SymmetrizedDependentComponents

SymmetrizedDependentComponents[comp,sym]

gives the list of components that are equivalent to the component comp by the symmetry sym.

Details

  • The component comp must be given as a list of positive integers.

Examples

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Basic Examples  (1)

Components of a depth-3 array related by symmetry to component {1,3,5}:

Components vanishing by symmetry are also related to other components:

Scope  (2)

This is an array with symmetry:

These are the dependent components associated to component {1,1,2}:

The respective values coincide by symmetry:

In an array with no symmetry, all components are independent:

Properties & Relations  (4)

Using Symmetric, SymmetrizedDependentComponents is essentially equivalent to Permutations:

SymmetrizedDependentComponents allows permuting only some elements:

SymmetrizedDependentComponents is an orbit computation under Permute action with the group associated to the symmetry permutations:

Take a symmetry for a depth-4 array:

There are 55 independent components in dimension 5:

Compute the respective dependent components and flatten the result:

The remaining components are all zero by symmetry:

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:

Complex phases are also possible:

Neat Examples  (1)

Plot random orbits of components of an array with symmetric blocks:

Take an array of depth 6 having 2 symmetric blocks of 3 levels:

Random orbits of the array, after flattening it to a matrix:

Wolfram Research (2012), SymmetrizedDependentComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html.

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

BibTeX

@misc{reference.wolfram_2023_symmetrizeddependentcomponents, author="Wolfram Research", title="{SymmetrizedDependentComponents}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}", note=[Accessed: 04-October-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_symmetrizeddependentcomponents, organization={Wolfram Research}, title={SymmetrizedDependentComponents}, year={2012}, url={https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}, note=[Accessed: 04-October-2023 ]}