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obladesSUn.cdf
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obladesSUn.cdf
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(* Content-type: application/vnd.wolfram.cdf.text *)
(*** Wolfram CDF File ***)
(* http://www.wolfram.com/cdf *)
(* CreatedBy='Mathematica 12.0' *)
(***************************************************************************)
(* *)
(* *)
(* Under the Wolfram FreeCDF terms of use, this file and its content are *)
(* bound by the Creative Commons BY-SA Attribution-ShareAlike license. *)
(* *)
(* For additional information concerning CDF licensing, see: *)
(* *)
(* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *)
(* *)
(* *)
(***************************************************************************)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 1088, 20]
NotebookDataLength[ 354622, 6319]
NotebookOptionsPosition[ 355244, 6319]
NotebookOutlinePosition[ 355583, 6334]
CellTagsIndexPosition[ 355540, 6331]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
DynamicModuleBox[{$CellContext`rows$$ = 3, $CellContext`cols$$ =
2, $CellContext`mat$$ = {{7, 6}, {7, 7}, {6, 5}}, $CellContext`coupling$$ =
1, $CellContext`mul$$ = Dynamic[
Apply[obladePackage`multiplicity, $CellContext`mat$$]], \
$CellContext`myRange$$, $CellContext`myPart$$, $CellContext`tp$$ =
0, $CellContext`tpval$$},
DynamicBox[ToBoxes[
Manipulate[
Which[$CellContext`tp$$ == 0,
Row[{
Invisible[$CellContext`doit],
(CheckAbort[
If[Length[#] > 800,
Abort[], #],
Print[$CellContext`limittext, 800]]& )[
TimeConstrained[
If[Setting[$CellContext`mul$$] == 0, $CellContext`notadmissibletext,
TableForm[{
PopupMenu[
Dynamic[$CellContext`coupling$$],
Range[1,
Setting[$CellContext`mul$$]]],
Show[
obladePackage`plotOblades[
$CellContext`myPart$$[$CellContext`mat$$, 1],
$CellContext`myPart$$[$CellContext`mat$$, 2],
$CellContext`myPart$$[$CellContext`mat$$,
3], $CellContext`coupling$$], ImageSize -> Full]}]], 1.5,
Print[1.5, 1.5, " seconds"]]]}], $CellContext`tp$$ ==
1, $CellContext`tpval$$ = obladePackage`tensorproduct[
$CellContext`myPart$$[$CellContext`mat$$, 1],
$CellContext`myPart$$[$CellContext`mat$$, 2]], $CellContext`tp$$ ==
2, $CellContext`tp$$ =
0; $CellContext`mat$$ = {{3, 2, 3, 2, 2}, {2, 3, 2, 2, 3}, {4, 5, 4, 2,
5}}; TableForm[{
PopupMenu[
Dynamic[$CellContext`coupling$$],
Range[1,
Setting[$CellContext`mul$$]]],
Show[
obladePackage`plotOblades[
$CellContext`myPart$$[$CellContext`mat$$, 1],
$CellContext`myPart$$[$CellContext`mat$$, 2],
$CellContext`myPart$$[$CellContext`mat$$,
3], $CellContext`coupling$$], ImageSize -> Full]}]],
Evaluate[
With[{$CellContext`makeRow$ = Function[{$CellContext`rowIndex$},
Map[
Function[{$CellContext`colIndex$},
InputField[
Dynamic[
Part[$CellContext`mat$$, $CellContext`rowIndex$, \
$CellContext`colIndex$]], Number, FieldSize -> 3]],
$CellContext`myRange$$[$CellContext`cols$$]], HoldAll]},
Row[{
TextGrid[{{"\[Omega]1", ""}, {}, {"\[Omega]2", ""}, {}, {
OverBar["\[Omega]3"], ""}}],
Grid[
Map[$CellContext`makeRow$,
$CellContext`myRange$$[$CellContext`rows$$]],
Dividers -> {{False, False, False}, {False, False, Red}}],
Invisible["xxxxxx"],
Button[
Style[
"Update components", FontSize -> 14, Bold,
Blue], $CellContext`mat$$; $CellContext`coupling$$ = 1;
Increment[$CellContext`cols$$];
Decrement[$CellContext`cols$$]; $CellContext`tp$$ = 0],
Column[{
Button[
Style["Previous pictograph", Bold],
If[$CellContext`coupling$$ > 1,
Decrement[$CellContext`coupling$$]; Null]],
Button[
Style["Next pictograph", Bold],
If[$CellContext`coupling$$ < Setting[$CellContext`mul$$],
Increment[$CellContext`coupling$$]; Null]]}],
Column[$CellContext`abouttabletext]}]]],
