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Please wait.\ \>", "Print"] }, Open ]], Cell[TextData[{ "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\_\(i\ j\ k\ l\)\)]], " = - ", Cell[BoxData[ \(TraditionalForm\`R\_\(j\ i\ k\ l\)\)]], " = - ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ l\ k\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`R\_\(k\ l\ i\ j\)\)]], "\n\nand satisfies the first Bianchi identity\n\n(2) ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(\(R\_\(i\ k\ l\ j\)\ + \)\ \)\)]], Cell[BoxData[ \(TraditionalForm\`R\_\(i\ l\ j\ k\)\)]], " = 0.\n\nCondition (1) yields that the group ring elements ", Cell[BoxData[ \(TraditionalForm\`R\_b\)]], " of ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " lie in the left ideal ", Cell[BoxData[ \(TraditionalForm\`I\_1\)]], " = \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[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],Perm[1,2,4,3],Perm[3,4,1,2]]", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[2, 1, 3, 4], Perm[1, 2, 4, 3], Perm[3, 4, 1, 2]]\ \>", "\<\ {( 2 1 3 4 ), ( 1 2 4 3 ), ( 3 4 1 2 )}\ \>"], "Output"] }, Open ]], Cell["The symmetry group:", "Text"], Cell[CellGroupData[{ Cell["symgrp = GeneratedGroup[gens]", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[1, 2, 3, 4], Perm[2, 1, 3, 4], Perm[1, 2, 4, 3], Perm[3, 4, 1, \ 2], Perm[2, 1, 4, 3], Perm[3, 4, 2, 1], Perm[4, 3, 1, 2], Perm[4, 3, 2, 1]]\ \>", "\<\ {( 1 2 3 4 ), ( 2 1 3 4 ), ( 1 2 4 3 ), ( 3 4 1 2 ), ( 2 1 4 3 ), ( 3 4 2 1 \ ), ( 4 3 1 2 ), ( 4 3 2 1 )}\ \>"], "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] - Perm[1, 2, 4, 3] - Perm[2, 1, 3, 4] + Perm[2, 1, 4, 3] + Perm[3, 4, 1, 2] - Perm[3, 4, 2, 1] - Perm[4, 3, 1, 2] + Perm[4, 3, 2, 1]\ \>", "\<\ ( 1 2 3 4 ) - ( 1 2 4 3 ) - ( 2 1 3 4 ) + ( 2 1 4 3 ) + ( 3 4 1 2 ) - ( 3 4 2 \ 1 ) - ( 4 3 1 2 ) + ( 4 3 2 1 )\ \>"], "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] - 8*Perm[1, 2, 4, 3] - 8*Perm[2, 1, 3, 4] + 8*Perm[2, 1, \ 4, 3] + 8*Perm[3, 4, 1, 2] - 8*Perm[3, 4, 2, 1] - 8*Perm[4, 3, 1, 2] + 8*Perm[4, 3, \ 2, 1]\ \>", "\<\ 8 ( 1 2 3 4 ) - 8 ( 1 2 4 3 ) - 8 ( 2 1 3 4 ) + 8 ( 2 1 4 3 ) + 8 ( 3 4 1 2 ) \ - 8 ( 3 4 2 1 ) - 8 ( 4 3 1 2 ) + 8 ( 4 3 2 1 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Because of (2) , the curvature tensor fulfils ", StyleBox["aR ", FontSlant->"Italic"], "= 0 with" }], "Text"], Cell[CellGroupData[{ Cell["a = Perm[1,2,3,4] + Perm[1,3,4,2] + Perm[1,4,2,3]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] + Perm[1, 3, 4, 2] + Perm[1, 4, 2, 3]\ \>", "\<\ ( 1 2 3 4 ) + ( 1 3 4 2 ) + ( 1 4 2 3 )\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Consequently, all ", Cell[BoxData[ \(TraditionalForm\`R\_b\)]], " of ", 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[ \(TraditionalForm\`S\_4\)]], "] | ", StyleBox["x\[CenterDot]a", FontSlant->"Italic"], " = 0 } of ", StyleBox["a", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\^*\)\)]], "." }], "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[ \(TraditionalForm\`I\_2\)]], "." }], "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]/8 - Perm[1, 2, 4, 3]/8 - Perm[2, 1, 3, 4]/8 + Perm[2, 1, 4, \ 3]/8 + Perm[3, 4, 1, 2]/8 - Perm[3, 4, 2, 1]/8 - Perm[4, 3, 1, 2]/8 + Perm[4, 3, \ 2, 1]/8\ \>", "\<\ ( 1 2 