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# balopat/CommutativityPlot.mathematica

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 CommutativityPlot[group_, OptionsPattern[]] := Module[{els, n, CycleLengths, ConjClasses, conjClasses, ConjClass, ConjClassId, colorList, conjugacyClassColor, regularGridColor, notCommutingColor, commutingColor, indexed, ConjClassSeparator, tickers, markers, mx}, els = GroupElements[group]; n = Length[els]; (* Initial sorting of elements provides a good approximation for \ sorting by conjugacy class, and provides a sorting for the conjugacy classes themselves *) CycleLengths[c_] := Sort[Length /@ c[[1]]]; els = SortBy[GroupElements[group], Function[g, {CycleLengths[g], g}]]; (* Calculating and sorting by conjugacy classes *) ConjClass[g_, el_] := GroupOrbits[g, {el}]; ConjClasses[g_] := DeleteDuplicates[ Table[ {ConjClass[g, GroupElements[g][[i]]]}, {i, 1, GroupOrder[g]} ] ]; conjClasses = ConjClasses[group]; ConjClassId[el_] := Position[conjClasses, {ConjClass[group, el]}][[1]][[1]]; els = SortBy[GroupElements[group], Function[g, {ConjClassId[g], g}]]; (************** Formattings ****************) colorList = ColorData[45, "ColorList"]; conjugacyClassColor = colorList[[5]]; regularGridColor = colorList[[10]]; commutingColor = colorList[[3]]; notCommutingColor = colorList[[7]]; indexed = Thread[{Range[n], els}]; (* Red grid lines for conjugacy classes *) ConjClassSeparator[p_] := Not[ p[[1]] + 1 > Length[indexed] || p[[1]] + 1 <= Length[indexed] && ConjClassId[indexed[[p[[1]]]][[2]]] == ConjClassId[indexed[[p[[1]] + 1]][[2]]]]; ticks = If[OptionValue["Ticks"], True, False]; permutationNames = OptionValue["PermutationNames"]; tickers = If[ticks, Function[p, If[ ConjClassSeparator[p], {p[[1]], conjugacyClassColor}, {p[[1]], {regularGridColor, Dashed}}]] /@ indexed, Function[p, {p[[1]], conjugacyClassColor}] /@ Select[indexed, ConjClassSeparator] ]; (* Setting up markers with nice formatting of the permutations *) markers = If[ticks, FormatCycle[c_] := If[c == Cycles[{{}}], "( )", StringReplace[ StringJoin[ToString /@ c[[1]]], {"{" -> "(", "}" -> ")", "," -> ""}]]; If[Length[permutationNames] > 0, Function[ x, {x[[1]], FormatCycle[x[[2]]] <> " " <> permutationNames[x[[2]]] } ] /@ indexed, Function[x, {x[[1]], FormatCycle[x[[2]]]} ] /@ indexed ], Automatic ];(* Function[x, If[OddQ[First[x]], x, {x[[1]],""} ]] /@ indexed;*) mx = ArrayReshape[Map[Function[ab, PermutationProduct[ab[[1]], ab[[2]]] == PermutationProduct[ab[[2]], ab[[1]]]], Tuples[els, 2]], {n, n}]; Return[ MatrixPlot[mx, Mesh -> {tickers, tickers}, ColorRules -> {True -> commutingColor, False -> notCommutingColor}, FrameTicks -> {markers, Map[Function[v, {v[[1]], Rotate[v[[2]], 90 Degree]}], markers]}]] ](* end module*) CommutatorPlot[QuaternionGroup, \ {"PermutationNames" -> quaternionPermutationNames}] CommutatorPlot[PauliGroup, {"PermutationNames" -> pauliGroupPermutationNames}] CommutatorPlot[DihedralGroup[10]] CommutatorPlot[SymmetricGroup[4]] CommutatorPlot[CyclicGroup[4]]