tensorProdIrrep.nb

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Decomposition of the tensor product 
of irreps of SU(n) into a direct sum\
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Given two irreducible representations of SU(n) specified by the components of \
their highest weights in the basis of fundamental weights, the program \
determines the decomposition of their tensor product into a sum of \
irreducible representations (irreps). 
The result is a list of pairs: the first element of each pair gives the \
highest weight of each irrep of the decomposition, the second element gives \
its multiplicity.
The program remembers the obtained results.
Example\
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Cell["The dynamic interface: Why two sections for the CDF Player ?", \
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Cell["\<\
The main function is tenspromatform (or equivalently, tensorproduct), and it \
is defined in the main program (package).
The code for the interface, that is used to generate a cdf file, comes next.
Now, why two versions ?\
\>", "Text",
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The first version of the interface code (for the CDF Player on the computer) \
updates correctly and automatically the given components of the weights, even \
after having increased or decreased their number (the value of n-1, for \
SU(n)), but unfortunately, while using the same code, the components are not \
updated automatically with the present IOS version (iPad) of the CDF Player : \
for this reason, in the second version, the user first enters all the \
components, and then he has to hit the \[OpenCurlyDoubleQuote]Update\
\[CloseCurlyDoubleQuote] button. 
Since the latter version also works on the computer (although it is not as \
nice as the first), this is the version that has been uploaded as standalone. 

The user enters the components of the weights (the predefined value of n is \
n=4). 
The decomposition of the tensor product is either dynamically updated (first \
version), or manually updated (hit the \[OpenCurlyDoubleQuote]Update\
\[CloseCurlyDoubleQuote] button: second version).
To change the value of n, one uses the two buttons : \
\[OpenCurlyDoubleQuote]Increase n\[CloseCurlyDoubleQuote] or \
\[OpenCurlyDoubleQuote]Decrease n\[CloseCurlyDoubleQuote].\
\>", "Text",
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   3.7153345225383587`*^9},ExpressionUUID->"0dddd434-3dc3-4cc4-9149-\
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Cell[BoxData[
 DynamicModuleBox[{$CellContext`rows$$ = 2, $CellContext`cols$$ = 
  3, $CellContext`mat$$ = {{3, 2, 1}, {2, 1, 2}}, $CellContext`myRange$$}, 
  DynamicBox[ToBoxes[
    Manipulate[
     Row[{"Product decomposition", 
       MatrixForm[$CellContext`mat$$], " = ", 
       (CheckAbort[
        If[Length[#] > 800, 
         Abort[], #], 
        Print[
        "Output too large: the number of inequivalent irreps should be \
smaller than ", 800]]& )[
        TimeConstrained[
         shorttensorProdIrrepPackage`tenspromatform[$CellContext`mat$$], 1.5, 
         
         Print[
         "More time needed: the computation is terminated if it takes more \
than ", 1.5, " seconds"]]]}], 
     Evaluate[
      With[{$CellContext`makeRow$ = Function[{$CellContext`rowIndex$}, 
          Map[
           Function[{$CellContext`colIndex$}, 
            InputField[
             Dynamic[
              
              Part[$CellContext`mat$$, $CellContext`rowIndex$, \
$CellContext`colIndex$]], Number, FieldSize -> 3]], 
           $CellContext`myRange$$[$CellContext`cols$$]], HoldAll]}, 
       Row[{
         Grid[
          Map[$CellContext`makeRow$, 
           $CellContext`myRange$$[$CellContext`rows$$]]], 
         Invisible["xxxxxx"], 
         Column[{
          "Tensor product of two irreducible representations specified by \
their higest weight.", 
           "Lines refer to the n-1 components of the chosen highest weights \
in the basis of fundamental weights (Dynkin basis)."}]}]]], 
     Row[{
       Panel[
        Style[$CellContext`cols$$ + 1, FontSize -> 14, Bold], 
        Style["SU(n), with n =", Bold, FontSize -> 14], Left, 
        BaselinePosition -> Bottom], 
       Panel[
        Style[
         Column[{
          "Product of irreducible", "representations of SU(n)", 
           "Decomposition into direct sum"}, Center], "Title"]]}, 
      Invisible["xxxxxx"]], 
     Column[{
      "The result is a list of pairs:", 
       "The first element gives the components of each term of the \
decomposition.", "the second element gives its multiplicity."}], 
     Button["Decrease n", 
      If[$CellContext`rows$$ > 
       1, $CellContext`mat$$ = Map[Drop[#, -1]& , $CellContext`mat$$]; 
       Decrement[$CellContext`cols$$]; Null]], 
     Button["Increase n", $CellContext`mat$$ = Transpose[
         Append[
          Transpose[$CellContext`mat$$], 
          Table[0, {1 + 1}]]]; Increment[$CellContext`cols$$]; Null]], 
    StandardForm],
   ImageSizeCache->{985., {285., 291.}}],
  SynchronousUpdating -> False,
  DynamicModuleValues:>{{DownValues[$CellContext`myRange$$] = {HoldPattern[
         $CellContext`myRange$$[
          Pattern[$CellContext`s, 
           Blank[]]]] :> Range[$CellContext`s]}}},
  Initialization:>{shorttensorProdIrrepPackage`tenspromatform[
      Pattern[Private`mat, 
       Blank[]]] := 
    Apply[shorttensorProdIrrepPackage`tensorproduct, Private`mat], 
    TagSet[shorttensorProdIrrepPackage`tenspromatform, 
     MessageName[shorttensorProdIrrepPackage`tenspromatform, "usage"], 
     "tenspromatform[{lambda,mu}] returns tensorproduct[lambda,mu]"], 
    shorttensorProdIrrepPackage`tensorproduct[{
       Pattern[Private`lam, 
        BlankSequence[]]}, {
       Pattern[Private`mu, 
        BlankSequence[]]}] := 
    If[Length[{Private`lam}] == Length[{Private`mu}], 
      Switch[
       ValueQ[shorttensorProdIrrepPackage`currentgroup], False, 
       shorttensorProdIrrepPackage`oblade[Length[{Private`lam}] + 1]; 
       shorttensorProdIrrepPackage`tensorproduct[{Private`lam}, {Private`mu}],
