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InvariantRing :: words

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

Synopsis

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.

See also

Ways to use words :

For the programmer

The object words is a method function.