Column[{
Panel[
Style[$CellContext`cols$$ + 1, FontSize -> 14, Bold],
Style["SU(n), with n =", Bold, FontSize -> 14], Left,
BaselinePosition -> Bottom],
Panel[
Style[
Column[{
"Pictographs for admissible triples",
"of irreducible representations", "of SU(n)", "O-blade version"},
Center], "Title"]]},
Invisible["xxxxxx"]],
Panel[$CellContext`mul$$,
Style["Multiplicity is", FontSize -> 14, Bold], Left],
Panel[$CellContext`coupling$$,
Style["The count index of this pictograph is", Bold], Left],
Column[$CellContext`aboutobladetext],
Button[
Style["SU(n): decrease n", FontSize -> 14, Bold],
If[$CellContext`rows$$ >
1, $CellContext`mat$$ =
Map[Drop[#, -1]& , $CellContext`mat$$]; $CellContext`coupling$$ = 1;
Decrement[$CellContext`cols$$]; Null]],
Button[
Style[
"SU(n): increase n", FontSize -> 14, Bold], $CellContext`mat$$ =
Transpose[
Append[
Transpose[$CellContext`mat$$],
Table[0, {3}]]]; $CellContext`coupling$$ = 1;
Increment[$CellContext`cols$$]; Null],
Button[
Style[
"Tensor product decomposition of \[Omega]1 \[CircleTimes] \[Omega]2 ",
FontSize -> 14, Bold, Blue], $CellContext`coupling$$ =
1; $CellContext`tp$$ = 1; Increment[$CellContext`cols$$];
Decrement[$CellContext`cols$$]; Null],
Button[
Style[
"An SU(6) example with multiplicity 592", FontSize -> 10, Bold, Brown,
TextAlignment -> Left], $CellContext`coupling$$ =
1; $CellContext`cols$$ = 6 - 1; $CellContext`tp$$ = 2;
Increment[$CellContext`cols$$]; Decrement[$CellContext`cols$$]; Null],
Button[
Style[
"Reset random SU(3) example", FontSize -> 10, Bold, Brown,
TextAlignment -> Left], $CellContext`cols$$ = 2; $CellContext`mat$$ =
With[{$CellContext`mat12 = Table[
RandomInteger[7], {2}, {3 - 1}]},
Append[$CellContext`mat12,
First[
RandomChoice[
Apply[
obladePackage`tensorproduct, $CellContext`mat12]]]]]; \
$CellContext`coupling$$ = 1; Increment[$CellContext`cols$$];
Decrement[$CellContext`cols$$]; $CellContext`tp$$ =
0], {{$CellContext`doit, 0, ""}, {0}, ControlType -> None},
TrackedSymbols :> {$CellContext`doit}], StandardForm],
ImageSizeCache->{1390., {650., 655.}}],
SynchronousUpdating -> False,
DynamicModuleValues:>{{DownValues[$CellContext`myRange$$] = {HoldPattern[
$CellContext`myRange$$[
Pattern[$CellContext`s,
Blank[]]]] :> Range[$CellContext`s]}}, {
DownValues[$CellContext`myPart$$] = {HoldPattern[
$CellContext`myPart$$[
Pattern[$CellContext`expr,
Blank[]],
Pattern[$CellContext`s,
BlankSequence[]]]] :> Part[$CellContext`expr, $CellContext`s]}}},
Initialization:>{obladePackage`multiplicity[
Pattern[Private`lambda,
Blank[]],
Pattern[Private`mu,
Blank[]],
Pattern[Private`nu,
Blank[]],
OptionsPattern[]] := Which[OptionValue[obladePackage`Kostant] == True,
obladePackage`multiplicityKostant[
Private`lambda, Private`mu, Private`nu],
OptionValue[obladePackage`Magic] == True,
obladePackage`multiplicityMagic[Private`lambda, Private`mu, Private`nu],
OptionValue[Volume] == True,
Private`multiplicityVolume[Private`lambda, Private`mu, Private`nu],
OptionValue[obladePackage`GenFun] == True,
obladePackage`multiplicityGenFun[
Private`lambda, Private`mu, Private`nu], OptionValue[obladePackage`LiE] ==
True,
obladePackage`multiplicityLiE[Private`lambda, Private`mu, Private`nu],
True,
obladePackage`multiplicityHive[Private`lambda, Private`mu, Private`nu]],
Options[obladePackage`multiplicity] = {
obladePackage`Kostant -> False, obladePackage`Magic -> False, Volume ->
False, obladePackage`GenFun -> False, obladePackage`LiE -> False},
TagSet[obladePackage`multiplicity,
MessageName[obladePackage`multiplicity, "usage"],
"multiplicity[lambda,mu,nu] gives the multiplicity of the term nu in the \
decomposition into irreducible components of a tensor product lambda \
