For a finite group action, we form a HashTable whose keys are the generators provided by the user. The value corresponding to a generator g is a HashTable containing all pairs a => b such that a*g == b. This represents the Schreier graph of the group relative to the generating set provided by the user.
The following example defines the permutation action of a symmetric group on three elements using only two generators, a transposition and a 3-cycle.
i1 : R = QQ[x_1..x_3] o1 = R o1 : PolynomialRing |
i2 : L = {matrix {{0,1,0},{1,0,0},{0,0,1}}, matrix {{0,0,1},{0,1,0},{1,0,0}} }
o2 = {| 0 1 0 |, | 0 0 1 |}
| 1 0 0 | | 0 1 0 |
| 0 0 1 | | 1 0 0 |
o2 : List
|
i3 : G = finiteAction(L, R)
o3 = R <- {| 0 1 0 |, | 0 0 1 |}
| 1 0 0 | | 0 1 0 |
| 0 0 1 | | 1 0 0 |
o3 : FiniteGroupAction
|
i4 : schreierGraph G
o4 = HashTable{0 => HashTable{| 0 0 1 | => | 0 0 1 |}}
| 0 1 0 | | 1 0 0 |
| 1 0 0 | | 0 1 0 |
| 0 0 1 | => | 0 0 1 |
| 1 0 0 | | 0 1 0 |
| 0 1 0 | | 1 0 0 |
| 0 1 0 | => | 1 0 0 |
| 0 0 1 | | 0 0 1 |
| 1 0 0 | | 0 1 0 |
| 0 1 0 | => | 1 0 0 |
| 1 0 0 | | 0 1 0 |
| 0 0 1 | | 0 0 1 |
| 1 0 0 | => | 0 1 0 |
| 0 0 1 | | 0 0 1 |
| 0 1 0 | | 1 0 0 |
| 1 0 0 | => | 0 1 0 |
| 0 1 0 | | 1 0 0 |
| 0 0 1 | | 0 0 1 |
1 => HashTable{| 0 0 1 | => | 1 0 0 |}
| 0 1 0 | | 0 1 0 |
| 1 0 0 | | 0 0 1 |
| 0 0 1 | => | 1 0 0 |
| 1 0 0 | | 0 0 1 |
| 0 1 0 | | 0 1 0 |
| 0 1 0 | => | 0 1 0 |
| 0 0 1 | | 1 0 0 |
| 1 0 0 | | 0 0 1 |
| 0 1 0 | => | 0 1 0 |
| 1 0 0 | | 0 0 1 |
| 0 0 1 | | 1 0 0 |
| 1 0 0 | => | 0 0 1 |
| 0 0 1 | | 1 0 0 |
| 0 1 0 | | 0 1 0 |
| 1 0 0 | => | 0 0 1 |
| 0 1 0 | | 0 1 0 |
| 0 0 1 | | 1 0 0 |
o4 : HashTable
|
The object schreierGraph is a method function.