3 4 ) ( 1 2 4 3 ) ( 2 1 3 4 ) ( 2 1 4 3 ) ( 3 4 1 2 ) ( 3 4 2 \ 1 ) ----------- - ----------- - ----------- + ----------- + ----------- - \ ----------- - 8 8 8 8 8 8 ( 4 3 1 2 ) ( 4 3 2 1 ) ----------- + ----------- 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[ \(TraditionalForm\`J\_2\)]], " where ", Cell[BoxData[ \(TraditionalForm\`J\_1\)]], " = (1 - ", StyleBox["f", FontSlant->"Italic"], ")\[CenterDot]\[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "] and ", Cell[BoxData[ \(TraditionalForm\`J\_2\)]], " = ", StyleBox["a", FontSlant->"Italic"], "\[CenterDot]\[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]. We have" }], "Text"], Cell[CellGroupData[{ Cell["h = Perm[1,2,3,4] - f", "Input"], Cell[OutputFormData["\<\ (7*Perm[1, 2, 3, 4])/8 + Perm[1, 2, 4, 3]/8 + Perm[2, 1, 3, 4]/8 - Perm[2, 1, \ 4, 3]/8 - Perm[3, 4, 1, 2]/8 + Perm[3, 4, 2, 1]/8 + Perm[4, 3, 1, 2]/8 - Perm[4, 3, \ 2, 1]/8\ \>", "\<\ 7 ( 1 2 3 4 ) ( 1 2 4 3 ) ( 2 1 3 4 ) ( 2 1 4 3 ) ( 3 4 1 2 ) ( 3 4 \ 2 1 ) ------------- + ----------- + ----------- - ----------- - ----------- + \ ----------- + 8 8 8 8 8 \ 8 ( 4 3 1 2 ) ( 4 3 2 1 ) ----------- - ----------- 8 8\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Now we determine the discrete Fourier transforms A and H of ", StyleBox["a ", FontSlant->"Italic"], "and ", StyleBox["h", FontSlant->"Italic"], ". They have the following block matrices:" }], "Text"], Cell[CellGroupData[{ Cell["(A1 = FourierTransform[Parti[4],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A2 = FourierTransform[Parti[3,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 1, 1}, {1, 1, 1}, {1, 1, 1}}\ \>", "\<\ 1 1 1 1 1 1 1 1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A3 = FourierTransform[Parti[2,2],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0, 0}, {0, 0}}\ \>", "\<\ 0 0 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A4 = FourierTransform[Parti[2,1,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, -1, 1}, {-1, 1, -1}, {1, -1, 1}}\ \>", "\<\ 1 -1 1 -1 1 -1 1 -1 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(A5 = FourierTransform[Parti[1,1,1,1],a]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{3}}\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H1 = FourierTransform[Parti[4],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1}}\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H2 = FourierTransform[Parti[3,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\ \>", "\<\ 1 0 0 0 1 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H3 = FourierTransform[Parti[2,2],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, -1/2}, {0, 0}}\ \>", "\<\ 1 -(-) 1 2 0 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H4 = FourierTransform[Parti[2,1,1],h]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\ \>", "\<\ 1 0 0 0 1 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["(H5 = FourierTransform[Parti[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,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["\<\ K5 = DecomposeR[{A5,H5}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1}}, HoldList[{{1}}]]\ \>", "\<\ {{{1}}, {{{1}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K5[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1}}\ \>", "\<\ 