Examples.nb

Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData["\<\"============================================================\
=====\"\>"], "Print"],

Cell[BoxData[
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    RowBox[{"2013", ",", "9", ",", "19"}], "}"}]}],
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  Editable->False]], "Print"],

Cell[BoxData["\<\"CopyRight (C) 2013, Teake Nutma, under the General Public \
License.\"\>"], "Print"],

Cell[BoxData["\<\"Based upon LiE version 2.2.2,\"\>"], "Print"],

Cell[BoxData["\<\"CopyRight (C) 1992-2002, Arjeh M. Cohen, Marc van Leeuwen, \
\"\>"], "Print"],

Cell[BoxData["\<\"and Bert Lisser, under the Lesser General Public License.\"\
\>"], "Print"],

Cell[BoxData["\<\"============================================================\
=====\"\>"], "Print"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["SetDefaultAlgebra", "Subsubsection"],

Cell[TextData[{
 "The default algebra can be set with ",
 StyleBox["SetDefaultAlgebra", "Input"],
 ":"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[TextData[{
 "This is the analogue of LiE\[CloseCurlyQuote]s ",
 StyleBox["setdefault",
  FontSlant->"Italic"],
 "."
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Cell[BoxData[
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Cell[BoxData["\<\"A2\"\>"], "Output"]
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Cell[TextData[{
 "Defaults can be removed by setting it to ",
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 ":"
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell["LieQuery", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["\<\"8\"\>"], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["\<\"O---O\\n1   2   \\nA2\"\>"], "Output"]
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Cell[CellGroupData[{

Cell["CallLieFunction", "Subsubsection"],

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Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["8"], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["8"], "Output"]
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Cell[CellGroupData[{