\[CircleTimes] mu of two irreps of the Lie group SU(n), given by the \
components of their highest weights along the basis of fundamental weights.\n\
This integer is also called Littlewood-Richardson coefficient. Equivalently \
it is the multiplicity of the trivial representation in the triple tensor \
product: lambda \[CircleTimes] mu \[CircleTimes] Conjugate[nu].\nThis \
function admits optional arguments.\nBy default it calls the function \
multiplicityHive, that uses the hive (oblade) method, and works for all \
SU(n).\nWith the option Kostant->True multiplicity calls the function \
multiplicityKostant that only works for those Lie groups for which the \
Kostant partition function is known.\nWith the option Magic->True \
multiplicity calls the function multiplicityMagic that uses the semi-magic \
algorithm and only works for SU(3). Very fast.\nWith the option Volume->True \
multiplicity calls the function volumeJ that uses the volume algorithm and \
only works for SU(3).Very fast.\nWith the option GenFun->True multiplicity \
calls the function multiplicityGenFun that uses a generating function that \
only works for SU(3). Slow.\nWith the option LiE->True multiplicity calls the \
function multiplicityLiE that uses the external program LiE.\n"],
TagSet[obladePackage`Kostant,
MessageName[obladePackage`Kostant, "usage"],
"An option for multiplicity"], obladePackage`multiplicityKostant[
Pattern[Private`lambda,
Blank[]],
Pattern[Private`mu,
Blank[]],
Pattern[Private`nu,
Blank[]]] :=
Which[obladePackage`currentgroup == 3, obladePackage`Kostant[{
Pattern[Private`x,
Blank[]],
Pattern[Private`y,
Blank[]]}] := If[
Or[Private`x < 0, Private`y < 0], 0, 1 + Min[Private`x, Private`y]];
Private`dynkintokac[{
Pattern[Private`a,
Blank[]],
Pattern[Private`b,
Blank[]]}] := Private`a {2/3, 1/3} + Private`b {1/3, 2/3};
Private`weyldot[
Pattern[Private`s,
Blank[]]][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] :=
Private`weyl[Private`s][{Private`la1, Private`la2} + {1, 1}] - {1, 1};
Private`weyl[1][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {Private`la1, Private`la2}; Private`weyl[2][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {-Private`la1, Private`la1 + Private`la2};
Private`weyl[3][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {Private`la1 + Private`la2, -Private`la2};
Private`weyl[4][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {Private`la2, -(Private`la1 + Private`la2)};
Private`weyl[5][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {-(Private`la1 + Private`la2), Private`la1};
Private`weyl[6][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]]}] := {-Private`la2, -Private`la1}; Private`sign[
Private`weyl[
Pattern[Private`s,
Blank[]]]] := Part[{1, -1, -1, 1, 1, -1}, Private`s]; If[
Apply[Nand,
Map[IntegerQ,
Private`dynkintokac[Private`lambda + Private`mu - Private`nu]]],
Print["Not an admissible intertwiner"]; 0,
Sum[Private`sign[
Private`weyl[Private`v]] Private`sign[
Private`weyl[Private`w]] obladePackage`Kostant[
Private`dynkintokac[
Private`weyldot[Private`v][Private`lambda] +
Private`weyldot[Private`w][Private`mu] - Private`nu]], {
Private`v, 1,
Factorial[3]}, {Private`w, 1,
Factorial[3]}]], obladePackage`currentgroup == 4,
obladePackage`Kostant[{
Pattern[Private`k1,
Blank[]],
Pattern[Private`k2,
Blank[]],
Pattern[Private`k3,
Blank[]]}] := Piecewise[{{0,
Or[
Private`k1 < 0, Private`k2 < 0, Private`k3 < 0]}, {(Private`k2 +
1) (Private`k2 + 2) ((Private`k2 + 3)/6),
And[
Private`k2 <= Private`k1, Private`k2 <= Private`k3]}, {(Private`k1 +
1) (Private`k1 + 2) ((3 Private`k2 - 2 Private`k1 + 3)/6),
Private`k1 <= Private`k2 <=
Private`k3}, {(Private`k1 + 1) (Private`k1 +
