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Please wait.\ \>", "Print"] }, Open ]], Cell[TextData[ "To calculate the plethysm [2] \[CircleDot] [4,2] by means of the determinant \ rule, we determine the following plethysms:\n\n[2] \[CircleDot][2]:"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(plet2p2\ = \ Plethysm[Parti[2], Parti[2]]\)], "Input"], Cell[OutputFormData["\<\ Parti[4] + Parti[2, 2]\ \>", "\<\ {4} + {2, 2}\ \>"], "Output"] }, Open ]], Cell[TextData["[2] \[CircleDot] [4]:"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(plet2p4\ = \ Plethysm[Parti[2], Parti[4]]\)], "Input"], Cell[OutputFormData["\<\ Parti[8] + Parti[4, 4] + Parti[6, 2] + Parti[4, 2, 2] + Parti[2, 2, 2, 2]\ \>", "\<\ {8} + {4, 4} + {6, 2} + {4, 2, 2} + {2, 2, 2, 2}\ \>"], "Output"] }, Open ]], Cell[TextData["[2] \[CircleDot] [5]:"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(plet2p5\ = \ Plethysm[Parti[2], Parti[5]]\)], "Input"], Cell[OutputFormData["\<\ Parti[10] + Parti[6, 4] + Parti[8, 2] + Parti[4, 4, 2] + Parti[6, 2, 2] + \ Parti[4, 2, 2, 2] + Parti[2, 2, 2, 2, 2]\ \>", "\<\ {10} + {6, 4} + {8, 2} + {4, 4, 2} + {6, 2, 2} + {4, 2, 2, 2} + {2, 2, 2, 2, \ 2}\ \>"], "Output"] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ \(Now\ we\ use\ the\ determinant\ rule\ to\ calculate\ [2]\ \[CircleDot]\ \([4, 2]\)\ . \n\n \([2]\)\ \[CircleDot]\ \([4, 2]\)\), " ", "=", " ", RowBox[{ RowBox[{"det", RowBox[{"(", GridBox[{ {\(\([2]\)\[CircleDot]\([4]\)\), \(\([2]\)\[CircleDot]\([5]\)\)}, {\(\([2]\)\[CircleDot]\([1]\)\), \(\([2]\)\[CircleDot]\([2]\)\)} }], ")"}]}], " ", "=", " ", RowBox[{ RowBox[{"det", RowBox[{"(", GridBox[{ {\(\([2]\)\[CircleDot]\([4]\)\), \(\([2]\)\[CircleDot]\([5]\)\)}, {\([2]\), \(\([2]\)\[CircleDot]\([2]\)\)} }], ")"}]}], "\n", "\n", " ", "=", " ", \(\((\ \([2]\)\ \[CircleDot]\ \([4]\)\ )\) \((\ \([2]\)\ \[CircleDot]\ \([2]\)\ )\)\ - \ \ \((\ \([2]\)\ \[CircleDot]\ \([5]\)\ )\)\ [2]\)}]}]}], " "}], TextForm]], "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[ \(sum1\ = \ \(RLRule[plet2p4, #]&\)\ /@\ plet2p2\)], "Input"], Cell[OutputFormData["\<\ Parti[12] + 2*Parti[6, 6] + Parti[7, 5] + 4*Parti[8, 4] + 2*Parti[9, 3] + \ 3*Parti[10, 2] + Parti[11, 1] + 2*Parti[4, 4, 4] + 2*Parti[5, 4, 3] + 6*Parti[6, 4, 2] + 2*Parti[6, 5, 1] + \ 3*Parti[7, 3, 2] + 3*Parti[7, 4, 1] + 4*Parti[8, 2, 2] + 2*Parti[8, 3, 1] + 2*Parti[9, 2, 1] + \ 4*Parti[4, 4, 2, 2] + Parti[4, 4, 3, 1] + 2*Parti[5, 3, 2, 2] + Parti[5, 3, 3, 1] + 3*Parti[5, 4, \ 2, 1] + Parti[5, 5, 1, 1] + 4*Parti[6, 2, 2, 2] + 3*Parti[6, 3, 2, 1] + 2*Parti[7, 2, 2, 1] + Parti[7, \ 3, 1, 1] + 3*Parti[4, 2, 2, 2, 2] + 2*Parti[4, 3, 2, 2, 1] + Parti[4, 3, 3, 1, 1] + \ 2*Parti[5, 2, 2, 2, 1] + Parti[5, 3, 2, 1, 1] + Parti[2, 2, 2, 2, 2, 2] + Parti[3, 2, 2, 2, 2, 1] + \ Parti[3, 3, 2, 2, 1, 1]\ \>", "\<\ {12} + 2 {6, 