        True, 
       If[
       Length[{Private`lam}] == shorttensorProdIrrepPackage`currentgroup - 1, 
        
        SortBy[
         Tally[
          Map[Reverse, 
           Private`tensorproductCC[{Private`lam}, {Private`mu}]]], Last], 
        shorttensorProdIrrepPackage`oblade[Length[{Private`lam}] + 1]; 
        shorttensorProdIrrepPackage`tensorproduct[{Private`lam}, {
          Private`mu}]]], 
      Print["Weights should have the same number of components"]], 
    TagSet[shorttensorProdIrrepPackage`tensorproduct, 
     MessageName[shorttensorProdIrrepPackage`tensorproduct, "usage"], 
     "tensorproduct[lambda,mu] returns the decomposition into irreducible \
components nu of a tensor product of two irreps lambda, mu, of the Lie group \
SU(n), given by the components of their highest weights along the basis of \
fundamental weights.  The last entry of each term is its multiplicity."], 
    shorttensorProdIrrepPackage`currentgroup = 4, 
    TagSet[shorttensorProdIrrepPackage`currentgroup, 
     MessageName[shorttensorProdIrrepPackage`currentgroup, "usage"], 
     "currentgroup returns the argument n of the group SU(n) currently in \
use."], shorttensorProdIrrepPackage`oblade[
      Pattern[Private`var, 
       Blank[]]] := 
    With[{Private`NN = Private`var}, Private`lambdaequations = And[
         Apply[And, 
          Table[Private`lambda[Private`p, 0, 0] == Sum[
             Private`j[
             Private`p, Private`NN - (Private`s + Private`p), Private`s], {
             Private`s, 0, Private`NN - Private`p}], {
           Private`p, 1, Private`NN - 1}]], 
         Apply[And, 
          Table[Private`lambda[0, Private`p, 0] == Sum[
             Private`j[
             Private`s, Private`p, Private`NN - (Private`s + Private`p)], {
             Private`s, 0, Private`NN - Private`p}], {
           Private`p, 1, Private`NN - 1}]], 
         Apply[And, 
          Table[Private`lambda[0, 0, Private`p] == Sum[
             Private`j[
             Private`s, Private`NN - (Private`s + Private`p), Private`p], {
             Private`s, 0, Private`NN - Private`p}], {
           Private`p, 1, Private`NN - 1}]]]; 
      Private`edgesEW[Private`NN] = Table[
         FoldList[Plus][
          Table[
           Private`j[
           Private`t, Private`NN - Private`s, Private`s - Private`t], {
           Private`s, Private`t, Private`NN - 1}]], {
         Private`t, 1, Private`NN - 1}]; Private`edgesNESW[Private`NN] = Table[
         FoldList[Plus][
          Table[
           Private`j[
           Private`s - Private`p, Private`p, Private`NN - Private`s], {
           Private`s, Private`p, Private`NN - 1}]], {
         Private`p, 1, Private`NN - 1}]; Private`edgesNWSE[Private`NN] = Table[
         FoldList[Plus][
          Table[
           Private`j[
           Private`NN - Private`s, Private`s - Private`p, Private`p], {
           Private`s, Private`p, Private`NN - 1}]], {
         Private`p, 1, Private`NN - 1}]; Private`edgesEWlabels[
         Pattern[Private`lev, 
          Blank[]]] := Table[
         Take[
          
          Table[(1/2) (4 + Private`p + Private`p^2) + (Private`lev - 1), {
           Private`p, Private`lev, Private`NN}], {
          Private`z, Private`z + 1}], {
         Private`z, 1, Private`NN - Private`lev}]; Private`edgesNESWlabels[
         Pattern[Private`lev, 
          Blank[]]] := Table[
         Take[
          
          Table[(1/2) (3 Private`p + Private`p^2) + ((1/2) (Private`lev - 1)) 
            Private`lev + (Private`lev - 1) Private`p, {
           Private`p, 1, Private`NN}], {Private`z, Private`z + 1}], {
         Private`z, 1, Private`NN - Private`lev}]; Private`edgesNWSElabels[
         Pattern[Private`lev, 
          Blank[]]] := Table[
         Take[
          Range[
          1 + ((1/2) (-Private`lev + Private`NN)) (1 - Private`lev + 
             Private`NN), 1 - Private`lev + 
           Private`NN + ((1/2) (-Private`lev + Private`NN)) (1 - Private`lev + 
             Private`NN)], {Private`z, Private`z + 1}], {
         Private`z, 1, Private`NN - Private`lev}]; Private`components = Join[
         Table[
          Table[
           Private`j[
           Private`p, Private`NN - (Private`s + Private`p), Private`s], {
           Private`s, 0, Private`NN - Private`p}], {
          Private`p, 1, Private`NN - 1}], 
         Table[
          Table[
           Private`j[
           Private`s, Private`p, Private`NN - (Private`s + Private`p)], {
           Private`s, 0, Private`NN - Private`p}], {
          Private`p, 1, Private`NN - 1}], 
         Table[
          Table[
           Private`j[
           Private`s, Private`NN - (Private`s + Private`p), Private`p], {
           Private`s, 0, Private`NN - Private`p}], {
          Private`p, 1, Private`NN - 1}]]; 
      Private`fundamentalcomponents = Union[
         Flatten[Private`components]]; Private`positivityconstraints = And[
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesEW[Private`NN]]]], 
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesNESW[Private`NN]]]], 
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesNWSE[Private`NN]]]]]; Private`buildlambdainput[{
          Pattern[Private`l1, 
           BlankSequence[]]}, {
          Pattern[Private`l2, 
           BlankSequence[]]}] := Flatten[{
          MapThread[Rule, {
            Table[
             Private`lambda[Private`p, 0, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l1}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, Private`p, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l2}}]}]; 
      Private`buildlambdainput[{
          Pattern[Private`l1, 
           BlankSequence[]]}, {
          Pattern[Private`l2, 
           BlankSequence[]]}, {
          Pattern[Private`l3, 
           BlankSequence[]]}] := Flatten[{
          MapThread[Rule, {
            Table[
             Private`lambda[Private`p, 0, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l1}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, Private`p, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l2}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, 0, Private`p], {