Cell["LieFunction", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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LiE  function func. It returns $Failed is the LiE function doesn't exist or \
hasn't been translated to a Mathematica name yet.\"\>", 
  "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3588057495-7624499"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["Dim"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["$Failed"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData["GroupCenter"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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     {
      RowBox[{"\<\"Adams\"\>", "\[Rule]", "\<\"Adams\"\>"}]},
     {
      RowBox[{"\<\"adjoint\"\>", "\[Rule]", "\<\"AdjointRepresentation\"\>"}]},
     {
      RowBox[{"\<\"alt_dom\"\>", "\[Rule]", "\<\"AlternatingDominant\"\>"}]},
     {
      RowBox[{"\<\"alt_tensor\"\>", 
       "\[Rule]", "\<\"AlternatingTensorPower\"\>"}]},
     {
      RowBox[{"\<\"alt_W_sum\"\>", 
       "\[Rule]", "\<\"AlternatingWeylSum\"\>"}]},
     {
      RowBox[{"\<\"block_mat\"\>", "\[Rule]", "\<\"BlockdiagonalMatrix\"\>"}]},
     {
      RowBox[{"\<\"branch\"\>", "\[Rule]", "\<\"Branch\"\>"}]},
     {
      RowBox[{"\<\"Bruhat_desc\"\>", 
       "\[Rule]", "\<\"BruhatDescendant\"\>"}]},
     {
      RowBox[{"\<\"Bruhat_leq\"\>", "\[Rule]", "\<\"BruhatLeq\"\>"}]},
     {
      RowBox[{"\<\"canonical\"\>", "\[Rule]", "\<\"CanonicalWeylWord\"\>"}]},
     {
      RowBox[{"\<\"Cartan\"\>", "\[Rule]", "\<\"Cartan\"\>"}]},
     {
      RowBox[{"\<\"Cartan_type\"\>", "\[Rule]", "\<\"CartanType\"\>"}]},
     {
      RowBox[{"\<\"center\"\>", "\[Rule]", "\<\"GroupCenter\"\>"}]},
     {
      RowBox[{"\<\"cent_roots\"\>", 
       "\[Rule]", "\<\"CentralizingRoots\"\>"}]},
     {
      RowBox[{"\<\"centr_type\"\>", "\[Rule]", "\<\"CentralizerType\"\>"}]},
     {
      RowBox[{"\<\"class_ord\"\>", "\[Rule]", "\<\"ConjugacyClassOrder\"\>"}]},
     {
      RowBox[{"\<\"closure\"\>", "\[Rule]", "\<\"Closure\"\>"}]},
     {
      RowBox[{"\<\"collect\"\>", "\[Rule]", "\<\"InverseBranch\"\>"}]},
     {
      RowBox[{"\<\"contragr\"\>", "\[Rule]", "\<\"Contragradient\"\>"}]},
     {
      RowBox[{"\<\"decomp\"\>", "\[Rule]", "\<\"Decomposition\"\>"}]},
     {
      RowBox[{"\<\"degree\"\>", "\[Rule]", "\<\"PolynomialDegree\"\>"}]},
     {
      RowBox[{"\<\"Demazure\"\>", "\[Rule]", "\<\"Demazure\"\>"}]},
     {
      RowBox[{"\<\"det_Cartan\"\>", "\[Rule]", "\<\"DetCartan\"\>"}]},
     {
      RowBox[{"\<\"diagram\"\>", "\[Rule]", "\<\"DynkinDiagram\"\>"}]},
     {
      RowBox[{"\<\"dim\"\>", "\[Rule]", "\<\"Dim\"\>"}]},
     {
      RowBox[{"\<\"dom_char\"\>", "\[Rule]", "\<\"DominantCharacter\"\>"}]},
     {
      RowBox[{"\<\"dominant\"\>", "\[Rule]", "\<\"Dominant\"\>"}]},
     {
      RowBox[{"\<\"dom_weights\"\>", "\[Rule]", "\<\"DominantWeights\"\>"}]},
     {
      RowBox[{"\<\"exponents\"\>", "\[Rule]", "\<\"Exponents\"\>"}]},
     {
      RowBox[{"\<\"filter_dom\"\>", "\[Rule]", "\<\"FilterDominant\"\>"}]},
     {
      RowBox[{"\<\"from_part\"\>", "\[Rule]", "\<\"FromPartition\"\>"}]},
     {
      RowBox[{"\<\"fundam\"\>", "\[Rule]", "\<\"FundamentalRoots\"\>"}]},
     {
      RowBox[{"\<\"high_root\"\>", "\[Rule]", "\<\"HighestRoot\"\>"}]},
     {
      RowBox[{"\<\"i_Cartan\"\>", "\[Rule]", "\<\"InverseCartan\"\>"}]},
     {
      RowBox[{"\<\"inprod\"\>", "\[Rule]", "\<\"InnerProduct\"\>"}]},
     {
      RowBox[{"\<\"KL_poly\"\>", 
       "\[Rule]", "\<\"KazhdanLusztigPolynomial\"\>"}]},
     {
      RowBox[{"\<\"Lie_code\"\>", "\[Rule]", "\<\"LieCode\"\>"}]},
     {
      RowBox[{"\<\"Lie_group\"\>", "\[Rule]", "\<\"LieGroup\"\>"}]},
     {
      RowBox[{"\<\"Lie_rank\"\>", "\[Rule]", "\<\"LieRank\"\>"}]},
     {
      RowBox[{"\<\"long_word\"\>", "\[Rule]", "\<\"LongestWord\"\>"}]},
     {
      RowBox[{"\<\"l_reduce\"\>", "\[Rule]", "\<\"LeftWeylReduce\"\>"}]},
     {
      RowBox[{"\<\"lr_reduce\"\>", "\[Rule]", "\<\"LeftRightWeylReduce\"\>"}]},
     {
      RowBox[{"\<\"LR_tensor\"\>", 