6} + {7, 5} + 4 {8, 4} + 2 {9, 3} + 3 {10, 2} + {11, 1} + 2 {4, \ 4, 4} + 2 {5, 4, 3} + 6 {6, 4, 2} + 2 {6, 5, 1} + 3 {7, 3, 2} + 3 {7, 4, 1} + 4 {8, 2, 2} + 2 {8, \ 3, 1} + 2 {9, 2, 1} + 4 {4, 4, 2, 2} + {4, 4, 3, 1} + 2 {5, 3, 2, 2} + {5, 3, 3, 1} + 3 {5, 4, 2, \ 1} + {5, 5, 1, 1} + 4 {6, 2, 2, 2} + 3 {6, 3, 2, 1} + 2 {7, 2, 2, 1} + {7, 3, 1, 1} + 3 {4, 2, \ 2, 2, 2} + 2 {4, 3, 2, 2, 1} + {4, 3, 3, 1, 1} + 2 {5, 2, 2, 2, 1} + {5, 3, 2, 1, 1} + {2, 2, 2, 2, 2, 2} \ + {3, 2, 2, 2, 2, 1} + {3, 3, 2, 2, 1, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sum2\ = \ RLRule[plet2p5, Parti[2]]\)], "Input"], Cell[OutputFormData["\<\ Parti[12] + Parti[6, 6] + Parti[7, 5] + 2*Parti[8, 4] + Parti[9, 3] + \ 2*Parti[10, 2] + Parti[11, 1] + Parti[4, 4, 4] + Parti[5, 4, 3] + 3*Parti[6, 4, 2] + Parti[6, 5, 1] + \ Parti[7, 3, 2] + Parti[7, 4, 1] + 2*Parti[8, 2, 2] + Parti[8, 3, 1] + Parti[9, 2, 1] + 2*Parti[4, 4, 2, 2] + \ Parti[4, 4, 3, 1] + Parti[5, 3, 2, 2] + Parti[5, 4, 2, 1] + 2*Parti[6, 2, 2, 2] + Parti[6, 3, \ 2, 1] + Parti[7, 2, 2, 1] + 2*Parti[4, 2, 2, 2, 2] + Parti[4, 3, 2, 2, 1] + Parti[5, 2, 2, 2, 1] + \ Parti[2, 2, 2, 2, 2, 2] + Parti[3, 2, 2, 2, 2, 1]\ \>", "\<\ {12} + {6, 6} + {7, 5} + 2 {8, 4} + {9, 3} + 2 {10, 2} + {11, 1} + {4, 4, 4} \ + {5, 4, 3} + 3 {6, 4, 2} + {6, 5, 1} + {7, 3, 2} + {7, 4, 1} + 2 {8, 2, 2} + {8, 3, 1} + {9, 2, 1} + 2 \ {4, 4, 2, 2} + {4, 4, 3, 1} + {5, 3, 2, 2} + {5, 4, 2, 1} + 2 {6, 2, 2, 2} + {6, 3, 2, 1} + {7, 2, 2, 1} \ + 2 {4, 2, 2, 2, 2} + {4, 3, 2, 2, 1} + {5, 2, 2, 2, 1} + {2, 2, 2, 2, 2, 2} + {3, 2, 2, 2, 2, 1}\ \ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(plet2p42\ = \ sum1\ - \ sum2\)], "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"plet2p42\" is \ similar to existing symbol \"plet2p2\".\ \>", "Message"], Cell[OutputFormData["\<\ Parti[6, 6] + 2*Parti[8, 4] + Parti[9, 3] + Parti[10, 2] + Parti[4, 4, 4] + \ Parti[5, 4, 3] + 3*Parti[6, 4, 2] + Parti[6, 5, 1] + 2*Parti[7, 3, 2] + 2*Parti[7, 4, 1] + \ 2*Parti[8, 2, 2] + Parti[8, 3, 1] + Parti[9, 2, 1] + 2*Parti[4, 4, 2, 2] + Parti[5, 3, 2, 2] + \ Parti[5, 3, 3, 1] + 2*Parti[5, 4, 2, 1] + Parti[5, 5, 1, 1] + 2*Parti[6, 2, 2, 2] + 2*Parti[6, \ 3, 2, 1] + Parti[7, 2, 2, 1] + Parti[7, 3, 1, 1] + Parti[4, 2, 2, 2, 2] + Parti[4, 3, 2, 2, 1] + Parti[4, \ 3, 3, 1, 1] + Parti[5, 2, 2, 2, 1] + Parti[5, 3, 2, 1, 1] + Parti[3, 3, 2, 2, 1, 1]\ \>", "\<\ {6, 6} + 2 {8, 4} + {9, 3} + {10, 2} + {4, 4, 4} + {5, 4, 3} + 3 {6, 4, 2} + \ {6, 5, 1} + 2 {7, 3, 2} + 2 {7, 4, 1} + 2 {8, 2, 2} + {8, 3, 1} + {9, 2, 1} + 2 {4, 4, 2, 2} + {5, 3, \ 2, 2} + {5, 3, 3, 1} + 2 {5, 4, 2, 1} + {5, 5, 1, 1} + 2 {6, 2, 2, 2} + 2 {6, 3, 2, 1} + {7, 2, 2, \ 1} + {7, 3, 1, 1} + {4, 2, 2, 2, 2} + {4, 3, 2, 2, 1} + {4, 3, 3, 1, 1} + {5, 2, 2, 2, 1} + {5, \ 3, 2, 1, 1} + {3, 3, 2, 2, 1, 1}\ \>"], "Output"] }, Open ]], Cell["The formula of Fritz S\[ADoubleDot]nger yields the same result.