Permutation actions on vectors

[Sihuang Hu]

Suppose that we have a group G=Sym(3). Then G naturally act on the vector space
V=VectorSpace(GF(2),3) by position. Now I want to compute the orbits of G acting on
V. It is easy to see that the orbits are
[ [(0,0,0)],
  [(1,0,0),(0,1,0),(0,0,1)],
  [(1,1,0),(1,0,1),(0,1,1)],
  [(1,1,1)],
].

I cannot find how to realize this group action "OnPosition" and compute its orbits in sage.
Do anyone know how to do this?

Thanks,
Sihuang

[Dima Pasechnik]

It would be more natural ro convert it to a matrix group, and then use
the natural action of this group.

>
> Thanks,
> Sihuang
>

[Nathann Cohen]

It would be more natural ro convert it to a matrix group, and then use
the natural action of this group.

This is related:

http://www.sagemath.org/doc/reference/combinat/sage/combinat/integer_vectors_mod_permgroup.html

Nathann 

[Sihuang Hu]

Thanks for your advice. I tried it, but still does not work.

sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: VS = VectorSpace(F,2)
sage: v = VS([1,1])
sage: G.orbit(v)
---------------------------------------------------------------------------
AttributeError: 'FinitelyGeneratedMatrixGroup_gap_with_category' object has no attribute 'orbit'


在 2015年1月13日星期二 UTC+1下午4:20:23，Dima Pasechnik写道：

[Sihuang Hu]

Yes, it is related, but not what I want.
Thanks.

在 2015年1月13日星期二 UTC+1下午4:45:23，Nathann Cohen写道：

[David Joyner]

On Tue, Jan 13, 2015 at 2:18 PM, Sihuang Hu <husi...@gmail.com> wrote:
> Yes, it is related, but not what I want.

Are you sure?

sage: L = [sorted(sorted(I.orbit(list(x)))) for x in GF(2)^3]
sage: set(map(tuple, L))
{([0, 0, 0],),
([0, 0, 1], [0, 1, 0], [1, 0, 0]),
([0, 1, 1], [1, 0, 1], [1, 1, 0]),
([1, 1, 1],)}

Is that what you want?

> Thanks.
>
> 在 2015年1月13日星期二 UTC+1下午4:45:23，Nathann Cohen写道：
>>>
>>> It would be more natural ro convert it to a matrix group, and then use
>>> the natural action of this group.
>>
>>
>> This is related:
>>
>>
>> http://www.sagemath.org/doc/reference/combinat/sage/combinat/integer_vectors_mod_permgroup.html
>>
>> Nathann
>

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[Sihuang Hu]

Thanks. This works for prime field GF(p). But for GF(q) where q is a prime power,
I think I have to transfer the elements in GF(q)^3 into lists of integers at first.

在 2015年1月13日星期二 UTC+1下午10:15:44，David Joyner写道：