             Private`p, 1, Private`NN - 1}], {Private`l3}}]}]; 
      Private`componentsEdgesAndnuc[{
          Pattern[Private`lam, 
           BlankSequence[]]}, {
          Pattern[Private`mu, 
           BlankSequence[]]}] := (
        Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}] = 
        Module[{Private`redsol, Private`locrules, Private`nuclist, 
           Private`fundcompval, Private`edgesEWValues, 
           Private`edgesNESWValues, Private`edgesNWSEValues}, 
          Private`redsol = Solve[
             And[
              ReplaceAll[Private`lambdaequations, 
               Private`buildlambdainput[{Private`lam}, {Private`mu}]], 
              Private`positivityconstraints], Integers]; 
          Private`nuclist = ReplaceAll[
             Table[
              Private`lambda[0, 0, Private`p], {
              Private`p, 1, Private`NN - 1}], 
             Private`redsol]; {{}, {}, {}, {}, Private`nuclist}]); 
      shorttensorProdIrrepPackage`currentgroup = Private`NN], 
    TagSet[shorttensorProdIrrepPackage`oblade, 
     MessageName[shorttensorProdIrrepPackage`oblade, "usage"], 
     "Evaluation of oblade[n] initializes all relevant data for SU(n). It \
returns n and set the variable currentgroup to n."], Private`lambdaequations = 
    And[Private`lambda[1, 0, 0] == 
      Private`j[1, 0, 3] + Private`j[1, 1, 2] + Private`j[1, 2, 1] + 
       Private`j[1, 3, 0], Private`lambda[2, 0, 0] == 
      Private`j[2, 0, 2] + Private`j[2, 1, 1] + Private`j[2, 2, 0], 
      Private`lambda[3, 0, 0] == Private`j[3, 0, 1] + Private`j[3, 1, 0], 
      Private`lambda[0, 1, 0] == 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1] + 
       Private`j[3, 1, 0], Private`lambda[0, 2, 0] == 
      Private`j[0, 2, 2] + Private`j[1, 2, 1] + Private`j[2, 2, 0], 
      Private`lambda[0, 3, 0] == Private`j[0, 3, 1] + Private`j[1, 3, 0], 
      Private`lambda[0, 0, 1] == 
      Private`j[0, 3, 1] + Private`j[1, 2, 1] + Private`j[2, 1, 1] + 
       Private`j[3, 0, 1], Private`lambda[0, 0, 2] == 
      Private`j[0, 2, 2] + Private`j[1, 1, 2] + Private`j[2, 0, 2], 
      Private`lambda[0, 0, 3] == Private`j[0, 1, 3] + Private`j[1, 0, 3]], 
    Private`edgesEW[3] = {{
       Private`j[1, 2, 0], Private`j[1, 1, 1] + Private`j[1, 2, 0]}, {
       Private`j[2, 1, 0]}}, Private`edgesEW[4] = {{
       Private`j[1, 3, 0], Private`j[1, 2, 1] + Private`j[1, 3, 0], 
       Private`j[1, 1, 2] + Private`j[1, 2, 1] + Private`j[1, 3, 0]}, {
       Private`j[2, 2, 0], Private`j[2, 1, 1] + Private`j[2, 2, 0]}, {
       Private`j[3, 1, 0]}}, Private`edgesEW[5] = {{
       Private`j[1, 4, 0], Private`j[1, 3, 1] + Private`j[1, 4, 0], 
       Private`j[1, 2, 2] + Private`j[1, 3, 1] + Private`j[1, 4, 0], 
       Private`j[1, 1, 3] + Private`j[1, 2, 2] + Private`j[1, 3, 1] + 
       Private`j[1, 4, 0]}, {
       Private`j[2, 3, 0], Private`j[2, 2, 1] + Private`j[2, 3, 0], 
       Private`j[2, 1, 2] + Private`j[2, 2, 1] + Private`j[2, 3, 0]}, {
       Private`j[3, 2, 0], Private`j[3, 1, 1] + Private`j[3, 2, 0]}, {
       Private`j[4, 1, 0]}}, Private`edgesEW[6] = {{
       Private`j[1, 5, 0], Private`j[1, 4, 1] + Private`j[1, 5, 0], 
       Private`j[1, 3, 2] + Private`j[1, 4, 1] + Private`j[1, 5, 0], 
       Private`j[1, 2, 3] + Private`j[1, 3, 2] + Private`j[1, 4, 1] + 
       Private`j[1, 5, 0], Private`j[1, 1, 4] + Private`j[1, 2, 3] + 
       Private`j[1, 3, 2] + Private`j[1, 4, 1] + Private`j[1, 5, 0]}, {
       Private`j[2, 4, 0], Private`j[2, 3, 1] + Private`j[2, 4, 0], 
       Private`j[2, 2, 2] + Private`j[2, 3, 1] + Private`j[2, 4, 0], 
       Private`j[2, 1, 3] + Private`j[2, 2, 2] + Private`j[2, 3, 1] + 
       Private`j[2, 4, 0]}, {
       Private`j[3, 3, 0], Private`j[3, 2, 1] + Private`j[3, 3, 0], 
       Private`j[3, 1, 2] + Private`j[3, 2, 1] + Private`j[3, 3, 0]}, {
       Private`j[4, 2, 0], Private`j[4, 1, 1] + Private`j[4, 2, 0]}, {
       Private`j[5, 1, 0]}}, Private`edgesNESW[3] = {{
       Private`j[0, 1, 2], Private`j[0, 1, 2] + Private`j[1, 1, 1]}, {
       Private`j[0, 2, 1]}}, Private`edgesNESW[4] = {{
       Private`j[0, 1, 3], Private`j[0, 1, 3] + Private`j[1, 1, 2], 
       Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1]}, {
       Private`j[0, 2, 2], Private`j[0, 2, 2] + Private`j[1, 2, 1]}, {
       Private`j[0, 3, 1]}}, Private`edgesNESW[5] = {{
       Private`j[0, 1, 4], Private`j[0, 1, 4] + Private`j[1, 1, 3], 
       Private`j[0, 1, 4] + Private`j[1, 1, 3] + Private`j[2, 1, 2], 
       Private`j[0, 1, 4] + Private`j[1, 1, 3] + Private`j[2, 1, 2] + 
       Private`j[3, 1, 1]}, {
       Private`j[0, 2, 3], Private`j[0, 2, 3] + Private`j[1, 2, 2], 
       Private`j[0, 2, 3] + Private`j[1, 2, 2] + Private`j[2, 2, 1]}, {
       Private`j[0, 3, 2], Private`j[0, 3, 2] + Private`j[1, 3, 1]}, {
       Private`j[0, 4, 1]}}, Private`edgesNESW[6] = {{
       Private`j[0, 1, 5], Private`j[0, 1, 5] + Private`j[1, 1, 4], 
       Private`j[0, 1, 5] + Private`j[1, 1, 4] + Private`j[2, 1, 3], 
       Private`j[0, 1, 5] + Private`j[1, 1, 4] + Private`j[2, 1, 3] + 
       Private`j[3, 1, 2], Private`j[0, 1, 5] + Private`j[1, 1, 4] + 
       Private`j[2, 1, 3] + Private`j[3, 1, 2] + Private`j[4, 1, 1]}, {
       Private`j[0, 2, 4], Private`j[0, 2, 4] + Private`j[1, 2, 3], 
       Private`j[0, 2, 4] + Private`j[1, 2, 3] + Private`j[2, 2, 2], 
       Private`j[0, 2, 4] + Private`j[1, 2, 3] + Private`j[2, 2, 2] + 
       Private`j[3, 2, 1]}, {
       Private`j[0, 3, 3], Private`j[0, 3, 3] + Private`j[1, 3, 2], 
       Private`j[0, 3, 3] + Private`j[1, 3, 2] + Private`j[2, 3, 1]}, {
       Private`j[0, 4, 2], Private`j[0, 4, 2] + Private`j[1, 4, 1]}, {
       Private`j[0, 5, 1]}}, Private`edgesNWSE[3] = {{
       Private`j[2, 0, 1], Private`j[1, 1, 1] + Private`j[2, 0, 1]}, {
       Private`j[1, 0, 2]}}, Private`edgesNWSE[4] = {{
       Private`j[3, 0, 1], Private`j[2, 1, 1] + Private`j[3, 0, 1], 
       Private`j[1, 2, 1] + Private`j[2, 1, 1] + Private`j[3, 0, 1]}, {
