Action of MatrixGroup on a MPolynomialRing

[simon-king-jena]

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

CC: @wdjoyner @malb

Component: commutative algebra

Keywords: matrix group, action, polynomial ring

Author: Simon King

Reviewer: David Loeffler, William Stein

Issue created by migration from https://trac.sagemath.org/ticket/4513

[simon-king-jena]

comment:1

Sorry, in the above code I forgot to copy/paste the line

sage: R.<x,y> = GF(3)[]

Moreover, for the reasons above, the ticket should belong to commutative algebra, not just algebra (I was clicking on the wrong button).

[simon-king-jena]

comment:2

The patch matrixgroupCall.patch is without doctests, and I am not sure if it couldn't be done better. So, it needs more work.

For example, Martin mentioned the possibility (off list) to create a pyx file with a Cython function, and then the call method would use that function. It might pay off here, since there are tight loops and since the method has to deal with tuples or lists. So Cdefining might speed things up.

Opinions?

At least, the following now works:

sage: R.<x,y>=GF(3)[]
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

[simon-king-jena]

call method for MatrixGroupelement (this time with doc test) and left_matrix_action for MPolynomial

[simon-king-jena]

comment:4

Attachment: matrixgroupCall.patch.gz

Some changes:

Nicolas Thiéry suggested to implement the action of matrix groups by a method for polynomials (I call the method left_matrix_action), and, for convenience, also provide a __call__ method for MatrixGroupElement that refers to left_matrix_action.

This has several advantages:

• The __call__ works for any class that has a left_matrix_action method, hence, it is not a-priori restricted to polynomials.
• I've put left_matrix_action into multi_polynomial.pyx, hence, I can use Cython.

I have two Cython concerns left:

1. The innerst loop is

prod([Im[k]**X[k] for k in xrange(n)])

where k is c'defined as int. Should this better be done in a for-loop, rather then creating a list and calling prod?
2. The variable X is of type polydict.ETuple, so I can not directly c'define X. One could do

   cdef tuple X
   for i from 0<i<l:
       X = tuple(Expo[i])

But would this be faster?