PermutationReplace

PermutationReplace[expr,perm]

replaces each part in expr by its image under the permutation perm.

PermutationReplace[expr,gr]

returns the list of images of expr under all elements of the permutation group gr.

Details

  • For an integer in expr present in the cycles of the permutation perm, the image is the integer to the right of , or the first integer of the cycle if is the last one. For an integer not present in the cycles of perm, the image is itself.
  • If g is a permutation object in expr, then the action is understood as right conjugation: PermutationProduct[InversePermutation[perm],g,perm]. This is equivalent to replacing the points in the cycles of g by their images under perm.
  • When applied to a permutation group expr, PermutationReplace operates on each individual generator, returning the same abstract group but acting on different points.
  • Both arguments are independently listable. If both arguments are lists then the second argument is threaded first.

Examples

open allclose all

Basic Examples  (2)

The image of integer 4 under Cycles[{{2,3,4,6}}] is integer 6:

Under the identity, permutation integers are not moved:

An action of a permutation on another permutation is understood as conjugation:

Images under all elements of a group:

Scope  (6)

The image of a point in the support of the permutation is the right neighbor of the point:

The image of the last point of a cycle is the first point of that cycle:

A point not present in the permutation support stays invariant:

PermutationReplace on arrays of integers returns the list of respective images:

PermutationReplace on other permutations is understood as conjugation:

On a permutation group, the generators are conjugated:

The second argument is listable:

If both arguments are lists then the second argument is threaded first:

Images under all elements of a group:

Properties & Relations  (4)

PermutationReplace is a right action with respect to PermutationProduct:

PermutationReplace on an identity permutation list coincides with PermutationList:

PermutationReplace on an identity permutation list produces the inverse result of Permute:

The orbit of a point under a permutation group is the union of the images of that point under the elements of the group:

Neat Examples  (1)

Graphical representation of the elements of groups by sorted lists of images:

Wolfram Research (2010), PermutationReplace, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationReplace.html.

Text

Wolfram Research (2010), PermutationReplace, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationReplace.html.

BibTeX

@misc{reference.wolfram_2021_permutationreplace, author="Wolfram Research", title="{PermutationReplace}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationReplace.html}", note=[Accessed: 23-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_permutationreplace, organization={Wolfram Research}, title={PermutationReplace}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationReplace.html}, note=[Accessed: 23-September-2021 ]}

CMS

Wolfram Language. 2010. "PermutationReplace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationReplace.html.

APA

Wolfram Language. (2010). PermutationReplace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationReplace.html