       Private`j[2, 0, 2], Private`j[1, 1, 2] + Private`j[2, 0, 2]}, {
       Private`j[1, 0, 3]}}, Private`edgesNWSE[5] = {{
       Private`j[4, 0, 1], Private`j[3, 1, 1] + Private`j[4, 0, 1], 
       Private`j[2, 2, 1] + Private`j[3, 1, 1] + Private`j[4, 0, 1], 
       Private`j[1, 3, 1] + Private`j[2, 2, 1] + Private`j[3, 1, 1] + 
       Private`j[4, 0, 1]}, {
       Private`j[3, 0, 2], Private`j[2, 1, 2] + Private`j[3, 0, 2], 
       Private`j[1, 2, 2] + Private`j[2, 1, 2] + Private`j[3, 0, 2]}, {
       Private`j[2, 0, 3], Private`j[1, 1, 3] + Private`j[2, 0, 3]}, {
       Private`j[1, 0, 4]}}, Private`edgesNWSE[6] = {{
       Private`j[5, 0, 1], Private`j[4, 1, 1] + Private`j[5, 0, 1], 
       Private`j[3, 2, 1] + Private`j[4, 1, 1] + Private`j[5, 0, 1], 
       Private`j[2, 3, 1] + Private`j[3, 2, 1] + Private`j[4, 1, 1] + 
       Private`j[5, 0, 1], Private`j[1, 4, 1] + Private`j[2, 3, 1] + 
       Private`j[3, 2, 1] + Private`j[4, 1, 1] + Private`j[5, 0, 1]}, {
       Private`j[4, 0, 2], Private`j[3, 1, 2] + Private`j[4, 0, 2], 
       Private`j[2, 2, 2] + Private`j[3, 1, 2] + Private`j[4, 0, 2], 
       Private`j[1, 3, 2] + Private`j[2, 2, 2] + Private`j[3, 1, 2] + 
       Private`j[4, 0, 2]}, {
       Private`j[3, 0, 3], Private`j[2, 1, 3] + Private`j[3, 0, 3], 
       Private`j[1, 2, 3] + Private`j[2, 1, 3] + Private`j[3, 0, 3]}, {
       Private`j[2, 0, 4], Private`j[1, 1, 4] + Private`j[2, 0, 4]}, {
       Private`j[1, 0, 5]}}, Private`edgesEWlabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Table[(1/2) (4 + Private`p + Private`p^2) + (Private`lev$ - 1), {
        Private`p, Private`lev$, 4}], {Private`z, Private`z + 1}], {
      Private`z, 1, 4 - Private`lev$}], 
    Attributes[Private`lev$] = {Temporary}, Private`edgesNESWlabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Table[(1/2) (3 Private`p + Private`p^2) + ((1/2) (Private`lev$ - 1)) 
         Private`lev$ + (Private`lev$ - 1) Private`p, {Private`p, 1, 4}], {
       Private`z, Private`z + 1}], {Private`z, 1, 4 - Private`lev$}], 
    Private`edgesNWSElabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Range[
       1 + ((1/2) (-Private`lev$ + 4)) (1 - Private`lev$ + 4), 1 - 
        Private`lev$ + 
        4 + ((1/2) (-Private`lev$ + 4)) (1 - Private`lev$ + 4)], {
       Private`z, Private`z + 1}], {Private`z, 1, 4 - Private`lev$}], 
    Private`components = {{
       Private`j[1, 3, 0], 
       Private`j[1, 2, 1], 
       Private`j[1, 1, 2], 
       Private`j[1, 0, 3]}, {
       Private`j[2, 2, 0], 
       Private`j[2, 1, 1], 
       Private`j[2, 0, 2]}, {
       Private`j[3, 1, 0], 
       Private`j[3, 0, 1]}, {
       Private`j[0, 1, 3], 
       Private`j[1, 1, 2], 
       Private`j[2, 1, 1], 
       Private`j[3, 1, 0]}, {
       Private`j[0, 2, 2], 
       Private`j[1, 2, 1], 
       Private`j[2, 2, 0]}, {
       Private`j[0, 3, 1], 
       Private`j[1, 3, 0]}, {
       Private`j[0, 3, 1], 
       Private`j[1, 2, 1], 
       Private`j[2, 1, 1], 
       Private`j[3, 0, 1]}, {
       Private`j[0, 2, 2], 
       Private`j[1, 1, 2], 
       Private`j[2, 0, 2]}, {
       Private`j[0, 1, 3], 
       Private`j[1, 0, 3]}}, Private`fundamentalcomponents = {
      Private`j[0, 1, 3], 
      Private`j[0, 2, 2], 
      Private`j[0, 3, 1], 
      Private`j[1, 0, 3], 
      Private`j[1, 1, 2], 
      Private`j[1, 2, 1], 
      Private`j[1, 3, 0], 
      Private`j[2, 0, 2], 
      Private`j[2, 1, 1], 
      Private`j[2, 2, 0], 
      Private`j[3, 0, 1], 
      Private`j[3, 1, 0]}, Private`positivityconstraints = 
    And[Private`j[1, 3, 0] >= 0, Private`j[1, 2, 1] + Private`j[1, 3, 0] >= 0,
       Private`j[1, 1, 2] + Private`j[1, 2, 1] + Private`j[1, 3, 0] >= 0, 
      Private`j[2, 2, 0] >= 0, Private`j[2, 1, 1] + Private`j[2, 2, 0] >= 0, 
      Private`j[3, 1, 0] >= 0, Private`j[0, 1, 3] >= 0, 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] >= 0, 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1] >= 0, 
      Private`j[0, 2, 2] >= 0, Private`j[0, 2, 2] + Private`j[1, 2, 1] >= 0, 
      Private`j[0, 3, 1] >= 0, Private`j[3, 0, 1] >= 0, 
      Private`j[2, 1, 1] + Private`j[3, 0, 1] >= 0, 
      Private`j[1, 2, 1] + Private`j[2, 1, 1] + Private`j[3, 0, 1] >= 0, 
      Private`j[2, 0, 2] >= 0, Private`j[1, 1, 2] + Private`j[2, 0, 2] >= 0, 
      Private`j[1, 0, 3] >= 0], Private`buildlambdainput[{
       Pattern[Private`l1$, 
        BlankSequence[]]}, {
       Pattern[Private`l2$, 
        BlankSequence[]]}] := Flatten[{
       MapThread[Rule, {
         Table[
          Private`lambda[Private`p, 0, 0], {Private`p, 1, 4 - 1}], {
         Private`l1$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, Private`p, 0], {Private`p, 1, 4 - 1}], {
         Private`l2$}}]}], Private`buildlambdainput[{
       Pattern[Private`l1$, 
        BlankSequence[]]}, {
       Pattern[Private`l2$, 
        BlankSequence[]]}, {
       Pattern[Private`l3$, 
        BlankSequence[]]}] := Flatten[{
       MapThread[Rule, {
         Table[
          Private`lambda[Private`p, 0, 0], {Private`p, 1, 4 - 1}], {
         Private`l1$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, Private`p, 0], {Private`p, 1, 4 - 1}], {
         Private`l2$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, 0, Private`p], {Private`p, 1, 4 - 1}], {
         Private`l3$}}]}], Attributes[Private`l1$] = {Temporary}, 
    Attributes[Private`l2$] = {Temporary}, 
    Attributes[Private`l3$] = {Temporary}, 
    Private`componentsEdgesAndnuc[{1, 1}, {2, 3}] = {{}, {}, {}, {}, {{4, 
      0}, {2, 1}, {5, 1}, {3, 2}, {3, 2}, {4, 3}, {1, 3}, {2, 4}}}, 
    Private`componentsEdgesAndnuc[{0, 1, 2}, {2, 3, 1}] = {{}, {}, {}, {}, {{
      1, 2, 0}, {3, 3, 0}, {1, 4, 0}, {1, 4, 0}, {2, 2, 1}, {2, 2, 1}, {0, 3, 
      1}, {0, 3, 1}, {4, 3, 1}, {2, 4, 1}, {2, 4, 1}, {0, 5, 1}, {1, 1, 2}, {
      3, 2, 2}, {3, 2, 2}, {1, 3, 2}, {1, 3, 2}, {1, 3, 2}, {3, 4, 2}, {1, 5, 
      2}, {2, 1, 3}, {2, 1, 3}, {0, 2, 3}, {0, 2, 3}, {4, 2, 3}, {2, 3, 3}, {
      2, 3, 3}, {0, 4, 3}, {1, 0, 4}, {3, 1, 4}, {1, 2, 4}, {1, 2, 4}, {2, 0, 
      5}, {0, 1, 5}}}, 
