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About SymmetricGroupRepresentation()

asked 6 years ago

sleeve chen gravatar image

updated 6 years ago

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 S3, 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: [010100001]

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

[1001]

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 6 years ago

eric_g gravatar image

updated 6 years ago

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: 6 years ago

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Last updated: Jun 17 '18