(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 236011, 6508] NotebookOptionsPosition[ 223565, 6134] NotebookOutlinePosition[ 224801, 6174] CellTagsIndexPosition[ 224665, 6168] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Credits", "Subsection", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.579077648693802*^9, 3.5790776504059*^9}}], Cell[TextData[{ "http://www.mcs.csueastbay.edu/~kbalasub/reprints/136.pdf\n\ http://www.math.umn.edu/~tlawson/old/18.704/symmetric1.pdf\n\ http://www.polyomino.f2s.com/david/haskell/charactersSn.html\n\ http://en.wikipedia.org/wiki/Symmetric_polynomial\n", StyleBox["http://www.personal.rhul.ac.uk/usah/080/QITNotes_files/Irreps_v06.\ pdf\n", FontVariations->{"CompatibilityType"->0}], "http://mathcircle.berkeley.edu/BMC3/SymPol.pdf\n\ http://faculty.math.tsinghua.edu.cn/~jzhou/SymmetricF.pdf\n\ Littlewood-Richardson program: modification of and parts from Daniel Bump \ (bump@math.stanford.edu) Copyright 1996.\n\ http://math.stackexchange.com/questions/114151\nhttp:// \ math.stackexchange.com/questions/395842\n\ http://math.stackexchange.com/questions/83214\n\ http://lipn.univ-paris13.fr/~toumazet/biblio/ARTICLES/NTB.pdf\[LineSeparator]\ http://young.sp2mi.univ-poitiers.fr/cgi-bin/form-prep/marc/LiE_form.act?\ action=LRR\[LineSeparator]http://mathoverflow.net/questions/22515/practical-\ ways-to-get-skew-schur-functions\n\ http://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule\n\ http://projecteuclid.org/download/pdf_1/euclid.bams/1183548016" }], "Text", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.577533907818513*^9, 3.5775339094591804`*^9}, 3.5779648110843143`*^9, {3.577964883421452*^9, 3.5779649475391197`*^9}, { 3.577965576535096*^9, 3.577965628862089*^9}, {3.5779657832209177`*^9, 3.577965784902014*^9}, {3.577965921363819*^9, 3.5779659607240705`*^9}, { 3.578050407713518*^9, 3.5780504131768303`*^9}, {3.5780504478808155`*^9, 3.578050448857871*^9}, {3.5780505104883966`*^9, 3.5780505461364355`*^9}, { 3.5790770973062644`*^9, 3.5790771006174536`*^9}, {3.5802053596327252`*^9, 3.5802053618515463`*^9}, {3.580994915083788*^9, 3.580994930579674*^9}, { 3.5845207533533297`*^9, 3.584520770218294*^9}, {3.6007758590824146`*^9, 3.600775859083414*^9}, 3.6227108298753824`*^9, {3.628937651325906*^9, 3.6289377015807805`*^9}, {3.628937756564925*^9, 3.628937774028924*^9}, { 3.628937807580843*^9, 3.628937814836258*^9}}], Cell[TextData[{ StyleBox["Special thanks to Marc van Leeuwen for several patient \ explanations, both by email and on Math StackExchange.", FontSlant->"Italic"], "\n", StyleBox["Also credit to ", FontSlant->"Italic"], "William J. Keith (Michigan Tech University Math Department) ", StyleBox["for guiding me in the study of t-core and t-quotients\ \[LineSeparator]Grateful mention of Prof. Joris Van der Jeugt for letting me \ attend his course 'Representation Theory' (R.U.Gent) fall 2014.\ \[LineSeparator]", FontSlant->"Italic"] }], "Text", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.577533907818513*^9, 3.5775339094591804`*^9}, 3.5779648110843143`*^9, {3.577964883421452*^9, 3.5779649475391197`*^9}, { 3.577965576535096*^9, 3.577965628862089*^9}, {3.5779657832209177`*^9, 3.577965784902014*^9}, {3.577965921363819*^9, 3.5779659607240705`*^9}, { 3.578050407713518*^9, 3.5780504131768303`*^9}, {3.5780504478808155`*^9, 3.578050448857871*^9}, {3.5780505104883966`*^9, 3.5780505461364355`*^9}, { 3.5790770973062644`*^9, 3.5790771006174536`*^9}, {3.5802053596327252`*^9, 3.5802053618515463`*^9}, {3.580994915083788*^9, 3.580994944362462*^9}, { 3.6289378291160746`*^9, 3.6289378440519295`*^9}, {3.628937890731599*^9, 3.628937926715657*^9}, {3.6289379818348103`*^9, 3.6289380100114217`*^9}, { 3.6289385621910048`*^9, 3.6289387208140774`*^9}, {3.6289388835723867`*^9, 3.6289388903567743`*^9}, {3.628948198907193*^9, 3.6289482034754543`*^9}, { 3.6749867363168*^9, 3.6749867363178*^9}, 3.6749871384147987`*^9}], Cell[TextData[{ "to do: cyclecleasses -> classSize ; 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", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["x", "v"], "z"], TraditionalForm]]], " : allows all symbolic manipulations using standard algebraic functions, \ but becomes large and slow for small to moderate arguments, and quite unfit \ for human consumption.