Let's look at the graph of a square. Since a square is a 2-cube, we can create the polytope and look at its graph:

polytope > $c=cube(2); polytope > $c->GRAPH->VISUAL;

To study the automorphisms of this graph, we create a `props::Graph`

object refering to the `C++`

class named `Graph`

(see the tutorial on graphs for more details):

polytope > $g=new props::Graph($c->GRAPH->ADJACENCY);

The picture of the graph shows that the node with label 0 is adjacent to the nodes 1 and 2, Node 1 is adjacent to 0 and 3, and so on. For the complete adjacency information you can print `$c→GRAPH→ADJACENCY`

or just the props::Graph object `$g`

:

polytope > print rows_labeled($g); 0:1 2 1:0 3 2:0 3 3:1 2

Now, we compute the generators of the automorphism group of this graph:

polytope > $aut=automorphisms($g);

In this case, the automorphism group has two generators:

polytope > print $aut; 0 2 1 3 1 0 3 2

Each generator is a permutation on the nodes. The first generator fixes the nodes 0 and 3, and exchanges the nodes 1 and 2, i.e., it describes the reflection along the diagonal through 0 and 3. The second generator is the reflection along the horizontal line.

In order to be able to work with the group, we create a new Group object, which lives in the application `group`

:

polytope > $autgroup=new group::Group(GENERATORS=>$aut);

Now we can ask for basic properties of the group, e.g., the number of elements:

polytope > print $autgroup->ORDER; 8

Sometimes, it is useful to know which elements of the group fix a specific set of indices, that is, we are interested in the subgroup which is the stabilizer of the given set. In the first case, we just fix the index 0:

polytope > $s0=new Set<Int>(0); polytope > $stab0=group::stabilizer_of_set($autgroup,$s0);

We learn that the node 0 is only fixed by the permutation `0 2 1 3`

:

polytope > print $stab0->ORDER; 2 polytope > print $stab0->GENERATORS; 0 2 1 3

In the second case, we look at the subgroup which leaves the set `{1,2}`

invariant:

polytope > $s12=new Set<Int>(1,2); polytope > $stab12=group::stabilizer_of_set($autgroup,$s12);

Now, we obtain a group of order 4:

polytope > print $stab12->ORDER; 4 polytope > print $stab12->GENERATORS; 3 1 2 0 0 2 1 3

Finally, we compute the orbits of the indices under the three different groups:

polytope > print group::compute_domain_orbits($stab0); {0} {1 2} {3} polytope > print group::compute_domain_orbits($stab12); {0 3} {1 2} polytope > print group::compute_domain_orbits($autgroup); {0 1 2 3}

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