    Private`componentsEdgesAndnuc[{0, 2, 0}, {0, 2, 1}] = {{}, {}, {}, {}, {{
      1, 0, 0}, {1, 2, 0}, {1, 4, 0}, {2, 0, 1}, {0, 1, 1}, {2, 2, 1}, {0, 3, 
      1}, {3, 0, 2}, {1, 1, 2}}}, 
    Private`componentsEdgesAndnuc[{1, 1, 0}, {2, 3, 0}] = {{}, {}, {}, {}, {{
      1, 4, 0}, {2, 2, 1}, {0, 3, 1}, {0, 5, 1}, {1, 3, 2}, {1, 1, 2}, {1, 3, 
      2}, {0, 2, 3}, {0, 4, 3}, {2, 1, 3}, {1, 2, 4}}}, 
    Private`componentsEdgesAndnuc[{1, 1, 2}, {2, 3, 1}] = {{}, {}, {}, {}, {{
      2, 1, 0}, {0, 2, 0}, {4, 2, 0}, {2, 3, 0}, {2, 3, 0}, {1, 2, 1}, {2, 3, 
      0}, {0, 4, 0}, {0, 4, 0}, {4, 4, 0}, {2, 5, 0}, {3, 3, 1}, {1, 4, 1}, {
      2, 5, 0}, {0, 6, 0}, {1, 4, 1}, {3, 1, 1}, {3, 1, 1}, {1, 2, 1}, {1, 2, 
      1}, {1, 2, 1}, {5, 2, 1}, {3, 3, 1}, {3, 3, 1}, {1, 4, 1}, {2, 2, 2}, {
      2, 2, 2}, {3, 3, 1}, {1, 4, 1}, {1, 4, 1}, {0, 3, 2}, {0, 3, 2}, {3, 5, 
      1}, {4, 3, 2}, {2, 4, 2}, {1, 6, 1}, {2, 4, 2}, {0, 5, 2}, {2, 0, 2}, {
      0, 1, 2}, {4, 1, 2}, {4, 1, 2}, {2, 2, 2}, {2, 2, 2}, {1, 1, 3}, {2, 2, 
      2}, {2, 2, 2}, {0, 3, 2}, {0, 3, 2}, {4, 3, 2}, {2, 4, 2}, {3, 2, 3}, {
      3, 2, 3}, {1, 3, 3}, {2, 4, 2}, {0, 5, 2}, {1, 3, 3}, {1, 3, 3}, {3, 4, 
      3}, {1, 5, 3}, {3, 0, 3}, {3, 0, 3}, {1, 1, 3}, {1, 1, 3}, {1, 1, 3}, {
      5, 1, 3}, {3, 2, 3}, {1, 3, 3}, {2, 1, 4}, {2, 1, 4}, {3, 2, 3}, {1, 3, 
      3}, {0, 2, 4}, {0, 2, 4}, {4, 2, 4}, {2, 3, 4}, {2, 3, 4}, {0, 4, 4}, {
      0, 0, 4}, {4, 0, 4}, {2, 1, 4}, {1, 0, 5}, {2, 1, 4}, {0, 2, 4}, {3, 1, 
      5}, {1, 2, 5}, {1, 2, 5}, {1, 0, 5}, {2, 0, 6}, {0, 1, 6}}}, 
    Private`componentsEdgesAndnuc[{1, 2, 0}, {2, 3, 0}] = {{}, {}, {}, {}, {{
      3, 2, 0}, {1, 3, 0}, {1, 5, 0}, {2, 3, 1}, {2, 1, 1}, {0, 2, 1}, {2, 3, 
      1}, {0, 4, 1}, {1, 2, 2}, {0, 6, 1}, {1, 4, 2}, {1, 0, 2}, {3, 1, 2}, {
      1, 2, 2}, {0, 1, 3}, {1, 4, 2}, {2, 2, 3}, {0, 3, 3}, {0, 5, 3}, {2, 0, 
      3}, {2, 2, 3}, {1, 1, 4}, {1, 3, 4}, {3, 0, 4}, {2, 1, 5}}}, 
    Private`componentsEdgesAndnuc[{1, 2, 0}, {2, 3, 1}] = {{}, {}, {}, {}, {{
      4, 2, 0}, {2, 3, 0}, {2, 3, 0}, {0, 4, 0}, {2, 5, 0}, {3, 3, 1}, {0, 6, 
      0}, {1, 4, 1}, {3, 1, 1}, {1, 2, 1}, {1, 2, 1}, {3, 3, 1}, {1, 4, 1}, {
      2, 2, 2}, {1, 4, 1}, {0, 3, 2}, {1, 6, 1}, {2, 4, 2}, {0, 5, 2}, {2, 0, 
      2}, {0, 1, 2}, {4, 1, 2}, {2, 2, 2}, {1, 1, 3}, {2, 2, 2}, {0, 3, 2}, {
      2, 4, 2}, {3, 2, 3}, {1, 3, 3}, {0, 5, 2}, {1, 3, 3}, {1, 5, 3}, {3, 0, 
      3}, {1, 1, 3}, {3, 2, 3}, {2, 1, 4}, {1, 3, 3}, {0, 2, 4}, {2, 3, 4}, {
      0, 4, 4}, {4, 0, 4}, {2, 1, 4}, {3, 1, 5}, {1, 2, 5}}}, 
    Private`componentsEdgesAndnuc[{3, 1, 1}, {1, 1, 0}] = {{}, {}, {}, {}, {{
      2, 2, 1}, {0, 3, 1}, {3, 0, 2}, {1, 1, 2}, {1, 1, 2}, {0, 0, 3}, {1, 3, 
      2}, {2, 1, 3}, {0, 2, 3}, {2, 1, 3}, {0, 2, 3}, {1, 0, 4}, {1, 0, 4}, {
      1, 2, 4}, {2, 0, 5}, {0, 1, 5}}}, 
    Private`componentsEdgesAndnuc[{3, 2, 0}, {3, 3, 0}] = {{}, {}, {}, {}, {{
      4, 2, 0}, {2, 3, 0}, {0, 4, 0}, {2, 5, 0}, {0, 6, 0}, {3, 3, 1}, {1, 4, 
      1}, {0, 8, 0}, {1, 6, 1}, {2, 4, 2}, {5, 0, 1}, {3, 1, 1}, {1, 2, 1}, {
      3, 3, 1}, {1, 4, 1}, {4, 1, 2}, {2, 2, 2}, {0, 3, 2}, {1, 6, 1}, {2, 4, 
      2}, {0, 5, 2}, {3, 2, 3}, {1, 3, 3}, {0, 7, 2}, {1, 5, 3}, {2, 3, 4}, {
      4, 1, 2}, {2, 2, 2}, {3, 0, 3}, {1, 1, 3}, {2, 4, 2}, {3, 2, 3}, {1, 3, 
      3}, {2, 1, 4}, {0, 2, 4}, {1, 5, 3}, {2, 3, 4}, {0, 4, 4}, {1, 2, 5}, {
      0, 6, 4}, {1, 4, 5}, {3, 2, 3}, {4, 0, 4}, {2, 1, 4}, {1, 0, 5}, {2, 3, 
      4}, {3, 1, 5}, {1, 2, 5}, {0, 1, 6}, {1, 4, 5}, {2, 2, 6}, {0, 3, 6}, {
      0, 5, 6}, {3, 1, 5}, {2, 0, 6}, {2, 2, 6}, {1, 1, 7}, {1, 3, 7}, {3, 0, 
      7}, {2, 1, 8}}}, 
    Private`componentsEdgesAndnuc[{3, 2, 3}, {3, 1, 0}] = {{}, {}, {}, {}, {{
      5, 3, 0}, {3, 4, 0}, {3, 4, 0}, {1, 5, 0}, {6, 1, 1}, {4, 2, 1}, {2, 3, 
      1}, {4, 2, 1}, {2, 3, 1}, {0, 4, 1}, {5, 0, 2}, {3, 1, 2}, {1, 2, 2}, {
      3, 1, 2}, {1, 2, 2}, {2, 0, 3}, {0, 1, 3}, {3, 6, 0}, {4, 4, 1}, {2, 5, 
      1}, {5, 2, 2}, {3, 3, 2}, {1, 4, 2}, {4, 1, 3}, {2, 2, 3}, {0, 3, 3}, {
      4, 4, 1}, {2, 5, 1}, {5, 2, 2}, {3, 3, 2}, {3, 3, 2}, {1, 4, 2}, {6, 0, 
      3}, {4, 1, 3}, {2, 2, 3}, {4, 1, 3}, {2, 2, 3}, {0, 3, 3}, {3, 0, 4}, {
      1, 1, 4}, {3, 0, 4}, {1, 1, 4}, {3, 5, 2}, {4, 3, 3}, {2, 4, 3}, {5, 1, 
      4}, {3, 2, 4}, {1, 3, 4}, {4, 3, 3}, {2, 4, 3}, {5, 1, 4}, {3, 2, 4}, {
      3, 2, 4}, {1, 3, 4}, {4, 0, 5}, {2, 1, 5}, {4, 0, 5}, {2, 1, 5}, {0, 2, 
      5}, {3, 4, 4}, {4, 2, 5}, {2, 3, 5}, {4, 2, 5}, {2, 3, 5}, {5, 0, 6}, {
      3, 1, 6}, {3, 1, 6}, {1, 2, 6}, {3, 3, 6}, {4, 1, 7}, {2, 2, 7}}}, 
    Private`componentsEdgesAndnuc[{3, 2, 3}, {3, 1, 
      3}] = {{}, {}, {}, {}, CompressedData["
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aN7fuXT2vbN4m30rv/4BNpkUBw==
      "]}, 
    Private`componentsEdgesAndnuc[{3, 2, 3}, {3, 3, 
      3}] = {{}, {}, {}, {}, CompressedData["
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cUvoFVGdOsOdylGseyIGctRIOWqf6ryi8E32W0JbrQyOfmZ1qkhZgVdKZyan
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xibg7RVdlc+4qUZNpYz+iqsWZETZGtsb82LkVLk8Wq/u73b4AJRY9pWGQiMA
WYuGbAG8Qiuc121BzpwzXHaUFSArMSwSqAGkJv5klh5SXaeics3BI0gY4gnV
Irc+JJJ0fkbLkFTkBho1mCzFoWNQmcPW2OvWolPnRFJlpIoax7qe0wCTMshG
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ZJ7btrKFp3JIt4ujeverlks+xoHxSSnOB75/5StRuGC/BfBY/BchHO6d/sMF
1jjX2sftx1+V3S4S
      "]}, 
    Private`componentsEdgesAndnuc[{1, 1, 2, 0}, {2, 3, 1, 
      0}] = {{}, {}, {}, {}, CompressedData["
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      "]}, 
    Private`componentsEdgesAndnuc[{1, 1, 2, 2}, {2, 3, 1, 
      0}] = {{}, {}, {}, {}, CompressedData["
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87OZH3dt7Pte9+VpP/AYhGLUWBnvbEN5WTr9tAkeSKxuIDAOeF5PS5eNZJK6
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      "]}, 
    Private`componentsEdgesAndnuc[{3, 2, 0, 0}, {3, 3, 0, 
      0}] = {{}, {}, {}, {}, {{0, 4, 2, 0}, {1, 2, 3, 0}, {2, 0, 4, 0}, {0, 2,
       5, 0}, {1, 0, 6, 0}, {0, 3, 3, 1}, {1, 1, 4, 1}, {0, 0, 8, 0}, {0, 1, 
      6, 1}, {0, 2, 4, 2}, {0, 5, 0, 1}, {1, 3, 1, 1}, {2, 1, 2, 1}, {0, 3, 3,