\nIt can however always be reduced to elementary S.P. \ by the standard \[OpenCurlyQuote]", StyleBox["SymmetricReduction", FontWeight->"Bold"], "\[CloseCurlyQuote] function.\n*\tcondensed format: ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalSigma]", "i"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SubscriptBox["a", "i"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["p", "n"], "i"], TraditionalForm]]], " where ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["p", "n"], "i"], TraditionalForm]]], " stands for the i-th partition of n in reverse lexicographic ordering. It \ codes the expanded format by ignoring the permutation of the \ (interchangeable) indices and extracting only the (orderless) exponents \ recast into a partition of n. This format loses the info on the actual number \ of variables used. It groups monomials according to the exponents : 5 ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ SubscriptBox["x", "1"], "3"], SuperscriptBox[ SubscriptBox["x", "3"], "2"]}], TraditionalForm]]], " becomes 5 ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["p", "5"], "3"], TraditionalForm]]], " since {3,2} is the 3rd partition of 5. The info that there are 3 or more \ variables \[OpenCurlyQuote]in play\[CloseCurlyQuote] (as shown by the indices \ 1 and 3) is lost.\n*\tunevaluated format ", StyleBox["uu[ arg , ", FontWeight->"Bold"], "# of variables", StyleBox[" v ]", FontWeight->"Bold"], " which can be taken as argument for symbolic transformation functions. Only \ the basic definition ", StyleBox["uu[ partition, v ]==0 /; Length[partition]>v", FontWeight->"Bold"], " is coded for (one cannot distribute 5 exponents over 4 variables). " }], "Text", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.5777742395513716`*^9, 3.577774318835666*^9}, { 3.5777743508993974`*^9, 3.577774390010274*^9}, {3.577774427058572*^9, 3.577774573970463*^9}, {3.577774613050089*^9, 3.577774906087525*^9}, { 3.577775411738783*^9, 3.5777755906976347`*^9}, {3.5777756548716087`*^9, 3.5777758579262805`*^9}, {3.577775891193183*^9, 3.5777759420701365`*^9}, { 3.577776035323717*^9, 3.577776105529538*^9}, {3.5777761466404915`*^9, 3.577776397181362*^9}, {3.577776448792719*^9, 3.5777765193891797`*^9}, { 3.5777765593282137`*^9, 3.5777766663948245`*^9}, 3.577776784196044*^9, { 3.5777774581154947`*^9, 3.5777774716780815`*^9}, {3.577783878822277*^9, 3.5777839501058083`*^9}, {3.5778050413347855`*^9, 3.57780504595105*^9}, { 3.5778050914946547`*^9, 3.5778051088866496`*^9}, {3.5778051561503525`*^9, 3.577805186606095*^9}, {3.577805225822338*^9, 3.5778052291595287`*^9}, { 3.5778052863498*^9, 3.5778054557964916`*^9}, {3.578050767928121*^9, 3.578050812126649*^9}}], Cell[TextData[{ "The S.P. ", StyleBox["p", FontWeight->"Bold"], " , ", StyleBox["h", FontWeight->"Bold"], " and ", StyleBox["e", FontWeight->"Bold"], " have the property of being threaded over their (first) argument: ", StyleBox["p[{3,1,1}, v]", FontWeight->"Bold"], " equals ", StyleBox["p[3, v] p[1, v]^2", FontWeight->"Bold"], " and the transformation rules work on this last unevaluated form with \ integer argument. The forward and backward conversions on ", StyleBox["uu", FontWeight->"Bold"], " are performed by the functions ", StyleBox["threadSP[ expr , uu ]", FontWeight->"Bold"], " and ", StyleBox["unthreadSP[ arg , uu ] ", FontWeight->"Bold"], "regrouping (products of) integer arguments uu[ n ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{ RowBox[{",", "v"}], "]"}], "i"], TraditionalForm]]], " uu[ m , ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"v", "]"}], "j"], TraditionalForm]]], " ... into partition arguments uu[{ n (i copies), m (j copies) ... }, v] \ or the reverse.