2) ((3 Private`k3 - Private`k1 + 3)/6), Private`k1 <= Private`k3 <=
Private`k1 + Private`k3 <=
Private`k2}, {(Private`k1 + 1) (Private`k1 +
2) ((3 Private`k3 - Private`k1 + 3)/6) - Binomial[
Private`k3 + Private`k1 - Private`k2 + 2, 3], Private`k1 <=
Private`k3 <= Private`k2 <=
Private`k1 + Private`k3}, {(Private`k3 + 1) (Private`k3 +
2) ((3 Private`k2 - 2 Private`k3 + 3)/6), Private`k3 <=
Private`k2 <=
Private`k1}, {(Private`k3 + 1) (Private`k3 +
2) ((3 Private`k1 - Private`k3 + 3)/6), Private`k3 <= Private`k1 <=
Private`k3 + Private`k1 <=
Private`k2}, {(Private`k3 + 1) (Private`k3 +
2) ((3 Private`k1 - Private`k3 + 3)/6) - Binomial[
Private`k1 + Private`k3 - Private`k2 + 2, 3], Private`k3 <=
Private`k1 <= Private`k2 <= Private`k3 + Private`k1}}];
Private`dynkintokac[{
Pattern[Private`a,
Blank[]],
Pattern[Private`b,
Blank[]],
Pattern[Private`c,
Blank[]]}] :=
Private`a {3/4, 1/2, 1/4} + Private`b {1/2, 1, 1/2} +
Private`c {1/4, 1/2, 3/4}; Private`weyldot[
Pattern[Private`s,
Blank[]]][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] :=
Private`weyl[
Private`s][{Private`la1, Private`la2, Private`la3} + {1, 1, 1}] - {1,
1, 1}; Private`sign[
Private`weyl[
Pattern[Private`s,
Blank[]]]] :=
Part[{1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1,
1, -1, -1, -1, 1}, Private`s]; Private`weyl[1][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {Private`la1, Private`la2, Private`la3};
Private`weyl[2][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la1, Private`la1 + Private`la2,
Private`la3}; Private`weyl[3][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la1 + Private`la2, -Private`la2, Private`la2 + Private`la3};
Private`weyl[4][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la1, Private`la2 + Private`la3, -Private`la3};
Private`weyl[5][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la2, -(Private`la1 + Private`la2), Private`la1 + Private`la2 +
Private`la3}; Private`weyl[6][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la1, Private`la1 + Private`la2 +
Private`la3, -Private`la3}; Private`weyl[7][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la1 - Private`la2, Private`la1,
Private`la2 + Private`la3}; Private`weyl[8][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la1 + Private`la2, Private`la3, -Private`la2 - Private`la3};
Private`weyl[9][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la1 + Private`la2 + Private`la3, -Private`la2 - Private`la3,
Private`la2}; Private`weyl[10][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la2, -Private`la1, Private`la1 +
Private`la2 + Private`la3}; Private`weyl[11][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la2, Private`la3, -(Private`la1 + Private`la2 + Private`la3)};
Private`weyl[12][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la2 + Private`la3, -(Private`la1 + Private`la2 + Private`la3),
Private`la1 + Private`la2}; Private`weyl[13][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la1 - Private`la2, Private`la1 +
Private`la2 + Private`la3, -(Private`la2 + Private`la3)};
Private`weyl[14][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la1 + Private`la2 + Private`la3, -Private`la3, -Private`la2};
Private`weyl[15][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-(Private`la1 + Private`la2 + Private`la3),
Private`la1, Private`la2}; Private`weyl[16][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la2, Private`la2 +
Private`la3, -(Private`la1 + Private`la2 + Private`la3)};
Private`weyl[17][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la2 + Private`la3, -Private`la3, -Private`la1 - Private`la2};
Private`weyl[18][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la2 - Private`la3, -Private`la1,
