# SymmetrizedIndependentComponents

SymmetrizedIndependentComponents[dims,sym]

gives the list of independent components of an array of dimensions dims with the symmetry sym.

# Details

• The dimensions dims must be given as a list of positive integers.
• The symmetry sym must be compatible with the list of dimensions dims.
• SymmetrizedIndependentComponents generalizes the combinatorial construction of subsets and multisets, the former corresponding to antisymmetry and the latter to symmetry.

# Examples

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## Basic Examples(1)

Independent components of a rank-3 antisymmetric array in dimension 4:

## Scope(4)

This is an array with symmetry:

These are its independent components:

Extract the values of the independent components:

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

Independent components of a rank-4 array whose symmetry is generated by a cycle:

Same situation, but having the generator with negative phase:

Use a nontrivial symmetry with complex generators:

## Applications(2)

Independent components of a general 3-dimensional stiffness tensor in elasticity theory:

Independent components of a general 4-dimensional Riemann curvature tensor, without taking into account its cyclic symmetry:

## Properties & Relations(5)

A skew-symmetric array in dimension is zero if its depth is larger than the dimension, and hence there are no independent components:

Independent components under rank- antisymmetry in dimension dim are equivalent to the list of -subsets of Range[dim]:

Independent components under rank- symmetry in dimension dim are equivalent to the list of -multisets of Range[dim]:

Given an array with symmetry, the values of its independent components can be extracted in several forms:

The computation of independent components can be simulated using representatives of the orbits of components under Permute action:

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

#### Text

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

#### CMS

Wolfram Language. 2012. "SymmetrizedIndependentComponents." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SymmetrizedIndependentComponents.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_symmetrizedindependentcomponents, author="Wolfram Research", title="{SymmetrizedIndependentComponents}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetrizedIndependentComponents.html}", note=[Accessed: 01-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_symmetrizedindependentcomponents, organization={Wolfram Research}, title={SymmetrizedIndependentComponents}, year={2012}, url={https://reference.wolfram.com/language/ref/SymmetrizedIndependentComponents.html}, note=[Accessed: 01-June-2023 ]}