# words -- associate a word in the generators of a group to each element

## Synopsis

• Usage:
words G
• Inputs:
• Outputs:
• , associating to each element in the group of the action a word of minimal length in (the indices of) the generators of the group

## Description

This function is provided by the package InvariantRing.

The following example computes, for each permutation in the symmetric group on three elements, a word of minimal length in the Coxeter generators.

 i1 : R = QQ[x_1..x_3] o1 = R o1 : PolynomialRing i2 : L = apply(2, i -> permutationMatrix(3, [i + 1, i + 2] ) ) o2 = {| 0 1 0 |, | 1 0 0 |} | 1 0 0 | | 0 0 1 | | 0 0 1 | | 0 1 0 | o2 : List i3 : S3 = finiteAction(L, R) o3 = R <- {| 0 1 0 |, | 1 0 0 |} | 1 0 0 | | 0 0 1 | | 0 0 1 | | 0 1 0 | o3 : FiniteGroupAction i4 : words S3 o4 = HashTable{| 0 0 1 | => {0, 1, 0}} | 0 1 0 | | 1 0 0 | | 0 0 1 | => {0, 1} | 1 0 0 | | 0 1 0 | | 0 1 0 | => {1, 0} | 0 0 1 | | 1 0 0 | | 0 1 0 | => {0} | 1 0 0 | | 0 0 1 | | 1 0 0 | => {1} | 0 0 1 | | 0 1 0 | | 1 0 0 | => {} | 0 1 0 | | 0 0 1 | o4 : HashTable

The computation of the words addressing each element in the group is actually performed by the method schreierGraph since the process of computing the Schreier graph of the group yields other useful information about the group.