SymmetrizedDependentComponents
SymmetrizedDependentComponents[comp,sym]
gives the list of components that are equivalent to the component comp by the symmetry sym.
Examples
open all close allBasic Examples (1)
Components of a depth-3 array related by symmetry to component {1,3,5}:
SymmetrizedDependentComponents[{1, 3, 5}, Antisymmetric[{1, 2, 3}]]Components vanishing by symmetry are also related to other components:
SymmetrizedDependentComponents[{1, 3, 3}, Antisymmetric[{1, 2, 3}]]Scope (2)
This is an array with symmetry:
A = {{{a, b}, {b, c}}, {{b, c}, {c, d}}};
TensorSymmetry[A]These are the dependent components associated to component {1,1,2}:
comps = SymmetrizedDependentComponents[{1, 1, 2}, %]The respective values coincide by symmetry:
Extract[A, comps]In an array with no symmetry, all components are independent:
SymmetrizedDependentComponents[{1, 3}, {}]Properties & Relations (4)
Using Symmetric, SymmetrizedDependentComponents is essentially equivalent to Permutations:
SymmetrizedDependentComponents[{1, 5, 2, 3}, Symmetric[{1, 2, 3, 4}]] == Sort@Permutations[{1, 5, 2, 3}]SymmetrizedDependentComponents allows permuting only some elements:
SymmetrizedDependentComponents[{1, 5, 2, 3}, Symmetric[{1, 2, 4}]]SymmetrizedDependentComponents is an orbit computation under Permute action with the group associated to the symmetry permutations:
g1 = Cycles[{{1, 2}}];
g2 = Cycles[{{1, 3}, {2, 4}}];
sym = {{g1, 1}, {g2, 1}};
G = PermutationGroup[{g1, g2}];SymmetrizedDependentComponents[{1, 2, 3, 4}, sym]GroupOrbits[G, {{1, 2, 3, 4}}, Permute]Take a symmetry for a depth-4 array:
sym = {{{2, 1, 3, 4}, -1}, {{3, 4, 1, 2}, 1}};There are 55 independent components in dimension 5:
Length[indeps = SymmetrizedIndependentComponents[{5, 5, 5, 5}, sym]]Compute the respective dependent components and flatten the result:
Length[deps = Flatten[SymmetrizedDependentComponents[#, sym]& /@ indeps, 1]]The remaining components are all zero by symmetry:
Length[zeros = Complement[Tuples[Range[5], 4], deps]]The relationship of the values of the dependent components to each other depends on the phases of the symmetry generators. For antisymmetry, the signs of the values alternate:
sym = Antisymmetric[{1, 2, 3}]array = SymmetrizedArray[pos_ :> Subscript[a, pos], {3, 3, 3}, sym]comps = SymmetrizedDependentComponents[{1, 2, 3}, sym]Extract[array, comps]Complex phases are also possible:
sym = {{{2, 3, 1}, Exp[2Pi I / 3]}}array = SymmetrizedArray[pos_ :> Subscript[a, pos], {3, 3, 3}, sym]comps = SymmetrizedDependentComponents[{1, 2, 3}, sym]Extract[array, comps]Neat Examples (1)
Plot random orbits of components of an array with symmetric blocks:
BlockSymmetric[n_, m_] := Join[{Symmetric[Range[m]]}, {Cycles[#], 1}& /@ Transpose /@ Partition[Partition[Range[n m], m], 2, 1]]orbit[n_ ? EvenQ, m_] := With[{pos = RandomInteger[{1, m n}, m n]},
Flatten[
SparseArray[
Thread[SymmetrizedDependentComponents[pos, BlockSymmetric[m, n]] -> 1]
],
Transpose@Partition[Range[m n], 2]
]
]Take an array of depth 6 having 2 symmetric blocks of 3 levels:
BlockSymmetric[2, 3]Random orbits of the array, after flattening it to a matrix:
RandomSeed[0];
Table[MatrixPlot[orbit[2, 3], FrameTicks -> None, ImageSize -> 100], {12}]Text
Wolfram Research (2012), SymmetrizedDependentComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html.
CMS
Wolfram Language. 2012. "SymmetrizedDependentComponents." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html.
APA
Wolfram Language. (2012). SymmetrizedDependentComponents. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html