Private`la1 + Private`la2}; Private`weyl[19][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la3, -(Private`la1 + Private`la2 + Private`la3), Private`la1};
Private`weyl[20][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-(Private`la1 + Private`la2 + Private`la3),
Private`la1 + Private`la2, -Private`la2}; Private`weyl[21][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {
Private`la3, -Private`la2 - Private`la3, -Private`la1};
Private`weyl[22][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la2 - Private`la3,
Private`la2, -Private`la1 - Private`la2}; Private`weyl[23][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la3, -Private`la1 - Private`la2,
Private`la1}; Private`weyl[24][{
Pattern[Private`la1,
Blank[]],
Pattern[Private`la2,
Blank[]],
Pattern[Private`la3,
Blank[]]}] := {-Private`la3, -Private`la2, -Private`la1}; If[
Apply[Nand,
Map[IntegerQ,
Private`dynkintokac[Private`lambda + Private`mu - Private`nu]]],
Print["Not an admissible intertwiner"]; 0,
Sum[Private`sign[
Private`weyl[Private`v]] Private`sign[
Private`weyl[Private`w]] obladePackage`Kostant[
Private`dynkintokac[
Private`weyldot[Private`v][Private`lambda] +
Private`weyldot[Private`w][Private`mu] - Private`nu]], {
Private`v, 1,
Factorial[4]}, {Private`w, 1,
Factorial[4]}]], obladePackage`currentgroup > 4,
Print["Kostant partition function not implemented, I continue using the \
hive method"];
obladePackage`multiplicityHive[Private`lambda, Private`mu, Private`nu]],
TagSet[obladePackage`multiplicityKostant,
MessageName[obladePackage`multiplicityKostant, "usage"],
"multiplicityKostant[lambda,mu,nu] gives multiplicity[lambda,mu,nu] by \
using the Kostant method.\nIt is equivalent to multiplicity[lambda,mu,nu, \
Kostant\[Rule]True].\nIt only works for those Lie groups for which the \
Kostant partition function is known. See the documentation of the command \
multiplicity.\nPrivate: it is called lr in other notebooks."],
obladePackage`currentgroup = 3,
TagSet[obladePackage`currentgroup,
MessageName[obladePackage`currentgroup, "usage"],
"currentgroup returns the argument n of the group SU(n) currently in \
use."], obladePackage`multiplicityHive[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}, {
Pattern[Private`nu,
BlankSequence[]]}] := Switch[
ValueQ[obladePackage`currentgroup], False,
obladePackage`setGroupSU[Length[{Private`lam}] + 1];
obladePackage`multiplicityHive[{Private`lam}, {Private`mu}, {
Private`nu}], True,
If[Length[{Private`lam}] == obladePackage`currentgroup - 1,
Length[
Private`tensorproductCC[{Private`lam}, {Private`mu},
Reverse[{Private`nu}]]],
obladePackage`setGroupSU[Length[{Private`lam}] + 1];
obladePackage`multiplicityHive[{Private`lam}, {Private`mu}, {
Private`nu}]]],
TagSet[obladePackage`multiplicityHive,
MessageName[obladePackage`multiplicityHive, "usage"],
"multiplicityHive[lambda,mu,nu] gives multiplicity[lambda,mu,nu] by \
using the hive (oblade) method, which is the default. See the documentation \
of the command multiplicity."], obladePackage`setGroupSU[
Pattern[Private`var,
Blank[]]] :=
With[{Private`NN = Private`var}, obladePackage`currentgroup = Private`NN;
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 = DeleteDuplicates[
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}}]}];
obladePackage`fundamentalcomponentsLabels =
ToString[Private`fundamentalcomponents]; 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 = Reduce[
And[
ReplaceAll[Private`lambdaequations,
Private`buildlambdainput[{Private`lam}, {Private`mu}]],
Private`positivityconstraints], Private`fundamentalcomponents,
Integers]; Private`locrules = ToRules[Private`redsol];
Private`fundcompval =
