Steenrod algebra bases

AUTHORS:

  • John H. Palmieri (2008-07-30): version 0.9

  • John H. Palmieri (2010-06-30): version 1.0

  • Simon King (2011-10-25): Fix the use of cached functions

This package defines functions for computing various bases of the Steenrod algebra, and for converting between the Milnor basis and any other basis.

This packages implements a number of different bases, at least at the prime 2. The Milnor and Serre-Cartan bases are the most familiar and most standard ones, and all of the others are defined in terms of one of these. The bases are described in the documentation for the function steenrod_algebra_basis(); also see the papers by Monks [Mon1998] and Wood [Woo1998] for more information about them. For commutator bases, see the preprint by Palmieri and Zhang [PZ2008].

  • 'milnor' – Milnor basis

  • 'serre-cartan' or 'adem' or 'admissible' – Serre-Cartan basis

Most of the rest of the bases are only defined when \(p=2\). The only exceptions are the \(P^s_t\)-bases and the commutator bases, which are defined at all primes.

  • 'wood_y' – Wood’s Y basis

  • 'wood_z' – Wood’s Z basis

  • 'wall', 'wall_long' – Wall’s basis

  • 'arnon_a', 'arnon_a_long' – Arnon’s A basis

  • 'arnon_c' – Arnon’s C 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 these with '_long' appended – various commutator bases

The main functions provided here are

  • steenrod_algebra_basis(). This computes a tuple representing basis elements for the Steenrod algebra in a given degree, at a given prime, with respect to a given basis. It is a cached function.

  • convert_to_milnor_matrix(). This returns the change-of-basis matrix, in a given degree, from any basis to the Milnor basis. It is a cached function.

  • convert_from_milnor_matrix(). This returns the inverse of the previous matrix.

INTERNAL DOCUMENTATION:

If you want to implement a new basis for the Steenrod algebra:

In the file steenrod_algebra.py:

For the class SteenrodAlgebra_generic, add functionality to the methods:

In the file steenrod_algebra_misc.py:

  • add functionality to get_basis_name: this should accept as input various synonyms for the basis, and its output should be a canonical name for the basis.

  • add a function BASIS_mono_to_string like milnor_mono_to_string or one of the other similar functions.

In this file steenrod_algebra_bases.py:

If the basis has an intrinsic way of defining a product, implement it in the file steenrod_algebra_mult.py and also in the product_on_basis method for SteenrodAlgebra_generic in steenrod_algebra.py.

sage.algebras.steenrod.steenrod_algebra_bases.arnonC_basis(bound=1)[source]

Arnon’s C basis in dimension \(n\).

INPUT:

  • n – nonnegative integer

  • bound – positive integer (optional)

OUTPUT: tuple of basis elements in dimension \(n\)

The elements of Arnon’s C basis are monomials of the form \(\text{Sq}^{t_1} ... \text{Sq}^{t_m}\) where for each \(i\), we have \(t_i \leq 2t_{i+1}\) and \(2^i | t_{m-i}\).

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import arnonC_basis
sage: arnonC_basis(7)
((7,), (2, 5), (4, 3), (4, 2, 1))

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import arnonC_basis
>>> arnonC_basis(Integer(7))
((7,), (2, 5), (4, 3), (4, 2, 1))

If optional argument bound is present, include only those monomials whose first term is at least as large as bound:

sage: arnonC_basis(7,3)
((7,), (4, 3), (4, 2, 1))

>>> from sage.all import *
>>> arnonC_basis(Integer(7),Integer(3))
((7,), (4, 3), (4, 2, 1))
sage.algebras.steenrod.steenrod_algebra_bases.atomic_basis(n, basis, **kwds)[source]

Basis for dimension \(n\) made of elements in ‘atomic’ degrees: degrees of the form \(2^i (2^j - 1)\).

This works at the prime 2 only.

INPUT:

  • n – nonnegative integer

  • basis – string, the name of the basis

  • profile – profile function (default: None). Together with truncation_type, specify the profile function to be used; None means the profile function for the entire Steenrod algebra. See sage.algebras.steenrod.steenrod_algebra and SteenrodAlgebra() for information on profile functions.

  • truncation_type – truncation type, either 0 or Infinity (default: Infinity if no profile function is specified, 0 otherwise).

