sleeve chen's profile - activity

I asked a similar question before,

https://ask.sagemath.org/question/427...

I am curious that is there any similar function for finding "edge symmetry"?

Note: the link I provided is for "node symmetry" in a network (graph).

I know we can use sage to find the group of automorphisms of a graph $G$:

G.automorphism_group().list()

However, the above way can only be used to the unweighted graph. So for example:

G = matrix([[0,10,0],
            [10,0,1],
            [0,1,0]])
G1 = Graph(G, weighted = True)
G1.show(edge_labels=True )  
G.automorphism_group().list()

The result is:

[(), (0,2)]

However, this result is not correct (correct for unweighted case). This is because $AD\neq DA$, where

$$D = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix},$$ which is a permutation matrix and

$$A = \begin{bmatrix} 0 & 10 & 0 \\ 10 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix},$$ which is an adjacency matrix.

Can we use SAGE to find the group of automorphisms of a graph?

I am a new student in SAGE. I read the following discussion:
evaluation of character of symmetric group
and then also read the manual.

However, I am still confused about some fundamental problem:
(I cannot find these function in "Sage Reference Manual: Groups, Release 8.2". Are both new functions?).

1. About

   SymmetricGroupRepresentation(partition, implementation='specht', ring=None, cache_matrices=True)
   

   I am confused about "partition". Suppose for $S_3$, and partition $=[2,1]$. What does it mean? (It seems $[1,2]$ is not valid)

2. About

   spc = SymmetricGroupRepresentation([2,1], 'specht')
   spc.representation_matrix(Permutation([1,2,3]))
   

   When I use

   spc.representation_matrix(Permutation([1,2]))
   

   error pops out. However, as far as I know, $(1,2)$ is a valid permutation, which represent the matrix representation: $$\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$

However, if I use 'orthogonal' instead of 'specht', the answer becomes

$$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$

But I test another permutation: $[1,2,3]$, the answer is the same. However, $[1,2]$ and $[1,2,3]$ are in the different conjugacy classes; they should not have the same character.

I cannot find "Permutation" in "Sage Reference Manual: Group". Where can I find this function?