ReplaceRepeated[Private`fundamentalcomponents, {Private`locrules}];
Private`nuclist = ReplaceRepeated[
Table[
Private`lambda[0, 0, Private`p], {
Private`p, 1, Private`NN - 1}], {Private`locrules}];
Private`edgesEWValues = Table[
ReplaceAll[
Private`edgesEW[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; Private`edgesNESWValues = Table[
ReplaceAll[
Private`edgesNESW[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; Private`edgesNWSEValues = Table[
ReplaceAll[
Private`edgesNWSE[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; {
Private`fundcompval, Private`edgesEWValues,
Private`edgesNESWValues, Private`edgesNWSEValues,
Private`nuclist}]); Private`componentsEdgesAndnuc[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}, {
Pattern[Private`nuc,
BlankSequence[]]}] := (
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}, {
Private`nuc}] =
Module[{Private`redsol, Private`locrules, Private`nuclist,
Private`fundcompval, Private`edgesEWValues,
Private`edgesNESWValues, Private`edgesNWSEValues},
Private`redsol = Reduce[
And[
ReplaceAll[Private`lambdaequations,
Private`buildlambdainput[{Private`lam}, {Private`mu}, {
Private`nuc}]], Private`positivityconstraints],
Private`fundamentalcomponents, Integers];
Private`redsol = And[Private`redsol,
Apply[And,
Table[
Private`lambda[0, 0, Private`p] ==
Part[{Private`nuc}, Private`p], {
Private`p, 1, Private`NN - 1}]]];
If[Private`redsol =!= False, Private`locrules =
ToRules[Private`redsol],
Return[{0, 0, 0, 0, 0}]];
Private`fundcompval =
ReplaceRepeated[Private`fundamentalcomponents, {Private`locrules}];
Private`nuclist = ReplaceRepeated[
Table[
Private`lambda[0, 0, Private`p], {
Private`p, 1, Private`NN - 1}], {Private`locrules}];
Private`edgesEWValues = Table[
ReplaceAll[
Private`edgesEW[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; Private`edgesNESWValues = Table[
ReplaceAll[
Private`edgesNESW[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; Private`edgesNWSEValues = Table[
ReplaceAll[
Private`edgesNWSE[Private`NN],
MapThread[Rule, {Private`fundamentalcomponents,
Part[Private`fundcompval, Private`s]}]], {Private`s, 1,
Length[Private`fundcompval]}]; {
Private`fundcompval, Private`edgesEWValues,
Private`edgesNESWValues, Private`edgesNWSEValues,
Private`nuclist}]);
Private`AdjMatbis = adjmatPackage`adjmatfct[Private`NN];
Private`plotSUNoblade[
Pattern[Private`mytripleofweights,
Blank[]],
Pattern[Private`loccoupling,
Blank[]]] :=
Module[{Private`adj, Private`vert = Private`mytripleofweights,
Private`grid, Private`gr2, Private`grla, Private`grmu, Private`grnu,
Private`grEW, Private`grNESW, Private`grNWSE},
Private`adj = Private`AdjMatbis;
Private`fullinfo = Private`componentsEdgesAndnuc[
Part[Private`mytripleofweights, 1],
Part[Private`mytripleofweights, 2],
Reverse[
Part[Private`mytripleofweights, 3]]];
Private`grid =
AdjacencyGraph[
Private`adj, VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grla =
AdjacencyGraph[Private`adj, VertexShapeFunction -> (Apply[Which,
Flatten[
Table[{#2 == Private`lambdalabel[Private`NN, Private`s], {
Text[
Style[
Part[Private`vert, 1, Private`s], 25, Blue], #,
Offset[{20, -30}]],
Disk[#, 0.03], {Black}}}, {Private`s, 1, Private`NN - 1}],
1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grmu =
AdjacencyGraph[Private`adj, VertexShapeFunction -> (Apply[Which,
Flatten[
Table[{#2 == Private`mulabel[Private`NN, Private`s], {
Text[
Style[
Part[Private`vert, 2, Private`s], 25, Blue], #,
Offset[{-30, -30}]],
Disk[#, 0.03], {Black}}}, {Private`s, 1, Private`NN - 1}],
1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grnu =
AdjacencyGraph[Private`adj, VertexShapeFunction -> (Apply[Which,
Flatten[
Table[{#2 == Private`nulabel[Private`NN, Private`s], {