OUTPUT: tuple of basis elements in dimension \(n\)

The atomic bases include Wood’s Y and Z bases, Wall’s basis, Arnon’s A basis, the \(P^s_t\)-bases, and the commutator bases. (All of these bases are constructed similarly, hence their constructions have been consolidated into a single function. Also, see the documentation for steenrod_algebra_basis() for descriptions of them.) For \(P^s_t\)-bases, you may also specify a profile function and truncation type; profile functions are ignored for the other bases.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import atomic_basis
sage: atomic_basis(6,'woody')
(((1, 0), (0, 1), (0, 0)), ((2, 0), (1, 0)), ((1, 1),))
sage: atomic_basis(8,'woodz')
(((2, 0), (0, 1), (0, 0)), ((0, 2), (0, 0)), ((1, 1), (1, 0)), ((3, 0),))
sage: atomic_basis(6,'woodz') == atomic_basis(6, 'woody')
True
sage: atomic_basis(9,'woodz') == atomic_basis(9, 'woody')
False

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import atomic_basis
>>> atomic_basis(Integer(6),'woody')
(((1, 0), (0, 1), (0, 0)), ((2, 0), (1, 0)), ((1, 1),))
>>> atomic_basis(Integer(8),'woodz')
(((2, 0), (0, 1), (0, 0)), ((0, 2), (0, 0)), ((1, 1), (1, 0)), ((3, 0),))
>>> atomic_basis(Integer(6),'woodz') == atomic_basis(Integer(6), 'woody')
True
>>> atomic_basis(Integer(9),'woodz') == atomic_basis(Integer(9), 'woody')
False

Wall’s basis:

sage: atomic_basis(8,'wall')
(((2, 2), (1, 0), (0, 0)), ((2, 0), (0, 0)), ((2, 1), (1, 1)), ((3, 3),))

>>> from sage.all import *
>>> atomic_basis(Integer(8),'wall')
(((2, 2), (1, 0), (0, 0)), ((2, 0), (0, 0)), ((2, 1), (1, 1)), ((3, 3),))

Arnon’s A basis:

sage: atomic_basis(7,'arnona')
(((0, 0), (1, 1), (2, 2)), ((0, 0), (2, 1)), ((1, 0), (2, 2)), ((2, 0),))

>>> from sage.all import *
>>> atomic_basis(Integer(7),'arnona')
(((0, 0), (1, 1), (2, 2)), ((0, 0), (2, 1)), ((1, 0), (2, 2)), ((2, 0),))

\(P^s_t\)-bases:

sage: atomic_basis(7,'pst_rlex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((2, 1), (0, 2)), ((0, 3),))
sage: atomic_basis(7,'pst_llex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
sage: atomic_basis(7,'pst_deg')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
sage: atomic_basis(7,'pst_revz')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))

>>> from sage.all import *
>>> atomic_basis(Integer(7),'pst_rlex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((2, 1), (0, 2)), ((0, 3),))
>>> atomic_basis(Integer(7),'pst_llex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
>>> atomic_basis(Integer(7),'pst_deg')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
>>> atomic_basis(Integer(7),'pst_revz')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))

Commutator bases:

sage: atomic_basis(7,'comm_rlex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((2, 1), (0, 2)), ((0, 3),))
sage: atomic_basis(7,'comm_llex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
sage: atomic_basis(7,'comm_deg')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
sage: atomic_basis(7,'comm_revz')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))

>>> from sage.all import *
>>> atomic_basis(Integer(7),'comm_rlex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((2, 1), (0, 2)), ((0, 3),))
>>> atomic_basis(Integer(7),'comm_llex')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
>>> atomic_basis(Integer(7),'comm_deg')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
>>> atomic_basis(Integer(7),'comm_revz')
(((0, 1), (1, 1), (2, 1)), ((0, 1), (1, 2)), ((0, 2), (2, 1)), ((0, 3),))
sage.algebras.steenrod.steenrod_algebra_bases.atomic_basis_odd(n, basis, p, **kwds)[source]

\(P^s_t\)-bases and commutator basis in dimension \(n\) at odd primes.

This function is called atomic_basis_odd in analogy with atomic_basis().

INPUT:

  • n – nonnegative integer

  • basis – string, the name of the basis

  • p – positive prime number

  • profile – profile function (default: None). Together with truncation_type, specify the profile function to be used; None means the profile function for the entire Steenrod algebra. See sage.algebras.steenrod.steenrod_algebra and SteenrodAlgebra() for information on profile functions.

  • truncation_type – truncation type, either 0 or Infinity (default: Infinity if no profile function is specified, 0 otherwise).

OUTPUT: tuple of basis elements in dimension \(n\)

The only possible difference in the implementations for \(P^s_t\) bases and commutator bases is that the former make sense, and require filtering, if there is a nontrivial profile function. This function is called by steenrod_algebra_basis(), and it will not be called for commutator bases if there is a profile function, so we treat the two bases exactly the same.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import atomic_basis_odd
sage: atomic_basis_odd(8, 'pst_rlex', 3)
(((), (((0, 1), 2),)),)

sage: atomic_basis_odd(18, 'pst_rlex', 3)
(((0, 2), ()), ((0, 1), (((1, 1), 1),)))
sage: atomic_basis_odd(18, 'pst_rlex', 3, profile=((), (2,2,2)))
(((0, 2), ()),)

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import atomic_basis_odd
>>> atomic_basis_odd(Integer(8), 'pst_rlex', Integer(3))
(((), (((0, 1), 2),)),)

>>> atomic_basis_odd(Integer(18), 'pst_rlex', Integer(3))
(((0, 2), ()), ((0, 1), (((1, 1), 1),)))
>>> atomic_basis_odd(Integer(18), 'pst_rlex', Integer(3), profile=((), (Integer(2),Integer(2),Integer(2))))
(((0, 2), ()),)
sage.algebras.steenrod.steenrod_algebra_bases.convert_from_milnor_matrix(n, basis, p=2, generic='auto')[source]

Change-of-basis matrix, Milnor to basis, in dimension \(n\).

