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