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

Synopsis

• Usage:

  words G

• Inputs:
  • G, an instance of the type FiniteGroupAction
• Outputs:
  • a hash table, 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

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.

See also

• group -- list all elements of the group of a finite group action
• schreierGraph -- Schreier graph of a finite group