       1}, {1, 1, 4, 1}, {0, 4, 1, 2}, {1, 2, 2, 2}, {2, 0, 3, 2}, {0, 1, 6, 
      1}, {0, 2, 4, 2}, {1, 0, 5, 2}, {0, 3, 2, 3}, {1, 1, 3, 3}, {0, 0, 7, 
      2}, {0, 1, 5, 3}, {0, 2, 3, 4}, {0, 4, 1, 2}, {1, 2, 2, 2}, {1, 3, 0, 
      3}, {2, 1, 1, 3}, {0, 2, 4, 2}, {0, 3, 2, 3}, {1, 1, 3, 3}, {1, 2, 1, 
      4}, {2, 0, 2, 4}, {0, 1, 5, 3}, {0, 2, 3, 4}, {1, 0, 4, 4}, {1, 1, 2, 
      5}, {0, 0, 6, 4}, {0, 1, 4, 5}, {0, 3, 2, 3}, {0, 4, 0, 4}, {1, 2, 1, 
      4}, {2, 1, 0, 5}, {0, 2, 3, 4}, {0, 3, 1, 5}, {1, 1, 2, 5}, {2, 0, 1, 
      6}, {0, 1, 4, 5}, {0, 2, 2, 6}, {1, 0, 3, 6}, {0, 0, 5, 6}, {0, 3, 1, 
      5}, {1, 2, 0, 6}, {0, 2, 2, 6}, {1, 1, 1, 7}, {0, 1, 3, 7}, {0, 3, 0, 
      7}, {0, 2, 1, 8}}}, 
    Private`componentsEdgesAndnuc[{3, 2, 1, 0}, {3, 3, 0, 
      0}] = {{}, {}, {}, {}, CompressedData["
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        BlankSequence[]]}, {
       Pattern[Private`mu$, 
        BlankSequence[]]}] := (
     Private`componentsEdgesAndnuc[{Private`lam$}, {Private`mu$}] = 
     Module[{Private`redsol$, Private`locrules$, Private`nuclist$, 
        Private`fundcompval$, Private`edgesEWValues$, 
        Private`edgesNESWValues$, Private`edgesNWSEValues$}, 
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            Private`buildlambdainput[{Private`lam$}, {Private`mu$}]], 
           Private`positivityconstraints], Integers]; 
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          Private`redsol$]; {{}, {}, {}, {}, Private`nuclist$}]), 
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    Attributes[Private`mu$] = {Temporary}, 
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    Attributes[Private`edgesNESWValues$] = {Temporary}, 
    Attributes[Private`edgesNWSEValues$] = {Temporary}, 
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        BlankSequence[]]}, {
       Pattern[Private`mu, 
        BlankSequence[]]}] := Part[
      Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 5], 
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The program decomposes the product of two irreducible representations of \
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 "The decomposition of the tensor product of irreducible representations is \
done by using an improved hive technique developed by the author and inspired \
from [1], [2], [3]. \nIn order to speed up the calculation we use a basis of \
elementary pictographs where syzygies (constraints) are eliminated before the \
actual reduction of the system of inequalities takes place.\nPictographs \
themselves (BZ-triangles, KT-honeycombs, O-blades), corresponding to a chosen \
space of intertwiners,  are not displayed here :  see the companion programs \
HYPERLINK. \n\n[1] ",
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Mathematica session. It does not contain any limitation.\n The main symbol \
exported by the package is the function tensorproduct (see also the related \
command tenspromatform).\n- A dynamic module that builds the user interface. \
This latter code is a bit convoluted because it is supposed to run not only \
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If you use this program in a scientific publication or talk, please give \
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DECOMPOSITION INTO A DIRECT SUM OF THE PRODUCT OF IRREDUCIBLE REPRESENTATIONS \
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      Private`fundamentalcomponents = Union[
         Flatten[Private`components]]; Private`positivityconstraints = And[
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesEW[Private`NN]]]], 
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesNESW[Private`NN]]]], 
         Apply[And, 
          Map[# >= 0& , 
           Flatten[
            Private`edgesNWSE[Private`NN]]]]]; Private`buildlambdainput[{
          Pattern[Private`l1, 
           BlankSequence[]]}, {
          Pattern[Private`l2, 
           BlankSequence[]]}] := Flatten[{
          MapThread[Rule, {
            Table[
             Private`lambda[Private`p, 0, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l1}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, Private`p, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l2}}]}]; 
      Private`buildlambdainput[{
          Pattern[Private`l1, 
           BlankSequence[]]}, {
          Pattern[Private`l2, 
           BlankSequence[]]}, {
          Pattern[Private`l3, 
           BlankSequence[]]}] := Flatten[{
          MapThread[Rule, {
            Table[
             Private`lambda[Private`p, 0, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l1}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, Private`p, 0], {
             Private`p, 1, Private`NN - 1}], {Private`l2}}], 
          MapThread[Rule, {
            Table[
             Private`lambda[0, 0, Private`p], {
             Private`p, 1, Private`NN - 1}], {Private`l3}}]}]; 
      Private`componentsEdgesAndnuc[{
          Pattern[Private`lam, 
           BlankSequence[]]}, {
          Pattern[Private`mu, 
           BlankSequence[]]}] := (
        Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}] = 
        Module[{Private`redsol, Private`locrules, Private`nuclist, 
           Private`fundcompval, Private`edgesEWValues, 
           Private`edgesNESWValues, Private`edgesNWSEValues}, 
          Private`redsol = Solve[
             And[
              ReplaceAll[Private`lambdaequations, 
               Private`buildlambdainput[{Private`lam}, {Private`mu}]], 
              Private`positivityconstraints], Integers]; 
          Private`nuclist = ReplaceAll[
             Table[
              Private`lambda[0, 0, Private`p], {
              Private`p, 1, Private`NN - 1}], 
             Private`redsol]; {{}, {}, {}, {}, Private`nuclist}]); 
      shorttensorProdIrrepPackage`currentgroup = Private`NN], 
    TagSet[shorttensorProdIrrepPackage`oblade, 
     MessageName[shorttensorProdIrrepPackage`oblade, "usage"], 
     "Evaluation of oblade[n] initializes all relevant data for SU(n). It \
