def steenrod_algebra_basis(n, basis='milnor', p=2, **kwds):
r"""
Basis for the Steenrod algebra in degree `n`.
INPUT:
- ``n`` - non-negative integer
- ``basis`` - string, which basis to use (optional, default = 'milnor')
- ``p`` - positive prime number (optional, default = 2)
- ``profile`` - profile function (optional, default None). This
is just passed on to the functions :func:`milnor_basis` and
:func:`pst_basis`.
- ``truncation_type`` - truncation type, either 0 or Infinity
(optional, default Infinity if no profile function is specified,
0 otherwise). This is just passed on to the function
:func:`milnor_basis`.
OUTPUT:
Tuple of objects representing basis elements for the Steenrod algebra
in dimension n.
.. note::
Users should use :func:`steenrod_algebra_basis` instead of
this function (which has a trailing underscore in its name):
:func:`steenrod_algebra_basis` is the cached version of this
one, and so will be faster.
The choices for the string ``basis`` are as follows; see the
documentation for :mod:`sage.algebras.steenrod.steenrod_algebra`
for details on each basis:
- 'milnor': Milnor basis.
- 'serre-cartan' or 'adem' or 'admissible': Serre-Cartan basis.
- 'pst', 'pst_rlex', 'pst_llex', 'pst_deg', 'pst_revz':
various `P^s_t`-bases.
- 'comm', 'comm_rlex', 'comm_llex', 'comm_deg', 'comm_revz', or
any of these with '_long' appended: various commutator bases.
The rest of these bases are only defined when `p=2`.
- 'wood_y': Wood's Y basis.
- 'wood_z': Wood's Z basis.
- 'wall' or 'wall_long': Wall's basis.
- 'arnon_a' or 'arnon_a_long': Arnon's A basis.
- 'arnon_c': Arnon's C basis.
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import steenrod_algebra_basis
sage: steenrod_algebra_basis(7,'milnor') # indirect doctest
((0, 0, 1), (1, 2), (4, 1), (7,))
sage: steenrod_algebra_basis(5) # milnor basis is the default
((2, 1), (5,))
Bases in negative dimensions are empty::
sage: steenrod_algebra_basis(-2, 'wall')
()
The third (optional) argument to 'steenrod_algebra_basis' is the
prime p::
sage: steenrod_algebra_basis(9, 'milnor', p=3)
(((1,), (1,)), ((0,), (2,)))
sage: steenrod_algebra_basis(9, 'milnor', 3)
(((1,), (1,)), ((0,), (2,)))
sage: steenrod_algebra_basis(17, 'milnor', 3)
(((2,), ()), ((1,), (3,)), ((0,), (0, 1)), ((0,), (4,)))
Other bases::
sage: steenrod_algebra_basis(7,'admissible')
((7,), (6, 1), (4, 2, 1), (5, 2))
sage: steenrod_algebra_basis(13,'admissible',p=3)
((1, 3, 0), (0, 3, 1))
sage: steenrod_algebra_basis(5,'wall')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
sage: steenrod_algebra_basis(5,'wall_long')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
sage: steenrod_algebra_basis(5,'pst-rlex')
(((0, 1), (2, 1)), ((1, 1), (0, 2)))
"""
from steenrod_algebra_misc import get_basis_name
try:
if n < 0 or int(n) != n:
return ()
except TypeError:
return ()
basis_name = get_basis_name(basis, p)
if basis_name.find('long') >= 0:
long = True
basis_name = basis_name.rsplit('_', 1)[0]
else:
long = False
profile = kwds.get("profile", None)
if (profile is not None and profile != () and profile != ((), ())
and basis != 'milnor' and basis.find('pst') == -1):
raise ValueError, "Profile functions may only be used with the Milnor or pst bases"
## Milnor basis
if basis_name == 'milnor':
return milnor_basis(n, p, **kwds)
## Serre-Cartan basis
elif basis_name == 'serre-cartan':
return serre_cartan_basis(n, p)
## Atomic bases, p odd:
elif p > 2 and (basis_name.find('pst') >= 0
or basis_name.find('comm') >= 0):
return atomic_basis_odd(n, basis_name, p, **kwds)
## Atomic bases, p=2
elif p == 2 and (basis_name == 'woody' or basis_name == 'woodz'
or basis_name == 'wall' or basis_name == 'arnona'
or basis_name.find('pst') >= 0
or basis_name.find('comm') >= 0):
return atomic_basis(n, basis_name, **kwds)
## Arnon 'C' basis
elif p == 2 and basis == 'arnonc':
return arnonC_basis(n)
else:
raise ValueError, "Unknown basis: %s at the prime %s" % (basis, p)
def steenrod_algebra_basis(n, basis='milnor', p=2, **kwds):
r"""
Basis for the Steenrod algebra in degree `n`.
