Are two permutations conjugate?

Returns TRUE if two permutations are conjugate and FALSE otherwise.

Usage

are_conjugate(x, y)
are_conjugate_single(a,b)

Arguments

x,y,a,b: Objects of class permutation, coerced to cycle form

Details

Two permutations are conjugate if and only if they have the same shape. Function are_conjugate() is vectorized and user-friendly; function are_conjugate_single() is lower-level and operates only on length-one permutations.

The reason that are_conjugate_single() is a separate function and not bundled inside are_conjugate() is that dealing with the identity permutation is a pain in the arse.

Value

Returns a vector of Booleans

Author

Robin K. S. Hankin

Note

The functionality detects conjugateness by comparing the shapes of two permutations; permutations are coerced to cycle form because function shape() does.

The group action of conjugation, that is \(x^y\) or y^-1 x y, is documented at conjugation.

 all(are_conjugate(x,x^y))

is always TRUE.

Examples


as.cycle("(123)(45)") %~% as.cycle("(89)(712)")  # same shape
#> [1] TRUE
as.cycle("(123)(45)") %~% as.cycle("(89)(7124)") # different shape
#> [1] FALSE

are_conjugate(rperm(20,3),rperm(20,3))
#>  [1] FALSE  TRUE FALSE  TRUE FALSE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE
#> [13] FALSE FALSE  TRUE FALSE FALSE  TRUE  TRUE FALSE

rperm(20,3) %~% as.cycle(1:3)
#>  [1] FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
#> [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE

z <- rperm(300,4)
stopifnot(all(are_conjugate(z,id)==is.id(z)))

z <- rperm(20)
stopifnot(all(z %~% capply(z,sample)))

data(megaminx)
stopifnot(all(are_conjugate(megaminx,megaminx^as.cycle(sample(129)))))