Text[
Style[
Part[Private`vert, 3, Private`NN - Private`s], 25,
Blue], #,
Offset[{-5,
Plus[25]}]],
Disk[#, 0.03], {Black}}}, {Private`s, 1, Private`NN - 1}],
1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grEW =
AdjacencyGraph[Private`adj, EdgeShapeFunction -> (Apply[Which,
Flatten[
Table[
Flatten[
Table[{
And[
First[#2] == First[Private`u], Last[#2] ==
Last[Private`u]], {
Text[
Style[
Part[
Reverse[
Part[
Part[Private`fullinfo, 2, Private`loccoupling],
Private`f]],
First[
Flatten[
Position[
Private`edgesEWlabels[Private`f], Private`u]]]], 20], (
Part[#, 1] + Part[#, 2])/2]}}, {Private`u,
Private`edgesEWlabels[Private`f]}], 1], {
Private`f, 1, Private`NN - 1}], 1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grNESW =
AdjacencyGraph[Private`adj, EdgeShapeFunction -> (Apply[Which,
Flatten[
Table[
Flatten[
Table[{
And[
First[#2] == First[Private`u], Last[#2] ==
Last[Private`u]], {
Text[
Style[
Part[
Part[
Part[Private`fullinfo, 3, Private`loccoupling],
Private`f],
First[
Flatten[
Position[
Private`edgesNESWlabels[Private`f], Private`u]]]], 20], (
Part[#, 1] + Part[#, 2])/2]}}, {Private`u,
Private`edgesNESWlabels[Private`f]}], 1], {
Private`f, 1, Private`NN - 1}], 1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False]; Private`grNWSE =
AdjacencyGraph[Private`adj, EdgeShapeFunction -> (Apply[Which,
Flatten[
Table[
Flatten[
Table[{
And[
First[#2] == First[Private`u], Last[#2] ==
Last[Private`u]], {
Text[
Style[
Part[
Reverse[
Part[
Part[Private`fullinfo, 4, Private`loccoupling],
Private`f]],
First[
Flatten[
Position[
Private`edgesNWSElabels[Private`f], Private`u]]]], 20], (
Part[#, 1] + Part[#, 2])/2]}}, {Private`u,
Private`edgesNWSElabels[Private`f]}], 1], {
Private`f, 1, Private`NN - 1}], 1]]& ), VertexCoordinates ->
adjmatPackage`vertexcoordtriangle[Private`NN], DirectedEdges ->
False];
Show[Private`grid, Private`grla, Private`grmu, Private`grnu,
Private`grEW, Private`grNESW, Private`grNWSE]];
Private`plotAllOblades[
Pattern[Private`myweights,
Blank[]]] := Table[
Private`plotSUNoblade[Private`myweights, Private`pos], {
Private`pos, 1,
Apply[obladePackage`multiplicity, Private`myweights]}];
Private`componentsAlongFundPict[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Part[
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 1];
obladePackage`edgesEWvalues[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Part[
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 2];
obladePackage`edgesNESWvalues[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Part[
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 3];
obladePackage`edgesNWSEvalues[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Part[
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 4];
Private`tensorproductCC[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Part[
Private`componentsEdgesAndnuc[{Private`lam}, {Private`mu}], 5];
obladePackage`tensorproduct[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}] := Sort[
Switch[
ValueQ[obladePackage`currentgroup], False,
obladePackage`setGroupSU[Length[{Private`lam}] + 1];
obladePackage`tensorproduct[{Private`lam}, {Private`mu}], True,
If[Length[{Private`lam}] == obladePackage`currentgroup - 1,
SortBy[
Tally[
Map[Reverse,
Private`tensorproductCC[{Private`lam}, {Private`mu}]]], Last],
obladePackage`setGroupSU[Length[{Private`lam}] + 1];
obladePackage`tensorproduct[{Private`lam}, {Private`mu}]]]];
obladePackage`tensorproduct[{
Pattern[Private`lam,
BlankSequence[]]}, {
Pattern[Private`mu,
BlankSequence[]]}, {
Pattern[Private`nu,
BlankSequence[]]}] := Switch[
ValueQ[obladePackage`currentgroup], False,
obladePackage`setGroupSU[Length[{Private`lam}] + 1];