INPUT:
- ``n`` - non-negative integer
- ``basis`` - string, which basis to use (optional, default = 'milnor')
- ``p`` - positive prime number (optional, default = 2)
- ``profile`` - profile function (optional, default None). This
is just passed on to the functions :func:`milnor_basis` and
:func:`pst_basis`.
- ``truncation_type`` - truncation type, either 0 or Infinity
(optional, default Infinity if no profile function is specified,
0 otherwise). This is just passed on to the function
:func:`milnor_basis`.
OUTPUT:
Tuple of objects representing basis elements for the Steenrod algebra
in dimension n.
.. note::
Users should use :func:`steenrod_algebra_basis` instead of
this function (which has a trailing underscore in its name):
:func:`steenrod_algebra_basis` is the cached version of this
one, and so will be faster.
The choices for the string ``basis`` are as follows; see the
documentation for :mod:`sage.algebras.steenrod.steenrod_algebra`
for details on each basis:
- 'milnor': Milnor basis.
- 'serre-cartan' or 'adem' or 'admissible': Serre-Cartan basis.
- 'pst', 'pst_rlex', 'pst_llex', 'pst_deg', 'pst_revz':
various `P^s_t`-bases.
- 'comm', 'comm_rlex', 'comm_llex', 'comm_deg', 'comm_revz', or
any of these with '_long' appended: various commutator bases.
The rest of these bases are only defined when `p=2`.
- 'wood_y': Wood's Y basis.
- 'wood_z': Wood's Z basis.
- 'wall' or 'wall_long': Wall's basis.
- 'arnon_a' or 'arnon_a_long': Arnon's A basis.
- 'arnon_c': Arnon's C basis.
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra_bases import steenrod_algebra_basis
sage: steenrod_algebra_basis(7,'milnor') # indirect doctest
((0, 0, 1), (1, 2), (4, 1), (7,))
sage: steenrod_algebra_basis(5) # milnor basis is the default
((2, 1), (5,))
Bases in negative dimensions are empty::
sage: steenrod_algebra_basis(-2, 'wall')
()
The third (optional) argument to 'steenrod_algebra_basis' is the
prime p::
sage: steenrod_algebra_basis(9, 'milnor', p=3)
(((1,), (1,)), ((0,), (2,)))
sage: steenrod_algebra_basis(9, 'milnor', 3)
(((1,), (1,)), ((0,), (2,)))
sage: steenrod_algebra_basis(17, 'milnor', 3)
(((2,), ()), ((1,), (3,)), ((0,), (0, 1)), ((0,), (4,)))
Other bases::
sage: steenrod_algebra_basis(7,'admissible')
((7,), (6, 1), (4, 2, 1), (5, 2))
sage: steenrod_algebra_basis(13,'admissible',p=3)
((1, 3, 0), (0, 3, 1))
sage: steenrod_algebra_basis(5,'wall')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
sage: steenrod_algebra_basis(5,'wall_long')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
sage: steenrod_algebra_basis(5,'pst-rlex')
(((0, 1), (2, 1)), ((1, 1), (0, 2)))
"""
from steenrod_algebra_misc import get_basis_name
try:
if n < 0 or int(n) != n:
return ()
except TypeError:
return ()
basis_name = get_basis_name(basis, p)
if basis_name.find('long') >= 0:
long = True
basis_name = basis_name.rsplit('_', 1)[0]
else:
long = False
profile = kwds.get("profile", None)
if (profile is not None and profile != () and profile != ((), ())
and basis != 'milnor' and basis.find('pst') == -1):
raise ValueError("Profile functions may only be used with the Milnor or pst bases")
## Milnor basis
if basis_name == 'milnor':
return milnor_basis(n,p, **kwds)
## Serre-Cartan basis
elif basis_name == 'serre-cartan':
return serre_cartan_basis(n,p)
## Atomic bases, p odd:
elif p > 2 and (basis_name.find('pst') >= 0
or basis_name.find('comm') >= 0):
return atomic_basis_odd(n, basis_name, p, **kwds)
## Atomic bases, p=2
elif p == 2 and (basis_name == 'woody' or basis_name == 'woodz'
or basis_name == 'wall' or basis_name == 'arnona'
or basis_name.find('pst') >= 0
or basis_name.find('comm') >= 0):
return atomic_basis(n, basis_name, **kwds)
## Arnon 'C' basis
elif p == 2 and basis == 'arnonc':
return arnonC_basis(n)
else:
raise ValueError("Unknown basis: %s at the prime %s" % (basis, p))