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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[ 20683, 771]*) (*NotebookOutlinePosition[ 21359, 795]*) (* CellTagsIndexPosition[ 21315, 791]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[ "Properties of the symmetrizers of tensors S\[CircleTimes]S and A\ \[CircleTimes]A"], "Title"], Cell["Bernd Fiedler, Leipzig, May 2001", "Subtitle"], Cell["\<\ Bernd Fiedler, Eichelbaumstr. 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: Character tables of S1...S17, DFT of S10 The evaluation of precomputed data is running. Please wait.\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["The problem", "Section"], Cell[TextData[{ "Basic elements for the forming of algebraic curvature tensors are product \ tensors S\[CircleTimes]S and A\[CircleTimes]A, where S, A \[Element] ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalT]\_2\)]], " are covariant tensors of order 2 which are symmetric or sew-symmetric, \ respectively. The symmetry classes of S\[CircleTimes]S and A\[CircleTimes]A \ are characterized by plethysms\n[2]\[CircleDot][2] \[Tilde] [4] + [", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], "] (1)\n[", Cell[BoxData[ \(TraditionalForm\`1\^2\)]], "]\[CircleDot][2] \[Tilde] [", Cell[BoxData[ \(TraditionalForm\`1\^4\)]], "] + [", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], "] (2)" }], "Text"], Cell[CellGroupData[{ Cell["Plethysm[Parti[2],Parti[2]]", "Input"], Cell[OutputFormData["\<\ Parti[4] + Parti[2, 2]\ \>", "\<\ {4} + {2, 2}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plethysm[Parti[1,1],Parti[2]]", "Input"], Cell[OutputFormData["\<\ Parti[2, 2] + Parti[1, 1, 1, 1]\ \>", "\<\ {2, 2} + {1, 1, 1, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The left ideals that belong to these plethysms can be decomposed into two \ minimal left ideals which belong to the partitions (4), (", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], ") or (", Cell[BoxData[ \(TraditionalForm\`1\^4\)]], "), (", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], "), respectively. The left ideals of (4) and (", Cell[BoxData[ \(TraditionalForm\`1\^4\)]], ") coincide with the 1-dimensional two-sided ideals of \ \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "] that are characterized by these partitions. The two-sided ideal \ \[GothicL] of \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "] that belongs to (", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], ") is a 4-dimensional ideal which can be decomposed into two minimal, \ 2-dimensional left ideals\n\[GothicL] = \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " \[CirclePlus] \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], " , (3)\nwhere ", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], " are the Young symmetrizers of the standard tableaux\n\n", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], " = ", Cell[BoxData[ FormBox[GridBox[{ {"1", "2"}, {"3", "4"} }], TraditionalForm]]], " and ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], " = ", Cell[BoxData[ FormBox[GridBox[{ {"1", "3"}, {"2", "4"} }], TraditionalForm]]], " .\n\nIn this notebook we investigate whether the minimal (", Cell[BoxData[ \(TraditionalForm\`2\^2\)]], ")-left ideals ", Cell[BoxData[ \(TraditionalForm\`\[GothicL]\_S\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[GothicL]\_A\)]], " which belong to tensors S\[CircleTimes]S and A\[CircleTimes]A are unequal \ to the left ideals \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " and \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The calculation", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "The tensors ", "S\[CircleTimes]S" }], "Subsection"], Cell[TextData[{ "The symmetry of