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[3]\ \>", "Input"], Cell[OutputFormData["\<\ 3\ \>", "\<\ 3\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K2 = DecomposeR[{A2,H2}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, HoldList[{{0, 0, 1}, {0, 0, 1}, {0, 0, 1}}, {{1, 0, -1}, {0, 0, 0}, {0, 0, \ 0}}, {{0, 0, 0}, {0, 1, -1}, {0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{{0, 0, 1}, {0, 0, 1}, {0, 0, 1}}, {{1, 0, -1}, {0, 0, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 1, -1}, {0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K2[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\ \>", "\<\ 1 0 0 0 1 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K4 = DecomposeR[{A4,H4}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, HoldList[{{0, 0, 1}, {0, 0, -1}, {0, 0, 1}}, {{1, 0, -1}, {0, 0, 0}, {0, 0, \ 0}}, {{0, 0, 0}, {0, 1, 1}, {0, 0, 0}}]]\ \>", "\<\ {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{{0, 0, 1}, {0, 0, -1}, {0, 0, 1}}, {{1, 0, -1}, {0, 0, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 1, 1}, {0, 0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K4[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\ \>", "\<\ 1 0 0 0 1 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ DefineMatrixRing[2]\ \>", "Input"], Cell[OutputFormData["\<\ 2\ \>", "\<\ 2\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ K3 = DecomposeR[{A3,H3}]\ \>", "Input"], Cell[OutputFormData["\<\ HoldList[{{1, 0}, {0, 0}}, HoldList[{{1, 0}, {0, 0}}]]\ \>", "\<\ {{{1, 0}, {0, 0}}, {{{1, 0}, {0, 0}}}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["MatrixForm[K3[[1]]]", "Input"], Cell[OutputFormData["\<\ {{1, 0}, {0, 0}}\ \>", "\<\ 1 0 0 0\ \>"], "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["(E2 = IdentityMatrix[2] - K3[[1]]) //MatrixForm", "Input"], Cell[OutputFormData["\<\ {{0, 0}, {0, 1}}\ \>", "\<\ 0 0 0 1\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["e = InvFourierTransform[Parti[2,2],E2]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4]/12 - Perm[1, 2, 4, 3]/12 - Perm[1, 4, 2, 3]/12 + Perm[1, 4, \ 3, 2]/12 - Perm[2, 1, 3, 4]/12 + Perm[2, 1, 4, 3]/12 - Perm[2, 3, 1, 4]/12 + Perm[2, \ 3, 4, 1]/12 + Perm[3, 2, 1, 4]/12 - Perm[3, 2, 4, 1]/12 + Perm[3, 4, 1, 2]/12 - Perm[3, \ 4, 2, 1]/12 + Perm[4, 1, 2, 3]/12 - Perm[4, 1, 3, 2]/12 - Perm[4, 3, 1, 2]/12 + Perm[4, \ 3, 2, 1]/12\ \>", "\<\ ( 1 2 3 4 ) ( 1 2 4 3 ) ( 1 4 2 3 ) ( 1 4 3 2 ) ( 2 1 3 4 ) ( 2 1 4 \ 3 ) ----------- - ----------- - ----------- + ----------- - ----------- + \ ----------- - 12 12 12 12 12 12 ( 2 3 1 4 ) ( 2 3 4 1 ) ( 3 2 1 4 ) ( 3 2 4 1 ) ( 3 4 1 2 ) ( 3 4 \ 2 1 ) ----------- + ----------- + ----------- - ----------- + ----------- - \ ----------- + 12 12 12 12 12 \ 12 ( 4 1 2 3 ) ( 4 1 3 2 ) ( 4 3 1 2 ) ( 4 3 2 1 ) ----------- - ----------- - ----------- + ----------- 12 12 12 12\ \>"], "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"}, {"2", "4"} }], ")"}], TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell["tabl = 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[tabl]", "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 ) - 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If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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