WOLFRAM

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)Summary of the most common use cases

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

Out[1]=1

Components vanishing by symmetry are also related to other components:

Out[2]=2

Scope  (2)Survey of the scope of standard use cases

This is an array with symmetry:

Out[1]=1

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

Out[2]=2

The respective values coincide by symmetry:

Out[3]=3

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

Out[1]=1

Properties & Relations  (4)Properties of the function, and connections to other functions

Using Symmetric, SymmetrizedDependentComponents is essentially equivalent to Permutations:

Out[1]=1

SymmetrizedDependentComponents allows permuting only some elements:

Out[2]=2

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

Out[2]=2
Out[3]=3

Take a symmetry for a depth-4 array:

There are 55 independent components in dimension 5:

Out[2]=2

Compute the respective dependent components and flatten the result:

Out[3]=3

The remaining components are all zero by symmetry:

Out[4]=4

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:

Out[1]=1
Out[2]=2
Out[3]=3
Out[4]=4

Complex phases are also possible:

Out[5]=5
Out[6]=6
Out[7]=7
Out[8]=8

Neat Examples  (1)Surprising or curious use cases

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

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

Out[3]=3

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

Out[4]=4
Wolfram Research (2012), SymmetrizedDependentComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html.
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.

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.

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

Wolfram Language. (2012). SymmetrizedDependentComponents. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html

BibTeX

@misc{reference.wolfram_2024_symmetrizeddependentcomponents, author="Wolfram Research", title="{SymmetrizedDependentComponents}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}", note=[Accessed: 07-January-2025 ]}

@misc{reference.wolfram_2024_symmetrizeddependentcomponents, author="Wolfram Research", title="{SymmetrizedDependentComponents}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}", note=[Accessed: 07-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_symmetrizeddependentcomponents, organization={Wolfram Research}, title={SymmetrizedDependentComponents}, year={2012}, url={https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}, note=[Accessed: 07-January-2025 ]}

@online{reference.wolfram_2024_symmetrizeddependentcomponents, organization={Wolfram Research}, title={SymmetrizedDependentComponents}, year={2012}, url={https://reference.wolfram.com/language/ref/SymmetrizedDependentComponents.html}, note=[Accessed: 07-January-2025 ]}