returns n and set the variable currentgroup to n."], Private`lambdaequations = 
    And[Private`lambda[1, 0, 0] == 
      Private`j[1, 0, 3] + Private`j[1, 1, 2] + Private`j[1, 2, 1] + 
       Private`j[1, 3, 0], Private`lambda[2, 0, 0] == 
      Private`j[2, 0, 2] + Private`j[2, 1, 1] + Private`j[2, 2, 0], 
      Private`lambda[3, 0, 0] == Private`j[3, 0, 1] + Private`j[3, 1, 0], 
      Private`lambda[0, 1, 0] == 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1] + 
       Private`j[3, 1, 0], Private`lambda[0, 2, 0] == 
      Private`j[0, 2, 2] + Private`j[1, 2, 1] + Private`j[2, 2, 0], 
      Private`lambda[0, 3, 0] == Private`j[0, 3, 1] + Private`j[1, 3, 0], 
      Private`lambda[0, 0, 1] == 
      Private`j[0, 3, 1] + Private`j[1, 2, 1] + Private`j[2, 1, 1] + 
       Private`j[3, 0, 1], Private`lambda[0, 0, 2] == 
      Private`j[0, 2, 2] + Private`j[1, 1, 2] + Private`j[2, 0, 2], 
      Private`lambda[0, 0, 3] == Private`j[0, 1, 3] + Private`j[1, 0, 3]], 
    Private`edgesEW[4] = {{
       Private`j[1, 3, 0], Private`j[1, 2, 1] + Private`j[1, 3, 0], 
       Private`j[1, 1, 2] + Private`j[1, 2, 1] + Private`j[1, 3, 0]}, {
       Private`j[2, 2, 0], Private`j[2, 1, 1] + Private`j[2, 2, 0]}, {
       Private`j[3, 1, 0]}}, Private`edgesNESW[4] = {{
       Private`j[0, 1, 3], Private`j[0, 1, 3] + Private`j[1, 1, 2], 
       Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1]}, {
       Private`j[0, 2, 2], Private`j[0, 2, 2] + Private`j[1, 2, 1]}, {
       Private`j[0, 3, 1]}}, Private`edgesNWSE[4] = {{
       Private`j[3, 0, 1], Private`j[2, 1, 1] + Private`j[3, 0, 1], 
       Private`j[1, 2, 1] + Private`j[2, 1, 1] + Private`j[3, 0, 1]}, {
       Private`j[2, 0, 2], Private`j[1, 1, 2] + Private`j[2, 0, 2]}, {
       Private`j[1, 0, 3]}}, Private`edgesEWlabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Table[(1/2) (4 + Private`p + Private`p^2) + (Private`lev$ - 1), {
        Private`p, Private`lev$, 4}], {Private`z, Private`z + 1}], {
      Private`z, 1, 4 - Private`lev$}], 
    Attributes[Private`lev$] = {Temporary}, Private`edgesNESWlabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Table[(1/2) (3 Private`p + Private`p^2) + (1/2) (Private`lev$ - 1) 
         Private`lev$ + (Private`lev$ - 1) Private`p, {Private`p, 1, 4}], {
       Private`z, Private`z + 1}], {Private`z, 1, 4 - Private`lev$}], 
    Private`edgesNWSElabels[
      Pattern[Private`lev$, 
       Blank[]]] := Table[
      Take[
       Range[
       1 + (1/2) (-Private`lev$ + 4) (1 - Private`lev$ + 4), 1 - Private`lev$ + 
        4 + (1/2) (-Private`lev$ + 4) (1 - Private`lev$ + 4)], {
       Private`z, Private`z + 1}], {Private`z, 1, 4 - Private`lev$}], 
    Private`components = {{
       Private`j[1, 3, 0], 
       Private`j[1, 2, 1], 
       Private`j[1, 1, 2], 
       Private`j[1, 0, 3]}, {
       Private`j[2, 2, 0], 
       Private`j[2, 1, 1], 
       Private`j[2, 0, 2]}, {
       Private`j[3, 1, 0], 
       Private`j[3, 0, 1]}, {
       Private`j[0, 1, 3], 
       Private`j[1, 1, 2], 
       Private`j[2, 1, 1], 
       Private`j[3, 1, 0]}, {
       Private`j[0, 2, 2], 
       Private`j[1, 2, 1], 
       Private`j[2, 2, 0]}, {
       Private`j[0, 3, 1], 
       Private`j[1, 3, 0]}, {
       Private`j[0, 3, 1], 
       Private`j[1, 2, 1], 
       Private`j[2, 1, 1], 
       Private`j[3, 0, 1]}, {
       Private`j[0, 2, 2], 
       Private`j[1, 1, 2], 
       Private`j[2, 0, 2]}, {
       Private`j[0, 1, 3], 
       Private`j[1, 0, 3]}}, Private`fundamentalcomponents = {
      Private`j[0, 1, 3], 
      Private`j[0, 2, 2], 
      Private`j[0, 3, 1], 
      Private`j[1, 0, 3], 
      Private`j[1, 1, 2], 
      Private`j[1, 2, 1], 
      Private`j[1, 3, 0], 
      Private`j[2, 0, 2], 
      Private`j[2, 1, 1], 
      Private`j[2, 2, 0], 
      Private`j[3, 0, 1], 
      Private`j[3, 1, 0]}, Private`positivityconstraints = 
    And[Private`j[1, 3, 0] >= 0, Private`j[1, 2, 1] + Private`j[1, 3, 0] >= 0,
       Private`j[1, 1, 2] + Private`j[1, 2, 1] + Private`j[1, 3, 0] >= 0, 
      Private`j[2, 2, 0] >= 0, Private`j[2, 1, 1] + Private`j[2, 2, 0] >= 0, 
      Private`j[3, 1, 0] >= 0, Private`j[0, 1, 3] >= 0, 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] >= 0, 
      Private`j[0, 1, 3] + Private`j[1, 1, 2] + Private`j[2, 1, 1] >= 0, 
      Private`j[0, 2, 2] >= 0, Private`j[0, 2, 2] + Private`j[1, 2, 1] >= 0, 
      Private`j[0, 3, 1] >= 0, Private`j[3, 0, 1] >= 0, 
      Private`j[2, 1, 1] + Private`j[3, 0, 1] >= 0, 
      Private`j[1, 2, 1] + Private`j[2, 1, 1] + Private`j[3, 0, 1] >= 0, 
      Private`j[2, 0, 2] >= 0, Private`j[1, 1, 2] + Private`j[2, 0, 2] >= 0, 
      Private`j[1, 0, 3] >= 0], Private`buildlambdainput[{
       Pattern[Private`l1$, 
        BlankSequence[]]}, {
       Pattern[Private`l2$, 
        BlankSequence[]]}] := Flatten[{
       MapThread[Rule, {
         Table[
          Private`lambda[Private`p, 0, 0], {Private`p, 1, 4 - 1}], {
         Private`l1$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, Private`p, 0], {Private`p, 1, 4 - 1}], {
         Private`l2$}}]}], Private`buildlambdainput[{
       Pattern[Private`l1$, 
        BlankSequence[]]}, {
       Pattern[Private`l2$, 
        BlankSequence[]]}, {
       Pattern[Private`l3$, 
        BlankSequence[]]}] := Flatten[{
       MapThread[Rule, {
         Table[
          Private`lambda[Private`p, 0, 0], {Private`p, 1, 4 - 1}], {
         Private`l1$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, Private`p, 0], {Private`p, 1, 4 - 1}], {
         Private`l2$}}], 
       MapThread[Rule, {
         Table[
          Private`lambda[0, 0, Private`p], {Private`p, 1, 4 - 1}], {
         Private`l3$}}]}], Attributes[Private`l1$] = {Temporary}, 
    Attributes[Private`l2$] = {Temporary}, 
    Attributes[Private`l3$] = {Temporary}, 
    Private`componentsEdgesAndnuc[{1, 0, 1}, {3, 0, 3}] = {{}, {}, {}, {}, {{
      3, 1, 1}, {2, 0, 2}, {4, 1, 2}, {3, 0, 3}, {2, 2, 2}, {3, 0, 3}, {4, 0, 
      4}, {1, 1, 3}, {2, 1, 4}}}, 
    Private`componentsEdgesAndnuc[{2, 0, 2}, {0, 0, 0}] = {{}, {}, {}, {}, {{
      2, 0, 2}}}, 