       "\[Rule]", "\<\"LittlewoodRichardson\"\>"}]},
     {
      RowBox[{"\<\"max_sub\"\>", "\[Rule]", "\<\"MaximalSubgroups\"\>"}]},
     {
      RowBox[{"\<\"n_comp\"\>", 
       "\[Rule]", "\<\"NumberOfSimpleComponents\"\>"}]},
     {
      RowBox[{"\<\"next_part\"\>", "\[Rule]", "\<\"NextPartition\"\>"}]},
     {
      RowBox[{"\<\"next_perm\"\>", "\[Rule]", "\<\"NextPermutation\"\>"}]},
     {
      RowBox[{"\<\"next_tabl\"\>", "\[Rule]", "\<\"NextTableau\"\>"}]},
     {
      RowBox[{"\<\"norm\"\>", "\[Rule]", "\<\"RootNorm\"\>"}]},
     {
      RowBox[{"\<\"n_tabl\"\>", "\[Rule]", "\<\"NumberOfTableaux\"\>"}]},
     {
      RowBox[{"\<\"n_vars\"\>", "\[Rule]", "\<\"NumberOfVariables\"\>"}]},
     {
      RowBox[{"\<\"orbit\"\>", "\[Rule]", "\<\"Orbit\"\>"}]},
     {
      RowBox[{"\<\"plethysm\"\>", "\[Rule]", "\<\"Plethysm\"\>"}]},
     {
      RowBox[{"\<\"pos_roots\"\>", "\[Rule]", "\<\"PositiveRoots\"\>"}]},
     {
      RowBox[{"\<\"p_tensor\"\>", "\[Rule]", "\<\"TensorPower\"\>"}]},
     {
      RowBox[{"\<\"print_tab\"\>", "\[Rule]", "\<\"PrintTableau\"\>"}]},
     {
      RowBox[{"\<\"reduce\"\>", "\[Rule]", "\<\"WeylReduce\"\>"}]},
     {
      RowBox[{"\<\"reflection\"\>", "\[Rule]", "\<\"Reflection\"\>"}]},
     {
      RowBox[{"\<\"res_mat\"\>", "\[Rule]", "\<\"RestrictionMatrix\"\>"}]},
     {
      RowBox[{"\<\"row_index\"\>", "\[Rule]", "\<\"RowIndex\"\>"}]},
     {
      RowBox[{"\<\"R_poly\"\>", "\[Rule]", "\<\"RPolynomial\"\>"}]},
     {
      RowBox[{"\<\"r_reduce\"\>", "\[Rule]", "\<\"RightWeylReduce\"\>"}]},
     {
      RowBox[{"\<\"RS\"\>", "\[Rule]", "\<\"RobinsonSchensted\"\>"}]},
     {
      RowBox[{"\<\"shape\"\>", "\[Rule]", "\<\"TableauShape\"\>"}]},
     {
      RowBox[{"\<\"sign_part\"\>", "\[Rule]", "\<\"PartitionSign\"\>"}]},
     {
      RowBox[{"\<\"spectrum\"\>", "\[Rule]", "\<\"ToralSpectrum\"\>"}]},
     {
      RowBox[{"\<\"support\"\>", "\[Rule]", "\<\"Support\"\>"}]},
     {
      RowBox[{"\<\"sym_char\"\>", "\[Rule]", "\<\"SymmetricCharacter\"\>"}]},
     {
      RowBox[{"\<\"sym_orbit\"\>", "\[Rule]", "\<\"SymmetricOrbit\"\>"}]},
     {
      RowBox[{"\<\"sym_tensor\"\>", 
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     {
      RowBox[{"\<\"tableaux\"\>", 
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     {
      RowBox[{"\<\"tensor\"\>", "\[Rule]", "\<\"LieTensor\"\>"}]},
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      RowBox[{"\<\"to_part\"\>", "\[Rule]", "\<\"ToPartition\"\>"}]},
     {
      RowBox[{"\<\"trans_part\"\>", 
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     {
      RowBox[{"\<\"unique\"\>", "\[Rule]", "\<\"CanonicalMatrix\"\>"}]},
     {
      RowBox[{"\<\"v_decomp\"\>", "\[Rule]", "\<\"VirtualDecomposition\"\>"}]},
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      RowBox[{"\<\"W_action\"\>", "\[Rule]", "\<\"WeylAction\"\>"}]},
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      RowBox[{"\<\"W_orbit\"\>", "\[Rule]", "\<\"WeylOrbit\"\>"}]},
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      RowBox[{"\<\"W_orbit_graph\"\>", 
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     {
      RowBox[{"\<\"W_orbit_size\"\>", "\[Rule]", "\<\"WeylOrbitSize\"\>"}]},
     {
      RowBox[{"\<\"W_order\"\>", "\[Rule]", "\<\"WeylOrder\"\>"}]},
     {
      RowBox[{"\<\"W_rt_action\"\>", "\[Rule]", "\<\"WeylRootAction\"\>"}]},
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      RowBox[{"\<\"W_rt_orbit\"\>", "\[Rule]", "\<\"WeylRootOrbit\"\>"}]},
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whose description is as follows:\\n\\ndim(grp)-> int\\ndim(g).   Returns the \
dimension of the Lie group g; equals dim(adjoint(g),g).\\n\\ndim(vec,grp)-> \
bin\\ndim (lambda,g) [lambda: weight].   Returns the dimension of the \
representation V_lambda.\\n\\ndim(pol,grp)-> bin\\ndim(p,g) [p: \
decomposition].   Returns the dimension of the g-module with decomposition \
polynomial p.\"\>", "MSG"]], "Print", "PrintUsage",
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Cell["SymmetricTensorPower", "Subsubsection"],

