def convert_to_milnor_matrix(n, basis, p=2):
r"""
Change-of-basis matrix, 'basis' to Milnor, in dimension
`n`, at the prime `p`.
INPUT:
- ``n`` - non-negative integer, the dimension
- ``basis`` - string, the basis from which to convert
- ``p`` - positive prime number (optional, default 2)
OUTPUT:
``matrix`` - change-of-basis matrix, a square matrix over ``GF(p)``
.. note::
This is called internally. It is not intended for casual
users, so no error checking is made on the integer `n`, the
basis name, or the prime. Also, users should call
:func:`convert_to_milnor_matrix` instead of this function
(which has a trailing underscore in its name), because the
former is the cached version of the latter.
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import convert_to_milnor_matrix
sage: convert_to_milnor_matrix(5, 'adem') # indirect doctest
[0 1]
[1 1]
sage: convert_to_milnor_matrix(45, 'milnor')
111 x 111 dense matrix over Finite Field of size 2
sage: convert_to_milnor_matrix(12,'wall')
[1 0 0 1 0 0 0]
[1 1 0 0 0 1 0]
[0 1 0 1 0 0 0]
[0 0 0 1 0 0 0]
[1 1 0 0 1 0 0]
[0 0 1 1 1 0 1]
[0 0 0 0 1 0 1]
The function takes an optional argument, the prime `p` over
which to work::
sage: convert_to_milnor_matrix(17,'adem',3)
[0 0 1 1]
[0 0 0 1]
[1 1 1 1]
[0 1 0 1]
sage: convert_to_milnor_matrix(48,'adem',5)
[0 1]
[1 1]
sage: convert_to_milnor_matrix(36,'adem',3)
[0 0 1]
[0 1 0]
[1 2 0]
"""
from sage.matrix.constructor import matrix
from sage.rings.all import GF
from steenrod_algebra import SteenrodAlgebra
if n == 0:
return matrix(GF(p), 1, 1, [[1]])
milnor_base = steenrod_algebra_basis(n, 'milnor', p)
rows = []
A = SteenrodAlgebra(basis=basis, p=p)
for poly in A.basis(n):
d = poly.milnor().monomial_coefficients()
for v in milnor_base:
entry = d.get(v, 0)
rows = rows + [entry]
d = len(milnor_base)
return matrix(GF(p), d, d, rows)
def convert_to_milnor_matrix(n, basis, p=2, generic='auto'):
r"""
Change-of-basis matrix, 'basis' to Milnor, in dimension
`n`, at the prime `p`.
INPUT:
- ``n`` - non-negative integer, the dimension
- ``basis`` - string, the basis from which to convert
- ``p`` - positive prime number (optional, default 2)
OUTPUT:
``matrix`` - change-of-basis matrix, a square matrix over ``GF(p)``
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import convert_to_milnor_matrix
sage: convert_to_milnor_matrix(5, 'adem') # indirect doctest
[0 1]
[1 1]
sage: convert_to_milnor_matrix(45, 'milnor')
111 x 111 dense matrix over Finite Field of size 2 (use the '.str()' method to see the entries)
sage: convert_to_milnor_matrix(12,'wall')
[1 0 0 1 0 0 0]
[1 1 0 0 0 1 0]
[0 1 0 1 0 0 0]
[0 0 0 1 0 0 0]
[1 1 0 0 1 0 0]
[0 0 1 1 1 0 1]
[0 0 0 0 1 0 1]
The function takes an optional argument, the prime `p` over
which to work::
sage: convert_to_milnor_matrix(17,'adem',3)
[0 0 1 1]
[0 0 0 1]
[1 1 1 1]
[0 1 0 1]
sage: convert_to_milnor_matrix(48,'adem',5)
[0 1]
[1 1]
sage: convert_to_milnor_matrix(36,'adem',3)
[0 0 1]
[0 1 0]
[1 2 0]
"""
from sage.matrix.constructor import matrix
from sage.rings.all import GF
from steenrod_algebra import SteenrodAlgebra
if generic == 'auto':
generic = False if p == 2 else True
if n == 0:
return matrix(GF(p), 1, 1, [[1]])
milnor_base = steenrod_algebra_basis(n, 'milnor', p, generic=generic)
rows = []
A = SteenrodAlgebra(basis=basis, p=p, generic=generic)
for poly in A.basis(n):
d = poly.milnor().monomial_coefficients()
for v in milnor_base:
entry = d.get(v, 0)
rows = rows + [entry]
d = len(milnor_base)
return matrix(GF(p), d, d, rows)
def convert_to_milnor_matrix(n, basis, p=2):
r"""
Change-of-basis matrix, 'basis' to Milnor, in dimension
`n`, at the prime `p`.
