 1 -(-) 0 0 1 0 2 1 -(-) 0 0 0 1 2 0 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H4 = FourierTransform[Parti[3,1,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, \ 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H5 = FourierTransform[Parti[2,2,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, -1/2, 1/4, -1/4}, {0, 1, 0, 0, 0}, {0, 0, 0, 1/2, -1/2}, {0, 0, 0, 1, \ 0}, {0, 0, 0, 0, 1}}\ \>", "\<\ 1 1 1 -(-) - -(-) 1 0 2 4 4 0 1 0 0 0 1 1 - -(-) 0 0 0 2 2 0 0 0 1 0 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H6 = FourierTransform[Parti[2,1,1,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, -1/4}, {0, 1, 0, 1/4}, {0, 0, 1, -1/4}, {0, 0, 0, 0}}\ \>", "\<\ 1 -(-) 1 0 0 4 1 - 0 1 0 4 1 -(-) 0 0 1 4 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H7 = FourierTransform[Parti[1,1,1,1,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0}}\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we calculate block matrices of the generating idempotent ", StyleBox["k", FontSlant->"Italic"], " of ", StyleBox["J", FontSlant->"Italic"], " by means of DecomposeR:" }], "Text"], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[1]\ \>", "Input"], Cell[OutputFormData["\<\ 1\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K1 = DecomposeR[{A1,B1,H1}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1}}, HoldList[{{1}}]]\ \>", "\<\ {{{1}}, {{{1}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K1[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1}}\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K7 = DecomposeR[{A7,B7,H7}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1}}, HoldList[{{1}}]]\ \>", "\<\ {{{1}}, {{{1}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K7[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1}}\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[4]\ \>", "Input"], Cell[OutputFormData["\<\ 4\ \>", "\<\ 4\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K2 = DecomposeR[{A2,B2,H2}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}, HoldList[{{1, 0, -1, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 0, 1}, {0, 0, 0, 1}}, {{0, 0, 1, -1}, {0, 0, 1, -1}, {0, 0, 1, -1}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 1, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}, {{{1, 0, -1, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 0, 1}, {0, 0, 0, 1}}, {{0, 0, 1, -1}, {0, 0, 1, -1}, {0, 0, 1, -1}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 1, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K2[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K6 = DecomposeR[{A6,B6,H6}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}, HoldList[{{0, 0, 1, 0}, {0, 0, -1, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}}, {{1, 0, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 1, 1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}, {{{0, 0, 1, 0}, {0, 0, -1, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}}, {{1, 0, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 1, 1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K6[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[5]\ \>", "Input"], Cell[OutputFormData["\<\ 5\ \>", "\<\ 5\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K3 = DecomposeR[{A3,B3,H3}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, \ {0, 0, 0, 0, 0}}, HoldList[{{0, 0, 0, 1, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, \ 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 0}}, {{0, 0, 1, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{1, 0, -1, -1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0}, {0, 1, -1, -1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, \ 0, 0, 0}}, {{{0, 0, 0, 1, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 0}}, {{0, 0, 1, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{1, 0, -1, -1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0}, {0, 1, -1, -1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K3[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, \ 0, 0, 0}}\ \>", "\<\ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K5 = DecomposeR[{A5,B5,H5}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, \ {0, 0, 0, 0, 1}}, HoldList[{{0, 0, 0, 0, 0}, {0, 0, -1/2, 1/4, 3/4}, {0, 0, 1/2, -1/4, -3/4}, {0, 0, 1/2, -1/4, -3/4}, {0, 0, -1/2, 1/4, \ 3/4}}, {{0, 0, 0, 0, 0}, {0, 0, -1/2, 1/4, -1/4}, {0, 0, 1/2, -1/4, 1/4}, {0, 0, -1/2, 1/4, -1/4}, {0, 0, 1/2, -1/4, 1/4}}, {{1, 0, 0, -1/4, -1/4}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, \ {0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0}, {0, 1, 1, -1/2, -1/2}, {0, 0, 0, 0, \ 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{0, 0, 0, 1/4, 1/4}, {0, 0, 0, 0, 0}, {0, 0, 0, 1/2, 1/2}, {0, 0, 0, 1, \ 1}, {0, 0, 0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, \ 0, 0, 1}}, 1 1 3 1 1 3 1 1 \ 3 {{{0, 0, 0, 0, 0}, {0, 0, -(-), -, -}, {0, 0, -, -(-), -(-)}, {0, 0, -, \ -(-), -(-)}, 2 4 4 2 4 4 2 4 \ 4 1 1 3 1 1 1 1 \ 1 1 {0, 0, -(-), -, -}}, {{0, 0, 0, 0, 0}, {0, 0, -(-), -, -(-)}, {0, 0, -, \ -(-), -}, 2 4 4 2 4 4 2 \ 4 4 1 1 1 1 1 1 {0, 0, -(-), -, -(-)}, {0, 0, -, -(-), -}}, 2 4 4 2 4 4 1 1 {{1, 0, 0, -(-), -(-)}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, \ 4 4 1 1 {0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0}, {0, 1, 1, -(-), -(-)}, {0, 0, 0, 0, \ 0}, 2 2 {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, 1 1 1 1 {{0, 0, 0, -, -}, {0, 0, 0, 0, 0}, {0, 0, 0, -, -}, {0, 0, 0, 1, 1}, {0, \ 0, 0, 0, 0}}} 4 4 2 2 }\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K5[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, \ 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[6]\ \>", "Input"], Cell[OutputFormData["\<\ 6\ \>", "\<\ 6\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K4 = DecomposeR[{A4,B4,H4}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}, HoldList[{{0, 0, 0, 1, -1/2, -1/2}, {0, 0, 0, 1, -1/2, -1/2}, {0, 0, 0, 0, \ 0, 0}, {0, 0, 0, 1, -1/2, -1/2}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, -1/2, 1/2}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1/2, -1/2}, {0, 0, 0, 0, -1/2, 1/2}}, {{0, 0, 1, 0, 1/2, -1/2}, {0, 0, -1, 0, -1/2, 1/2}, {0, 0, 1, 0, 1/2, \ -1/2}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, \ 1/2, 1/2}, {0, 0, 0, 0, 1/2, 1/2}, {0, 0, 0, 0, 1/2, 1/2}}, {{1, 0, -1, -1, 0, 1}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, \ 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 1, 1, -1, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, \ 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, \ 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}, 1 1 1 1 {{{0, 0, 0, 1, -(-), -(-)}, {0, 0, 0, 1, -(-), -(-)}, {0, 0, 0, 0, 0, 0}, 2 2 2 2 1 1 {0, 0, 0, 1, -(-), -(-)}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, 2 2 1 1 {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, -(-), -}, {0, 0, 0, \ 0, 0, 0}, 2 2 1 1 1 1 {0, 0, 0, 0, -, -(-)}, {0, 0, 0, 0, -(-), -}}, 2 2 2 2 1 1 1 1 1 1 {{0, 0, 1, 0, -, -(-)}, {0, 0, -1, 0, -(-), -}, {0, 0, 1, 0, -, -(-)}, 2 2 2 2 2 2 {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, 1\ 1 {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, \ -, -}, 2\ 2 1 1 1 1 {0, 0, 0, 0, -, -}, {0, 0, 0, 0, -, -}}, 2 2 2 2 {{1, 0, -1, -1, 0, 1}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, \ 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 1, 1, -1, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, \ 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K4[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, \ 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}\ \>", "\<\ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Thus the only non-vanishing block matrix of the Fourier transform E of ", StyleBox["e", FontSlant->"Italic"], " is" }], "Text"], Cell[CellGroupData[{ Cell["(E3 = IdentityMatrix[5] - K3[[1]]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, \ 0, 0, 1}}\ \>", "\<\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["e = InvFourierTransform[Parti[3,2],E3]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5]/24 - Perm[1, 2, 4, 3, 5]/24 - Perm[1, 2, 4, 5, 3]/24 + Perm[1, 2, 5, 4, 3]/24 - Perm[1, 4, 2, 3, 5]/24 - Perm[1, 4, 2, 5, 3]/24 + Perm[1, 4, 3, 2, 5]/24 + Perm[1, 4, 5, 2, 3]/24 - Perm[2, 1, 3, 4, 5]/24 + Perm[2, 1, 4, 3, 5]/24 + Perm[2, 1, 4, 5, 3]/24 - Perm[2, 1, 5, 4, 3]/24 - Perm[2, 3, 1, 4, 5]/24 + Perm[2, 3, 4, 1, 5]/24 + Perm[2, 3, 4, 5, 1]/24 - Perm[2, 3, 5, 4, 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, 2, 1, 4, 5]/24 - Perm[3, 2, 4, 1, 5]/24 - Perm[3, 2, 4, 5, 1]/24 + Perm[3, 2, 5, 4, 1]/24 + Perm[3, 4, 1, 2, 5]/24 - Perm[3, 4, 2, 1, 5]/24 - Perm[3, 4, 2, 5, 1]/24 + Perm[3, 4, 5, 2, 1]/24 + Perm[4, 1, 2, 3, 5]/24 + Perm[4, 1, 2, 5, 3]/24 - Perm[4, 1, 3, 2, 5]/24 - Perm[4, 1, 5, 2, 3]/24 - Perm[4, 3, 1, 2, 5]/24 + Perm[4, 3, 2, 1, 5]/24 + Perm[4, 3, 2, 5, 1]/24 - Perm[4, 3, 5, 2, 1]/24 - Perm[4, 5, 1, 2, 3]/24 + Perm[4, 5, 2, 1, 3]/24 + Perm[4, 5, 2, 3, 1]/24 - Perm[4, 5, 3, 2, 