I am new in Julia. I want to do a program where I can use group Actions.
For example: Take a vector $(a,b,c,d)$ in $\mathbb{R}^{4}$, and consider the action of elements of $S_{4}$ on this, such as the cycle $(1,2,3,4)$. I would like to have a program which computes:
$$(1,2,3,4)(a,b,c,d) = (d,a,b,c)$$
It would be great if this would be possible for any permutation. Do you have any ideas which packages must I download? and How must I write it?
thank you for your help.
You might like the package Permutations, https://github.com/scheinerman/Permutations.jl.
julia> using Permutations
julia> p = Permutation([2,3,4,1]) # an element of S_4
(1,2,3,4) # printed in cycle notation
julia> p(1) # apply it to one element
2
julia> two_row(p) # alternative way to print this
2×4 Matrix{Int64}:
1 2 3 4
2 3 4 1
julia> inv(p)
(1,4,3,2)
julia> input = [:a, :b, :c, :d]; # an array of any 4 things
julia> [input[p(i)] for i in eachindex(input)] # permute those?
4-element Vector{Symbol}:
:b
:c
:d
:a
julia> ans == input[p.(eachindex(input))] # another way to write the same
true
julia> input[inv(p).(eachindex(input))] # with inv(p)
4-element Vector{Symbol}:
:d
:a
:b
:c
julia> Permutation([2,3,1,4,6,5])
(1,2,3)(4)(5,6) # it really prints the cycles