INPUT:
- ``n`` - non-negative integer, the dimension
- ``basis`` - string, the basis from which to convert
- ``p`` - positive prime number (optional, default 2)
OUTPUT:
``matrix`` - change-of-basis matrix, a square matrix over ``GF(p)``
.. note::
This is called internally. It is not intended for casual
users, so no error checking is made on the integer `n`, the
basis name, or the prime. Also, users should call
:func:`convert_to_milnor_matrix` instead of this function
(which has a trailing underscore in its name), because the
former is the cached version of the latter.
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import convert_to_milnor_matrix
sage: convert_to_milnor_matrix(5, 'adem') # indirect doctest
[0 1]
[1 1]
sage: convert_to_milnor_matrix(45, 'milnor')
111 x 111 dense matrix over Finite Field of size 2
sage: convert_to_milnor_matrix(12,'wall')
[1 0 0 1 0 0 0]
[1 1 0 0 0 1 0]
[0 1 0 1 0 0 0]
[0 0 0 1 0 0 0]
[1 1 0 0 1 0 0]
[0 0 1 1 1 0 1]
[0 0 0 0 1 0 1]
The function takes an optional argument, the prime `p` over
which to work::
sage: convert_to_milnor_matrix(17,'adem',3)
[0 0 1 1]
[0 0 0 1]
[1 1 1 1]
[0 1 0 1]
sage: convert_to_milnor_matrix(48,'adem',5)
[0 1]
[1 1]
sage: convert_to_milnor_matrix(36,'adem',3)
[0 0 1]
[0 1 0]
[1 2 0]
"""
from sage.matrix.constructor import matrix
from sage.rings.all import GF
from steenrod_algebra import SteenrodAlgebra
if n == 0:
return matrix(GF(p), 1, 1, [[1]])
milnor_base = steenrod_algebra_basis(n,'milnor',p)
rows = []
A = SteenrodAlgebra(basis=basis, p=p)
for poly in A.basis(n):
d = poly.milnor().monomial_coefficients()
for v in milnor_base:
entry = d.get(v, 0)
rows = rows + [entry]
d = len(milnor_base)
return matrix(GF(p),d,d,rows)
def convert_to_milnor_matrix(n, basis, p=2):
r"""
Change-of-basis matrix, 'basis' to Milnor, in dimension
`n`, at the prime `p`.
INPUT:
- ``n`` - non-negative integer, the dimension
- ``basis`` - string, the basis from which to convert
- ``p`` - positive prime number (optional, default 2)
OUTPUT:
``matrix`` - change-of-basis matrix, a square matrix over ``GF(p)``
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import convert_to_milnor_matrix
sage: convert_to_milnor_matrix(5, 'adem') # indirect doctest
[0 1]
[1 1]
sage: convert_to_milnor_matrix(45, 'milnor')
111 x 111 dense matrix over Finite Field of size 2
sage: convert_to_milnor_matrix(12,'wall')
[1 0 0 1 0 0 0]
[1 1 0 0 0 1 0]
[0 1 0 1 0 0 0]
[0 0 0 1 0 0 0]
[1 1 0 0 1 0 0]
[0 0 1 1 1 0 1]
[0 0 0 0 1 0 1]
The function takes an optional argument, the prime `p` over
which to work::
sage: convert_to_milnor_matrix(17,'adem',3)
[0 0 1 1]
[0 0 0 1]
[1 1 1 1]
[0 1 0 1]
sage: convert_to_milnor_matrix(48,'adem',5)
[0 1]
[1 1]
sage: convert_to_milnor_matrix(36,'adem',3)
[0 0 1]
[0 1 0]
[1 2 0]
"""
from sage.matrix.constructor import matrix
from sage.rings.all import GF
from steenrod_algebra import SteenrodAlgebra
if n == 0:
return matrix(GF(p), 1, 1, [[1]])
milnor_base = steenrod_algebra_basis(n,'milnor',p)
rows = []
A = SteenrodAlgebra(basis=basis, p=p)
for poly in A.basis(n):
d = poly.milnor().monomial_coefficients()
for v in milnor_base:
entry = d.get(v, 0)
rows = rows + [entry]
d = len(milnor_base)
return matrix(GF(p),d,d,rows)