    Private`componentsEdgesAndnuc[{2, 2, 3}, {1, 0, 3}] = {{}, {}, {}, {}, {{
      5, 1, 0}, {3, 2, 0}, {5, 3, 0}, {6, 1, 1}, {4, 2, 1}, {6, 3, 1}, {7, 1, 
      2}, {5, 2, 2}, {3, 2, 0}, {1, 3, 0}, {3, 4, 0}, {4, 2, 1}, {2, 3, 1}, {
      4, 4, 1}, {5, 2, 2}, {3, 3, 2}, {1, 5, 0}, {2, 3, 1}, {0, 4, 1}, {2, 5, 
      1}, {3, 3, 2}, {1, 4, 2}, {0, 6, 1}, {1, 4, 2}, {4, 0, 1}, {2, 1, 1}, {
      4, 2, 1}, {5, 0, 2}, {3, 1, 2}, {5, 2, 2}, {6, 0, 3}, {4, 1, 3}, {6, 2, 
      3}, {2, 3, 1}, {3, 1, 2}, {1, 2, 2}, {3, 3, 2}, {4, 1, 3}, {2, 2, 3}, {
      4, 3, 3}, {1, 4, 2}, {2, 2, 3}, {0, 3, 3}, {2, 4, 3}, {0, 5, 3}, {3, 1, 
      2}, {2, 0, 3}, {4, 1, 3}, {3, 0, 4}, {5, 1, 4}, {2, 2, 3}, {1, 1, 4}, {
      3, 2, 4}, {1, 3, 4}, {3, 0, 4}, {4, 0, 5}, {2, 1, 5}}}, 
    Private`componentsEdgesAndnuc[{2, 3, 3}, {3, 1, 
      2}] = {{}, {}, {}, {}, CompressedData["
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      "]}, 
    Private`componentsEdgesAndnuc[{3, 0, 0}, {2, 3, 3}] = {{}, {}, {}, {}, {{
      4, 4, 0}, {2, 5, 0}, {3, 5, 1}, {5, 2, 1}, {3, 3, 1}, {4, 3, 2}, {1, 4, 
      1}, {2, 4, 2}, {3, 4, 3}, {6, 0, 2}, {4, 1, 2}, {5, 1, 3}, {2, 2, 2}, {
      3, 2, 3}, {4, 2, 4}, {0, 3, 2}, {1, 3, 3}, {2, 3, 4}, {3, 3, 5}}}, 
    Private`componentsEdgesAndnuc[{3, 0, 1}, {1, 2, 1}] = {{}, {}, {}, {}, {{
      2, 0, 0}, {4, 1, 0}, {2, 2, 0}, {3, 0, 1}, {2, 2, 0}, {0, 3, 0}, {1, 1, 
      1}, {3, 2, 1}, {1, 3, 1}, {2, 1, 2}, {1, 3, 1}, {0, 2, 2}, {2, 3, 2}, {
      1, 2, 3}, {0, 4, 2}, {3, 0, 1}, {1, 1, 1}, {4, 0, 2}, {2, 1, 2}, {2, 1, 
      2}, {0, 2, 2}, {3, 1, 3}, {1, 2, 3}, {1, 2, 3}, {2, 2, 4}, {0, 3, 4}, {
      1, 0, 3}, {2, 0, 4}, {0, 1, 4}, {1, 1, 5}}}, 
    Private`componentsEdgesAndnuc[{3, 1, 2}, {2, 0, 2}] = {{}, {}, {}, {}, {{
      3, 1, 0}, {1, 2, 0}, {2, 0, 1}, {0, 1, 1}, {3, 3, 0}, {4, 1, 1}, {2, 2, 
      1}, {3, 0, 2}, {1, 1, 2}, {4, 3, 1}, {5, 1, 2}, {3, 2, 2}, {4, 0, 3}, {
      2, 1, 3}, {1, 4, 0}, {2, 2, 1}, {0, 3, 1}, {1, 1, 2}, {2, 4, 1}, {3, 2, 
      2}, {1, 3, 2}, {2, 1, 3}, {0, 5, 1}, {1, 3, 2}, {0, 2, 3}, {2, 2, 1}, {
      3, 0, 2}, {1, 1, 2}, {0, 0, 3}, {3, 2, 2}, {4, 0, 3}, {2, 1, 3}, {1, 0, 
      4}, {4, 2, 3}, {5, 0, 4}, {3, 1, 4}, {1, 3, 2}, {2, 1, 3}, {0, 2, 3}, {
      2, 3, 3}, {3, 1, 4}, {1, 2, 4}, {0, 4, 3}, {1, 2, 4}, {2, 1, 3}, {1, 0, 
      4}, {3, 1, 4}, {2, 0, 5}, {4, 1, 5}, {1, 2, 4}, {0, 1, 5}, {2, 2, 5}, {
      0, 3, 5}, {2, 0, 5}, {3, 0, 6}, {1, 1, 6}}}, 
    Private`componentsEdgesAndnuc[{3, 1, 2}, {2, 1, 
      2}] = {{}, {}, {}, {}, CompressedData["
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/B2itLDQoLgQmkI1DtlaVBz4WiyP3mi1i8VWRGl8WWi6OZIbmqW3C3pxVUOX
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eICPX9unBfw=
      "]}, Private`componentsEdgesAndnuc[{
       Pattern[Private`lam$, 
        BlankSequence[]]}, {
       Pattern[Private`mu$, 
        BlankSequence[]]}] := (
     Private`componentsEdgesAndnuc[{Private`lam$}, {Private`mu$}] = 
     Module[{Private`redsol$, Private`locrules$, Private`nuclist$, 
        Private`fundcompval$, Private`edgesEWValues$, 
        Private`edgesNESWValues$, Private`edgesNWSEValues$}, 
       Private`redsol$ = Solve[
          And[
           ReplaceAll[Private`lambdaequations, 
            Private`buildlambdainput[{Private`lam$}, {Private`mu$}]], 
           Private`positivityconstraints], Integers]; 
       Private`nuclist$ = ReplaceAll[
          Table[
           Private`lambda[0, 0, Private`p], {Private`p, 1, 4 - 1}], 
          Private`redsol$]; {{}, {}, {}, {}, Private`nuclist$}]), 
    Attributes[Private`lam$] = {Temporary}, 
    Attributes[Private`mu$] = {Temporary}, 
    Attributes[Private`redsol$] = {Temporary}, 
    Attributes[Private`locrules$] = {Temporary}, 
    Attributes[Private`nuclist$] = {Temporary}, 
    Attributes[Private`fundcompval$] = {Temporary}, 
    Attributes[Private`edgesEWValues$] = {Temporary}, 
    Attributes[Private`edgesNESWValues$] = {Temporary}, 
    Attributes[Private`edgesNWSEValues$] = {Temporary}, 
    Private`tensorproductCC[{
       Pattern[Private`lam, 
        BlankSequence[]]}, {
       Pattern[Private`mu, 
        BlankSequence[]]}] := Part[
      Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 5], 
    Attributes[$CellContext`makeRow$] = {Temporary}, 
    Attributes[$CellContext`rowIndex$] = {Temporary}, 
    Attributes[$CellContext`colIndex$] = {Temporary}}]], "Output",
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   3.715334251343973*^9}},ExpressionUUID->"f94497b9-c71f-44ac-b6a5-\
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}, Closed]]
}, Open  ]],

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The program decomposes the product of two irreducible representations of \
SU(n) into a direct sum.\
\>", "Text",
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Cell["DETAILS", "Subsection",
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Cell[TextData[{
 "The decomposition of the tensor product of irreducible representations is \
done by using an improved hive technique developed by the author and inspired \
from [1], [2], [3]. \nIn order to speed up the calculation we use a basis of \
elementary pictographs where syzygies (constraints) are eliminated before the \
actual reduction of the system of inequalities takes place.\nPictographs \
themselves (BZ-triangles, KT-honeycombs, O-blades), corresponding to a chosen \
space of intertwiners,  are not displayed here :  see the companion programs \
\[OpenCurlyDoubleQuote]obladesSUn\[CloseCurlyDoubleQuote] and \
\[OpenCurlyDoubleQuote]hobeycombsSUn\[CloseCurlyDoubleQuote]. \n\n[1] ",
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exterior algebra of the adjoint representation, A.D. Berenstein and A.Z. \
Zelevinsky (1992)\"\>",
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      "http://emis.ams.org/journals/JACO/Volume1_1/h7x307752t449r38.fulltext.\
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pdf", "Hyperlink"]& ]],
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     BaseStyle->{"Hyperlink"},
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*)