(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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). 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It implements the elementary, homogeneous, \ power-sum, monomial, and Schur symmetric functions.\n\nIf all you need is the \ elementary symmetric functions, you can use SymmetricPolynomial in ", StyleBox["Mathematica", FontSlant->"Italic"], "'s built-in package Algebra`SymmetricPolynomials. If you need more than \ that, read on!\n\nA nice introduction/reminder about symmetric functions, and \ a vastly more powerful Maple symmetric functions package is available at ", ButtonBox["http://www.math.lsa.umich.edu/~jrs/maple.html#SF", ButtonData:>{ URL[ "http://www.math.lsa.umich.edu/~jrs/maple.html#SF"], None}, ButtonStyle->"Hyperlink"], ". Also check out ", StyleBox["The Symmetric Group", FontSlant->"Italic"], " (GTM 207) by Bruce Sagan.\n\nCurrent versions of the package and this \ documentation are available from ", ButtonBox["http://www.math.umn.edu/~drake", ButtonData:>{ URL[ "http://www.math.umn.edu/~drake/"], None}, ButtonStyle->"Hyperlink"], ".\n\n5 December 2004" }], "Text"], Cell[TextData[{ "Since this notebook is all about a certain package, let's begin by loading \ that package. The functions have single-letter names, which you might have \ already defined (although if you're a symmetric functions kind of person, you \ probably don't use 'e' or 'm' for a throwaway variable). That's why I \ recommend using the Clear statement just before loading the package.\n\nMake \ sure SymmetricFunctions.m is in a directory that ", StyleBox["Mathematica", FontSlant->"Italic"], " searches for packages. The current directory is a safe bet." }], "Text"], Cell[BoxData[{ \(\(Unprotect[e, h, p, m, s];\)\), "\[IndentingNewLine]", \(\(Clear[e, h, p, m, s];\)\), "\[IndentingNewLine]", \(<< SymmetricFunctions`\)}], "Input"], Cell["\<\ Check out the little help blurbs for these functions. The complete \ list is in the Catalog section below.\ \>", "Text"], Cell[BoxData[{ \(\(?m\)\), "\[IndentingNewLine]", \(\(?s\)\)}], "Input"], Cell["Okay, let's actually do something:", "Text"], Cell[BoxData[{ \(\(vars\ = \ {a, b, c, d, e};\)\), "\[IndentingNewLine]", \(h[vars, \ 3]\), "\[IndentingNewLine]", \(s[vars, {2, 1}]\)}], "Input"], Cell["\<\ Check an identity relating e's and h's. You should get zero for \ each term. On a 1.4GHz Athlon, this takes about 30 seconds for 9 variables. \ \ \>", "Text"], Cell[BoxData[ \(Table[ Sum[\((\(-1\))\)^r\ h[vars, \ n - r] e[vars, r], {r, 0, n}] // Simplify, {n, 1, Length[vars]}]\)], "Input"], Cell[TextData[{ "I've defined Schur functions using the Jacobi-Trudi determinant with e's, \ since those are builtin ", StyleBox["Mathematica", FontSlant->"Italic"], " functions and are fast. Let's check that the Jacobi-Trudi determinant \ with h's agrees with our definition. Again, you should get all zeros:" }], "Text"], Cell[BoxData[{ \(newschur[l_, p_]\ := Det[Table[ h[l, p[\([i]\)] - i + j], {i, 1, Length[p]}, {j, 1, Length[p]}]]\), "\[IndentingNewLine]", \(Map[s[vars, #1] - newschur[vars, #1] &, \ Partitions[5]] // Simplify\)}], "Input"], Cell[TextData[{ "The generating function for the h's is basically like the generating \ function for all partitions. In ", StyleBox["Mathematica", FontSlant->"Italic"], " notation, we would write it\n\n\t", StyleBox["H[t_] := Sum[ h[vars,n] t^n, {n,0,Infinity}]", FontFamily->"Courier"], "\n\nLet's expand the series, find the nth coefficient, and subtract off \ h[vars,n] to see if we got it right. Can you see that I like subtracting \ things to see if they're equal?" }], "Text"], Cell[BoxData[{ \(H[t_]\ := \ Product[1/\((1 - vars[\([i]\)]\ t)\), {i, 1, Length[vars]}]\), "\[IndentingNewLine]", \(Table[ Coefficient[Series[H[t], {t, 0, n}], t^n] - h[vars, n] // Simplify, {n, 1, 