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Cell[BoxData[
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weight, result: decomposition]. (symmetric tensor power)  Returns the \
decomposition polynomial of the n-th symmetric tensor power of V_lambda.  See \
also alt_tensor and plethysm.\\n\\nsym_tensor(int,pol,grp)-> \
pol\\nsym_tensor(n,p,g) [p,result: decomposition].   This is similar to \
sym_tensor(n,,g), but with the irreducible module V_lambda replaced by the \
module with decomposition polynomial p.\"\>", "MSG"]], "Print", "PrintUsage",
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Cell[CellGroupData[{

Cell["DynkinDiagram", "Subsubsection"],

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Cell[CellGroupData[{

Cell["CartanMatrix / CartanProduct", "Subsubsection"],

Cell["\<\
CartanMatrix and CartanProduct both point to \[OpenCurlyDoubleQuote]cartan\
\[CloseCurlyDoubleQuote].\
\>", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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\\\"Cartan\\\", whose description is as follows:\\n\\nCartan(vec,vec,grp)-> \
int\\nCartan(alpha,beta,g) [alpha,beta: root].   Returns the `Cartan product' \
<alpha,beta>, i.e., the integral value 2(alpha,beta)=(beta,beta), where beta \
must be a root, and alpha is any root vector. [This is is not an inner \
product because the function is not linear in beta. The function is linear in \
alpha however. See also inprod and norm.\\n\\nCartan(grp)-> mat\\nCartan(g) \
[result: lin(root; weight)].   Returns the Cartan matrix of g, which is the \
transformation matrix from the root lattice to the weight lattice, using the \
bases of fundamental roots and fundamental weights respectively. Hence the \
i-th row of the Cartan matrix equals the i-th fundamental root, expressed as \
weight vector. The labeling of the fundamental roots is as indicated by \
diagram(g). When g is semisimple, the (i,j)-entry of the Cartan matrix is \
<alpha_i,alpha_j>. If g contains a central torus, so that the semisimple rank \
s of g is differs from the Lie rank r, then the Cartan matrix is not square, \
as it is an s x r matrix, but all entries beyond column s are zero.\"\>", 
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Cell[CellGroupData[{

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\"Cartan\\\", whose description is as follows:\\n\\nCartan(vec,vec,grp)-> int\
\\nCartan(alpha,beta,g) [alpha,beta: root].   Returns the `Cartan product' \
<alpha,beta>, i.e., the integral value 2(alpha,beta)=(beta,beta), where beta \
must be a root, and alpha is any root vector. [This is is not an inner \
product because the function is not linear in beta. The function is linear in \
alpha however. See also inprod and norm.\\n\\nCartan(grp)-> mat\\nCartan(g) \
[result: lin(root; weight)].   Returns the Cartan matrix of g, which is the \
transformation matrix from the root lattice to the weight lattice, using the \
bases of fundamental roots and fundamental weights respectively. Hence the \
i-th row of the Cartan matrix equals the i-th fundamental root, expressed as \
weight vector. The labeling of the fundamental roots is as indicated by \
diagram(g). When g is semisimple, the (i,j)-entry of the Cartan matrix is \
<alpha_i,alpha_j>. If g contains a central torus, so that the semisimple rank \
s of g is differs from the Lie rank r, then the Cartan matrix is not square, \
as it is an s x r matrix, but all entries beyond column s are zero.\"\>", 
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