a tensor S\[CircleTimes]S can be described by the \ symmetrizer\ne := ", Cell[BoxData[ \(TraditionalForm\`\(1\/8\ \)\)]], "(id + (1 2))\[CenterDot](id + (3 4))\[CenterDot](id + (1 3)(2 4))" }], "Text"], Cell[CellGroupData[{ Cell["uu = 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["vv = 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["ww = 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[CellGroupData[{ Cell["ee = PermProd[uu,PermProd[vv,ww]]", "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[CellGroupData[{ Cell["PermProd[ee,ee] - 8 ee //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["ee = 1/8 ee //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[CellGroupData[{ Cell["PermProd[ee,ee] - ee //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Thus the above group ring element ", StyleBox["ee", FontWeight->"Bold"], " is an idempotent. Now we calculate the products e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "and e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["tabl1 = DefTableau[{1,2},{3,4}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 2], TabRow[3, 4]]\ \>", "\<\ {1, 2} {3, 4}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tabl2 = DefTableau[{1,3},{2,4}]", "Input"], Cell[OutputFormData["\<\ Tableau[TabRow[1, 3], TabRow[2, 4]]\ \>", "\<\ {1, 3} {2, 4}\ \>"], "Output"] }, Open ]], Cell["The Young symmetrizers of these tableaux read", "Text"], Cell[CellGroupData[{ Cell["yt1 = YoungSymmetrizer[tabl1]", "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[CellGroupData[{ Cell["yt2 = YoungSymmetrizer[tabl2]", "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[{ "If the minimal left ideal ", Cell[BoxData[ \(TraditionalForm\`\[GothicL]\_S\)]], " would coincide with one of the left ideals \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " and \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], ", then one of the products e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "and e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "has to vanish. However, both products are unequal to 0." }], "Text"], Cell[CellGroupData[{ Cell["eeyt1 = PermProd[ee,yt1] //Expand", "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[CellGroupData[{ Cell["eeyt2 = PermProd[ee,yt2] //Expand", "Input"], Cell[OutputFormData["\<\ -Perm[1, 3, 2, 4]/2 + Perm[1, 3, 4, 2]/2 - Perm[1, 4, 2, 3]/2 + Perm[1, 4, 3, \ 2]/2 - Perm[2, 3, 1, 4]/2 + Perm[2, 3, 4, 1]/2 - Perm[2, 4, 1, 3]/2 + Perm[2, 4, 3, 1]/2 + Perm[3, 1, 2, 4]/2 - Perm[3, 1, 4, 2]/2 + Perm[3, 2, 1, 4]/2 - Perm[3, 2, 4, 1]/2 + Perm[4, 1, 2, 3]/2 - Perm[4, 1, 3, 2]/2 + Perm[4, 2, 1, 3]/2 - Perm[4, 2, 3, 1]/2\ \>", "\<\ -( 1 3 2 4 ) ( 1 3 4 2 ) ( 1 4 2 3 ) ( 1 4 3 2 ) ( 2 3 1 4 ) ( 2 3 \ 4 1 ) ------------ + ----------- - ----------- + ----------- - ----------- + \ ----------- - 2 2 2 2 2 2 ( 2 4 1 3 ) ( 2 4 3 1 ) ( 3 1 2 4 ) ( 3 1 4 2 ) ( 3 2 1 4 ) ----------- + ----------- + ----------- - ----------- + ----------- - 2 2 2 2 2 ( 3 2 4 1 ) ( 4 1 2 3 ) ( 4 1 3 2 ) ( 4 2 1 3 ) ( 4 2 3 1 ) ----------- + ----------- - ----------- + ----------- - ----------- 2 2 2 2 2\ \>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Remark:", FontWeight->"Bold"], " We have the following interesting results: e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " fulfills e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " and e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], " is nilpotent." }], "Text"], Cell[CellGroupData[{ Cell["eeyt1 - yt1", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["PermProd[eeyt2,eeyt2]", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["The tensors A\[CircleTimes]A"], "Subsection"], Cell[TextData[{ "The symmetry of a tensor A\[CircleTimes]A can be described by the \ symmetrizer\ne := ", Cell[BoxData[ \(TraditionalForm\`\(1\/8\ \)\)]], "(id - (1 2))\[CenterDot](id - (3 4))\[CenterDot](id + (1 3)(2 4))" }], "Text"], Cell[CellGroupData[{ Cell["uuu = 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["vvv = 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["www = 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[CellGroupData[{ Cell["eee = PermProd[uuu,PermProd[vvv,www]]", "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[CellGroupData[{ Cell["PermProd[eee,eee] - 8 eee //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["eee = 1/8 eee //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[CellGroupData[{ Cell["PermProd[eee,eee] - eee //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Thus the above group ring element ", StyleBox["eee", FontWeight->"Bold"], " is an idempotent. Now we calculate the products e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "and e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "." }], "Text"], Cell[TextData[{ "If the minimal left ideal ", Cell[BoxData[ \(TraditionalForm\`\[GothicL]\_A\)]], " would coincide with one of the left ideals \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], " and \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], ", then one of the products e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "and e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "has to vanish. We will see that e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_1\)\)]], "= 0 but e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], "\[NotEqual] 0." }], "Text"], Cell[CellGroupData[{ Cell["eeeyt1 = PermProd[eee,yt1] //Expand", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"eeeyt1\" is similar to existing symbol \"eeyt1\".\ \>", "Message"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["eeeyt2 = PermProd[eee,yt2] //Expand", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"eeeyt2\" is similar to existing symbol \"eeyt2\".\ \>", "Message"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4] - Perm[1, 2, 4, 3] + Perm[1, 3, 2, 4]/2 - Perm[1, 3, 4, 2]/2 \ - Perm[1, 4, 2, 3]/2 + Perm[1, 4, 3, 2]/2 - Perm[2, 1, 3, 4] + Perm[2, 1, 4, \ 3] - Perm[2, 3, 1, 4]/2 + Perm[2, 3, 4, 1]/2 + Perm[2, 4, 1, 3]/2 - Perm[2, 4, 3, 1]/2 - Perm[3, 1, 2, 4]/2 + Perm[3, 1, 4, 2]/2 + Perm[3, 2, 1, 4]/2 - Perm[3, 2, 4, 1]/2 + Perm[3, 4, 1, 2] - Perm[3, 4, 2, \ 1] + Perm[4, 1, 2, 3]/2 - Perm[4, 1, 3, 2]/2 - Perm[4, 2, 1, 3]/2 + Perm[4, 2, 3, 1]/2 - Perm[4, 3, 1, 2] + Perm[4, 3, 2, 1]\ \>", "\<\ ( 1 3 2 4 ) ( 1 3 4 2 ) ( 1 4 2 3 ) ( 1 4 3 \ 2 ) ( 1 2 3 4 ) - ( 1 2 4 3 ) + ----------- - ----------- - ----------- + \ ----------- - 2 2 2 2 ( 2 3 1 4 ) ( 2 3 4 1 ) ( 2 4 1 3 ) ( 2 1 3 4 ) + ( 2 1 4 3 ) - ----------- + ----------- + ----------- - 2 2 2 ( 2 4 3 1 ) ( 3 1 2 4 ) ( 3 1 4 2 ) ( 3 2 1 4 ) ( 3 2 4 1 ) ----------- - ----------- + ----------- + ----------- - ----------- + 2 2 2 2 2 ( 4 1 2 3 ) ( 4 1 3 2 ) ( 4 2 1 3 ) ( 3 4 1 2 ) - ( 3 4 2 1 ) + ----------- - ----------- - ----------- + 2 2 2 ( 4 2 3 1 ) ----------- - ( 4 3 1 2 ) + ( 4 3 2 1 ) 2\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Thus we have shown that ", Cell[BoxData[ \(TraditionalForm\`\[GothicL]\_A\)]], " = \[DoubleStruckCapitalK][", Cell[BoxData[ \(TraditionalForm\`S\_4\)]], "]\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], ".\n\nRemark: e\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`y\_\(t\_2\)\)]], " is essentially idempotent." }], "Text"], Cell[CellGroupData[{ Cell["PermProd[eeeyt2,eeeyt2] - 12 eeeyt2 //Expand", "Input"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{748, 615}, WindowMargins->{{0, 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. 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