6}]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "Let's do the same thing for the generating function for the p's, which is \ usually defined by\n\n\t", StyleBox["P[t_] := Sum[ p[vars,n]/n t^n, {n,0,Infinity}]", FontFamily->"Courier"], "\n\n(note that we use p[vars,n]/n, not just p[vars,n])." }], "Text"], Cell[BoxData[{ \(P[t_]\ := \ Log[Product[ 1/\((1 - vars[\([i]\)]\ t)\), {i, 1, Length[vars]}]]\), "\[IndentingNewLine]", \(Table[ Coefficient[Series[P[t], {t, 0, n}], t^n] - p[vars, n]/n // Simplify, {n, 1, 6}]\)}], "Input"], Cell["\<\ The Schur functions are an orthonormal basis with respect to the \ usual symmetric function inner product. Let's check that. You should get an \ identity matrix below:\ \>", "Text"], Cell[BoxData[{ \(\(n = 5;\)\), "\[IndentingNewLine]", \(\(P = Partitions[n];\)\), "\n", \(\(vars = Table[x[i], {i, 1, n}];\)\), "\n", \(Table[ sfip[s[vars, P[\([j]\)]], s[vars, P[\([i]\)]], vars], {i, 1, Length[P]}, {j, 1, Length[P]}] // MatrixForm\)}], "Input"], Cell[TextData[{ "The Schur functions can be written as\n\n\t", Cell[BoxData[ \(TraditionalForm\`s\_\[Lambda]\)]], "= \[Sum] ", Cell[BoxData[ \(TraditionalForm\`K\_\[Lambda]\[Mu]\)]], Cell[BoxData[ \(TraditionalForm\`m\_\[Mu]\)]], "\n\nwhere the sum extends over all \[Mu] that are less than or equal to \ \[Lambda] in the dominance order. Also, since the inner product of ", Cell[BoxData[ \(TraditionalForm\`h\_\[Alpha]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_\[Beta]\)]], " is zero unless \[Alpha]=\[Beta] (in which case it equals 1), we can take \ the inner product of ", Cell[BoxData[ \(TraditionalForm\`h\_\[Mu]\)]], "with ", Cell[BoxData[ \(TraditionalForm\`s\_\[Lambda]\)]], "and all the terms in the sum will be zero except for one term, and we'll \ get the Kostka number for shape \[Lambda] and content \[Mu]. Let's load up my \ Tableaux package (which computes the Kostka numbers by their definition as \ the number of semistand tableaux) and see if these two things agree." }], "Text"], Cell[BoxData[{ \(<< Tableaux`\), "\[IndentingNewLine]", \(KostkaNumber[{5, 3}, {3, 3, 2}]\), "\[IndentingNewLine]", \(\(vars = Table[x[i], {i, 1, 8}];\)\), "\[IndentingNewLine]", \(sfip[h[vars, {3, 3, 2}], s[vars, {5, 3}], vars]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Catalog of functions", "Section"], Cell[TextData[{ "Just evaluate the cell below to see all the functions available, then \ click to see the help blurb. See the package file for the actual definitions \ of these functions, or use two question marks: ", StyleBox["??h", FontFamily->"Courier"] }], "Text"], Cell[BoxData[ \(\(?SymmetricFunctions`*\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Credits, etc", "Section"], Cell[TextData[{ "The SymmetricFunctions package and this documentation are by Dan Drake (", ButtonBox["drake@math.umn.edu", ButtonData:>{ URL[ "mailto:drake@math.umn.edu?subject=SymmetricFunctions"], None}, ButtonStyle->"Hyperlink"], "). My research specialty has little to do with symmetric functions; I just \ wrote this because there aren't any ", StyleBox["Mathematica", FontSlant->"Italic"], " packages for it, and I needed to figure out a tiny symmetric function \ problem.\n\nThe SymmetricFunctions package is \[Copyright] 2004 Dan Drake and \ may be redistributed and/or modified under the terms of the GNU General \ Public License. See SymmetricFunctions.m for details." }], "Text"] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{661, 877}, WindowMargins->{{Automatic, 60}, {Automatic, 44}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "grad", \ "drake", "research", "op"}, "SymmetricFunctions.nb.ps", CharacterEncoding -> \ "iso8859-1"], "Magnification"->1}, ShowSelection->True ] (******************************************************************* 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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