\nThe S.P. ", StyleBox["m", FontWeight->"Bold"], " (monomial) and ", StyleBox["s", FontWeight->"Bold"], " (Schur) do not share this property, and always need a partition as first \ argument. Transformations thus need a threading step in order to get \ acceptable arguments." }], "Text", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.5777768190505104`*^9, 3.5777768826353064`*^9}, { 3.577776924801244*^9, 3.577777161105074*^9}, {3.5777793774672203`*^9, 3.5777794391871557`*^9}, {3.5777795416584992`*^9, 3.577779710506572*^9}, 3.5778054755086193`*^9, {3.577805577051427*^9, 3.577805588659091*^9}, { 3.5778059809685297`*^9, 3.577806001144684*^9}, 3.5810072698924417`*^9}], Cell[TextData[{ "Transformation Functions:\n", StyleBox["u[ arg , ", FontWeight->"Bold"], "# of variables", StyleBox[" v ]", FontWeight->"Bold"], " produces the expanded format \[CapitalSigma] a ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ SubscriptBox["x", "1"], "i"], " ", SuperscriptBox[ SubscriptBox["x", "2"], "j"]}], TraditionalForm]]], " ... ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["x", "v"], "z"], TraditionalForm]]], " ,\n", StyleBox["expr2pow[ arg ]", FontWeight->"Bold"], " converts expanded format into condensed format ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalSigma]", "i"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SubscriptBox["a", "i"], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["p", "n"], "i"], TraditionalForm]]], " ,\n", StyleBox["pow2m[ condensed , v ", FontWeight->"Bold"], ":optional ", StyleBox["]", FontWeight->"Bold"], " converts the condensed format into monomial S.P. but if no ", StyleBox["v", FontWeight->"Bold"], " (# of variables) is entered, then it defaults to the partition size ", StyleBox["n", FontWeight->"Bold"], " common to all terms ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["p", "n"], "i"], TraditionalForm]]], " \n", StyleBox["uu[ arg , v ]", FontWeight->"Bold"], " is an inactive (unevaluated) representation except for a partition \ \[Lambda] as argument when |\[Lambda]|> ", StyleBox["v", FontWeight->"Bold"], " (producing 0) ;\n\n", StyleBox["u2w[ arg , v ]", FontWeight->"Bold"], " converts ", StyleBox["uu[ arg , v ]", FontWeight->"Bold"], " into ", StyleBox["ww[ arg , v ]", FontWeight->"Bold"], " if such transformation is known and available; it is intended to be \ implemented as a substitution rule, example: ", StyleBox["( expression containing uu[ arg , v ] ) /. uu-> u2w //Expand", FontWeight->"Bold"], "\n\n", StyleBox["monomProd2Sum[ arg ]", FontWeight->"Bold"], " implements the decomposition of powers and products of mm[ par, v ] into \ sums of them. ( http://math.stackexchange.com/questions/83214 )\n", StyleBox["schurProd2Sum[ arg ]", FontWeight->"Bold"], " implements the decomposition of powers and products of ss[ par, v ] into \ sums of them by calling the L-R rule package.\n\n", "The following grid shows the available conversions:\n(entries in ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " need a partition-type argument, entries in black can also take an integer \ argument) " }], "Text", CellMargins->{{66, 0}, {2, 10}}, CellChangeTimes->{{3.5777785515178013`*^9, 3.5777785721122103`*^9}, { 3.5777786183480654`*^9, 3.577779088103716*^9}, {3.577779187635451*^9, 3.5777791926042967`*^9}, {3.5777797538046193`*^9, 3.577779780305467*^9}, { 3.577781091933731*^9, 3.577781378111946*^9}, 3.5777839687782803`*^9, { 3.5778056437232404`*^9, 3.5778057436659565`*^9}, {3.577805793129786*^9, 3.5778059050651884`*^9}, {3.577966040850653*^9, 3.577966093786681*^9}, { 3.5779661457706547`*^9, 3.5779661875460434`*^9}, {3.577966322609769*^9, 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