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# symmetricGroupActors -- permutation action of the symmetric group

- Usage:
    `symmetricGroupActors(R)`
- Inputs:
  - R, a polynomial ring,
- Outputs:
  - a list,

## Description

Returns a list of of matrices, each representing an element of the symmetric group permuting the variables of the polynomial ring in the input. This simplifies the setup for symmetric group actions with the action command.

The output list contains one element for each conjugacy class of the symmetric group. The conjugacy classes are determined by their cycle type and are in bijection with the partitions of $n$, where $n$ is the number of variables. Therefore the first element of the list will always be a cycle of length $n$, and the last element will be the identity.

    i1 : R=QQ[x_1..x_4]

    o1 = R

    o1 : PolynomialRing

    i2 : symmetricGroupActors(R)

    o2 = {| x_2 x_3 x_4 x_1 |, | x_2 x_3 x_1 x_4 |, | x_2 x_1 x_4 x_3 |, | x_2
         ------------------------------------------------------------------------
         x_1 x_3 x_4 |, | x_1 x_2 x_3 x_4 |}

    o2 : List

    i3 : partitions 4

    o3 = {Partition{4}, Partition{3, 1}, Partition{2, 2}, Partition{2, 1, 1},
         ------------------------------------------------------------------------
         Partition{1, 1, 1, 1}}

    o3 : List

## See also

- BettiCharacters Example 1 -- Specht ideals / subspace arrangements
- BettiCharacters Example 2 -- Symbolic powers of star configurations

## Ways to use symmetricGroupActors:

- `symmetricGroupActors(PolynomialRing)`

## For the programmer

The object symmetricGroupActors is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BettiCharacters.m2:2859:0.