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(plet\ = \ Plethysm[Parti[2], Parti[4, 2]]\)], "Input"], Cell[OutputFormData["\<\ Parti[6, 6] + 2*Parti[8, 4] + Parti[9, 3] + Parti[10, 2] + Parti[4, 4, 4] + \ Parti[5, 4, 3] + 3*Parti[6, 4, 2] + Parti[6, 5, 1] + 2*Parti[7, 3, 2] + 2*Parti[7, 4, 1] + \ 2*Parti[8, 2, 2] + Parti[8, 3, 1] + Parti[9, 2, 1] + 2*Parti[4, 4, 2, 2] + Parti[5, 3, 2, 2] + \ Parti[5, 3, 3, 1] + 2*Parti[5, 4, 2, 1] + Parti[5, 5, 1, 1] + 2*Parti[6, 2, 2, 2] + 2*Parti[6, \ 3, 2, 1] + Parti[7, 2, 2, 1] + Parti[7, 3, 1, 1] + Parti[4, 2, 2, 2, 2] + Parti[4, 3, 2, 2, 1] + Parti[4, \ 3, 3, 1, 1] + Parti[5, 2, 2, 2, 1] + Parti[5, 3, 2, 1, 1] + Parti[3, 3, 2, 2, 1, 1]\ \>", "\<\ {6, 6} + 2 {8, 4} + {9, 3} + {10, 2} + {4, 4, 4} + {5, 4, 3} + 3 {6, 4, 2} + \ {6, 5, 1} + 2 {7, 3, 2} + 2 {7, 4, 1} + 2 {8, 2, 2} + {8, 3, 1} + {9, 2, 1} + 2 {4, 4, 2, 2} + {5, 3, \ 2, 2} + {5, 3, 3, 1} + 2 {5, 4, 2, 1} + {5, 5, 1, 1} + 2 {6, 2, 2, 2} + 2 {6, 3, 2, 1} + {7, 2, 2, \ 1} + {7, 3, 1, 1} + {4, 2, 2, 2, 2} + {4, 3, 2, 2, 1} + {4, 3, 3, 1, 1} + {5, 2, 2, 2, 1} + {5, \ 3, 2, 1, 1} + {3, 3, 2, 2, 1, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(plet\ === \ plet2p42\)], "Input"], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", WindowSize->{771, 364}, 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. 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, 62, 0, 105, "Title"], Cell[1796, 53, 56, 0, 64, "Subtitle"], Cell[1855, 55, 135, 3, 71, "Subsubtitle"], Cell[1993, 60, 109, 3, 33, "Text"], Cell[CellGroupData[{ Cell[2127, 67, 49, 1, 30, "Input"], Cell[2179, 70, 614, 14, 229, "Print"] }, Open ]], Cell[2808, 87, 173, 2, 71, "Text"], Cell[CellGroupData[{ Cell[3006, 93, 75, 1, 30, "Input"], Cell[3084, 96, 88, 4, 27, "Output"] }, Open ]], Cell[3187, 103, 47, 0, 33, "Text"], Cell[CellGroupData[{ Cell[3259, 107, 75, 1, 30, "Input"], Cell[3337, 110, 175, 4, 27, "Output"] }, Open ]], Cell[3527, 117, 47, 0, 33, "Text"], Cell[CellGroupData[{ Cell[3599, 121, 75, 1, 30, "Input"], Cell[3677, 124, 254, 7, 27, "Output"] }, Open ]], Cell[3946, 134, 1244, 28, 119, "Text"], Cell[CellGroupData[{ Cell[5215, 166, 80, 1, 30, "Input"], Cell[5298, 169, 1321, 32, 127, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6656, 206, 69, 1, 30, "Input"], Cell[6728, 209, 1008, 24, 87, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7773, 238, 63, 1, 30, "Input"], Cell[7839, 241, 142, 3, 42, "Message"], Cell[7984, 246, 1054, 24, 87, "Output"] }, Open ]], Cell[9053, 273, 79, 0, 33, "Text"], Cell[CellGroupData[{ Cell[9157, 277, 75, 1, 30, "Input"], Cell[9235, 280, 1054, 24, 87, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10326, 309, 54, 1, 30, "Input"], Cell[10383, 312, 62, 4, 27, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)