#4513: [with patch, needs work] Action of MatrixGroup on a MPolynomialRing
-----------------------------------+----------------------------------------
Reporter: SimonKing | Owner: SimonKing
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-4.5.1
Component: commutative algebra | Keywords: matrix group, action,
polynomial ring
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
-----------------------------------+----------------------------------------
Changes (by SimonKing):
* upstream: => N/A
Old description:
> A group of n by n matrices over a field K acts on a polynomial ring with
> n variables over K. However, this is not implemented yet.
>
> Off list, David Joyner suggested to implement it with a `__call__` method
> in `matrix_group_element.py`. Then, the following should work:
> {{{
> sage: M=Matrix(GF(3),[[1,2],[1,1]])
> sage: G=MatrixGroup([M])
> sage: g=G.0
> sage: p=x*y^2
> sage: g(p)
> x^3 + x^2*y - x*y^2 - y^3
> sage: _==(x+2*y)*(x+y)^2
> True
> }}}
>
> Although it concerns `matrix_group_element.py`, I believe this ticket
> belongs to Commutative Algebra, for two reasons:
> 1. An efficient implementation probably requires knowledge of the guts
> of MPolynomialElement.
> 2. My long-term goal is to re-implement my algorithms for the
> computation of non-modular invariant rings. The current implementation is
> in the `finvar.lib` library of Singular -- the slow Singular interpreter
> sometimes is a bottle necks.
>
> One more general technical question: It is `matrix_group_element.py`,
> hence seems to be pure python. Is it possible to define an additional
> method in some `.pyx` file using Cython? I don't know if this would be
> reasonable to do here, but perhaps this could come in handy at some
> point...
New description:
A group of n by n matrices over a field K acts on a polynomial ring with n
variables over K. However, this is not implemented yet.
The following should work:
{{{
sage: M = Matrix(GF(3),[[1,2],[1,1]])
sage: N = Matrix(GF(3),[[2,2],[2,1]])
sage: G = MatrixGroup([M,N])
sage: m = G.0
sage: n = G.1
sage: R.<x,y> = GF(3)[]
sage: m*x
x + y
sage: x*m
x - y
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
}}}
On the other hand, we still want to have the usual action on vectors or
matrices:
{{{
sage: x = vector([1,1])
sage: x*m
(2, 0)
sage: m*x
(0, 2)
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
}}}
{{{
sage: x = matrix([[1,2],[1,1]])
sage: x*m
[0 1]
[2 0]
sage: m*x
[0 1]
[2 0]
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
}}}
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4513#comment:15>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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