1]/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, 4, 1, 2, 3]/24 - Perm[5, 4, 2, 1, 3]/24 - Perm[5, 4, 2, 3, 1]/24 + Perm[5, 4, 3, 2, 1]/24\ \>", "\<\ ( 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 ) ------------- - ------------- - ------------- + ------------- - ------------- \ - 24 24 24 24 24 ( 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 \ ) ------------- + ------------- + ------------- - ------------- + \ ------------- + 24 24 24 24 24 ( 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 \ ) ------------- - ------------- - ------------- + ------------- + \ ------------- - 24 24 24 24 24 ( 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 \ ) ------------- - ------------- - ------------- + ------------- + \ ------------- + 24 24 24 24 24 ( 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 \ ) ------------- - ------------- - ------------- + ------------- + \ ------------- - 24 24 24 24 24 ( 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 \ ) ------------- - ------------- + ------------- + ------------- + \ ------------- - 24 24 24 24 24 ( 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 \ ) ------------- - ------------- - ------------- + ------------- + \ ------------- - 24 24 24 24 24 ( 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 \ ) ------------- - ------------- + ------------- + ------------- - \ ------------- + 24 24 24 24 24 ( 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 \ ) ------------- + ------------- - ------------- - ------------- + \ ------------- - 24 24 24 24 24 ( 5 4 2 1 3 ) ( 5 4 2 3 1 ) ( 5 4 3 2 1 ) ------------- - ------------- + ------------- 24 24 24\ \>"], "Output"] }, Open ]], Cell[TextData[{ "This idempotent ", StyleBox["e", FontSlant->"Italic"], " is equal to the (normalized) Young symmetrizer of the tableau ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "3", "5"}, {"2", "4", " "} }], ")"}], TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell["tabl = 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["ys = YoungSymmetrizer[tabl]", "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["e === Expand[ys/24]", "Input"], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{695, 606}, WindowMargins->{{2, 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, 135, 3, 240, "Title"], Cell[1869, 56, 56, 0, 64, "Subtitle"], Cell[1928, 58, 135, 3, 71, "Subsubtitle"], Cell[CellGroupData[{ Cell[2088, 65, 31, 0, 30, "Input"], Cell[2122, 67, 794, 18, 297, "Print"] }, Open ]], Cell[2931, 88, 1636, 48, 261, "Text"], Cell[4570, 138, 53, 0, 33, "Text"], Cell[CellGroupData[{ Cell[4648, 142, 89, 2, 30, "Input"], Cell[4740, 146, 170, 4, 27, "Output"] }, Open ]], Cell[4925, 153, 35, 0, 33, "Text"], Cell[CellGroupData[{ Cell[4985, 157, 46, 0, 30, "Input"], Cell[5034, 159, 363, 9, 47, "Output"] }, Open ]], Cell[5412, 171, 108, 3, 33, "Text"], Cell[CellGroupData[{ Cell[5545, 178, 53, 0, 30, "Input"], Cell[5601, 180, 110, 4, 27, "Output"] }, Open ]], Cell[5726, 187, 46, 0, 33, "Text"], Cell[CellGroupData[{ Cell[5797, 191, 45, 0, 30, "Input"], Cell[5845, 193, 366, 10, 47, "Output"] }, Open ]], Cell[6226, 206, 104, 2, 33, "Text"], Cell[CellGroupData[{ Cell[6355, 212, 34, 0, 30, "Input"], Cell[6392, 214, 395, 8, 47, "Output"] }, Open ]], Cell[6802, 225, 322, 14, 33, "Text"], Cell[CellGroupData[{ Cell[7149, 243, 72, 0, 30, "Input"], Cell[7224, 245, 162, 4, 27, "Output"] }, Open ]], Cell[7401, 252, 729, 32, 33, "Text"], Cell[8133, 286, 322, 14, 33, "Text"], Cell[CellGroupData[{ Cell[8480, 304, 72, 0, 30, "Input"], Cell[8555, 306, 162, 4, 27, "Output"] }, Open ]], Cell[8732, 313, 729, 32, 33, "Text"], Cell[9464, 347, 397, 17, 33, "Text"], Cell[9864, 366, 124, 5, 33, "Text"], Cell[CellGroupData[{ Cell[10013, 375, 37, 0, 30, "Input"], Cell[10053, 377, 621, 13, 87, "Output"] }, Open ]], Cell[10689, 393, 1164, 53, 33, "Text"], Cell[CellGroupData[{ Cell[11878, 450, 40, 0, 30, "Input"], Cell[11921, 452, 633, 14, 87, "Output"] }, Open ]], Cell[12569, 469, 240, 8, 33, "Text"], Cell[CellGroupData[{ Cell[12834, 481, 65, 0, 30, "Input"], Cell[12902, 483, 60, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12999, 492, 67, 0, 30, "Input"], Cell[13069, 494, 168, 10, 87, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13274, 509, 67, 0, 30, "Input"], Cell[13344, 511, 264, 13, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13645, 529, 69, 0, 30, "Input"], Cell[13717, 531, 350, 16, 127, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14104, 552, 69, 0, 30, "Input"], Cell[14176, 554, 263, 13, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14476, 572, 71, 0, 30, "Input"], Cell[14550, 574, 184, 10, 87, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14771, 589, 73, 0, 30, "Input"], Cell[14847, 591, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14944, 600, 65, 0, 30, "Input"], Cell[15012, 602, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15109, 611, 67, 0, 30, "Input"], Cell[15179, 613, 168, 10, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15384, 628, 67, 0, 30, "Input"], Cell[15454, 630, 264, 13, 121, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15755, 648, 69, 0, 30, "Input"], Cell[15827, 650, 349, 16, 141, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16213, 671, 69, 0, 30, "Input"], Cell[16285, 673, 263, 13, 121, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16585, 691, 71, 0, 30, "Input"], Cell[16659, 693, 185, 10, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16881, 708, 73, 0, 30, "Input"], Cell[16957, 710, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17054, 719, 65, 0, 30, "Input"], Cell[17122, 721, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17219, 730, 67, 0, 30, "Input"], Cell[17289, 732, 168, 10, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17494, 747, 67, 0, 30, "Input"], Cell[17564, 749, 579, 24, 221, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18180, 778, 69, 0, 30, "Input"], Cell[18252, 780, 315, 16, 141, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18604, 801, 69, 0, 30, "Input"], Cell[18676, 803, 450, 24, 221, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19163, 832, 71, 0, 30, "Input"], Cell[19237, 834, 366, 18, 181, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19640, 857, 73, 0, 30, "Input"], Cell[19716, 859, 60, 4, 41, "Output"] }, Open ]], Cell[19791, 866, 217, 8, 33, "Text"], Cell[CellGroupData[{ Cell[20033, 878, 45, 3, 48, "Input"], Cell[20081, 883, 56, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20174, 892, 53, 3, 48, "Input"], Cell[20230, 897, 102, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20369, 906, 36, 0, 30, "Input"], Cell[20408, 908, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20505, 917, 53, 3, 48, "Input"], Cell[20561, 922, 102, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20700, 931, 36, 0, 30, "Input"], Cell[20739, 933, 60, 4, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20836, 942, 45, 3, 48, "Input"], Cell[20884, 947, 56, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20977, 956, 53, 3, 48, "Input"], Cell[21033, 961, 702, 16, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[21772, 982, 36, 0, 30, "Input"], Cell[21811, 984, 168, 10, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22016, 999, 53, 3, 48, "Input"], Cell[22072, 1004, 696, 16, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22805, 1025, 36, 0, 30, "Input"], Cell[22844, 1027, 168, 10, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23049, 1042, 45, 3, 48, "Input"], Cell[23097, 1047, 56, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23190, 1056, 53, 3, 48, "Input"], Cell[23246, 1061, 1029, 26, 147, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24312, 1092, 36, 0, 30, "Input"], Cell[24351, 1094, 234, 13, 121, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24622, 1112, 53, 3, 48, "Input"], Cell[24678, 1117, 2068, 56, 307, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[26783, 1178, 36, 0, 30, "Input"], Cell[26822, 1180, 234, 13, 121, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[27093, 1198, 45, 3, 48, "Input"], Cell[27141, 1203, 56, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[27234, 1212, 53, 3, 48, "Input"], Cell[27290, 1217, 2826, 69, 427, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30153, 1291, 36, 0, 30, "Input"], Cell[30192, 1293, 315, 16, 141, "Output"] }, Open ]], Cell[30522, 1312, 154, 5, 33, "Text"], Cell[CellGroupData[{ Cell[30701, 1321, 64, 0, 30, "Input"], Cell[30768, 1323, 234, 13, 121, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[31039, 1341, 55, 0, 30, "Input"], Cell[31097, 1343, 3612, 74, 407, "Output"] }, Open ]], Cell[34724, 1420, 335, 12, 44, "Text"], Cell[CellGroupData[{ Cell[35084, 1436, 49, 0, 30, "Input"], Cell[35136, 1438, 109, 6, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35282, 1449, 44, 0, 30, "Input"], Cell[35329, 1451, 1982, 47, 207, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37348, 1503, 36, 0, 30, "Input"], Cell[37387, 1505, 62, 4, 27, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)