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About SymmetricGroupRepresentation()

asked 2018-06-16 17:18:04 -0500

sleeve chen gravatar image

updated 2018-06-17 00:35:01 -0500

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?

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answered 2018-06-17 03:11:34 -0500

eric_g gravatar image

updated 2018-06-17 03:13:45 -0500

You may find answers to your questions by typing, in a Sage session,

 SymmetricGroupRepresentation?

as well as

Partition?

and

Permutation?

In particular, you will see that the documentation returned by Partition? says: A partition p of a nonnegative integer n is a non-increasing list of positive integers (the parts of the partition) with total sum n

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Asked: 2018-06-16 17:15:58 -0500

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Last updated: Jun 17