INPUT:

  • n – nonnegative integer, the dimension

  • basis – string, the basis to which to convert

  • p – positive prime number (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.

EXAMPLES:

sage: # needs sage.modules
sage: from sage.algebras.steenrod.steenrod_algebra_bases import convert_from_milnor_matrix, convert_to_milnor_matrix
sage: convert_from_milnor_matrix(12, 'wall')
[1 0 0 1 0 0 0]
[0 0 1 1 0 0 0]
[0 0 0 1 0 1 1]
[0 0 0 1 0 0 0]
[1 0 1 0 1 0 0]
[1 1 1 0 0 0 0]
[1 0 1 0 1 0 1]
sage: convert_from_milnor_matrix(38, 'serre_cartan')
72 x 72 dense matrix over Finite Field of size 2 (use the '.str()' method to see the entries)
sage: x = convert_to_milnor_matrix(20, 'wood_y')
sage: y = convert_from_milnor_matrix(20, 'wood_y')
sage: x*y
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

>>> from sage.all import *
>>> # needs sage.modules
>>> from sage.algebras.steenrod.steenrod_algebra_bases import convert_from_milnor_matrix, convert_to_milnor_matrix
>>> convert_from_milnor_matrix(Integer(12), 'wall')
[1 0 0 1 0 0 0]
[0 0 1 1 0 0 0]
[0 0 0 1 0 1 1]
[0 0 0 1 0 0 0]
[1 0 1 0 1 0 0]
[1 1 1 0 0 0 0]
[1 0 1 0 1 0 1]
>>> convert_from_milnor_matrix(Integer(38), 'serre_cartan')
72 x 72 dense matrix over Finite Field of size 2 (use the '.str()' method to see the entries)
>>> x = convert_to_milnor_matrix(Integer(20), 'wood_y')
>>> y = convert_from_milnor_matrix(Integer(20), 'wood_y')
>>> x*y
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

The function takes an optional argument, the prime \(p\) over which to work:

sage: convert_from_milnor_matrix(17, 'adem', 3)                                 # needs sage.modules
[2 1 1 2]
[0 2 0 1]
[1 2 0 0]
[0 1 0 0]

>>> from sage.all import *
>>> convert_from_milnor_matrix(Integer(17), 'adem', Integer(3))                                 # needs sage.modules
[2 1 1 2]
[0 2 0 1]
[1 2 0 0]
[0 1 0 0]
sage.algebras.steenrod.steenrod_algebra_bases.convert_to_milnor_matrix(basis, p=2, generic='auto')[source]

Change-of-basis matrix, ‘basis’ to Milnor, in dimension \(n\), at the prime \(p\).

INPUT:

  • n – nonnegative integer, the dimension

  • basis – string, the basis from which to convert

  • p – positive prime number (default: 2)

OUTPUT:

matrix – change-of-basis matrix, a square matrix over \(\GF{p}\)

EXAMPLES:

sage: # needs sage.modules
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]

>>> from sage.all import *
>>> # needs sage.modules
>>> from sage.algebras.steenrod.steenrod_algebra_bases import convert_to_milnor_matrix
>>> convert_to_milnor_matrix(Integer(5), 'adem')  # indirect doctest
[0 1]
[1 1]
>>> convert_to_milnor_matrix(Integer(45), 'milnor')
111 x 111 dense matrix over Finite Field of size 2 (use the '.str()' method to see the entries)
>>> convert_to_milnor_matrix(Integer(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: # needs sage.modules
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.all import *
>>> # needs sage.modules
>>> convert_to_milnor_matrix(Integer(17), 'adem', Integer(3))
[0 0 1 1]
[0 0 0 1]
[1 1 1 1]
[0 1 0 1]
>>> convert_to_milnor_matrix(Integer(48), 'adem', Integer(5))
[0 1]
[1 1]
>>> convert_to_milnor_matrix(Integer(36), 'adem', Integer(3))
[0 0 1]
[0 1 0]
[1 2 0]
sage.algebras.steenrod.steenrod_algebra_bases.milnor_basis(n, p=2, **kwds)[source]

Milnor basis in dimension \(n\) with profile function profile.

INPUT:

  • n – nonnegative integer

  • p – positive prime number (default: 2)

  • profile – profile function (default: None). Together with truncation_type, specify the profile function to be used; None means the profile function for the entire Steenrod algebra. See sage.algebras.steenrod.steenrod_algebra and SteenrodAlgebra for information on profile functions.

  • truncation_type – truncation type, either 0 or Infinity (default: Infinity if no profile function is specified, 0 otherwise)

OUTPUT: tuple of mod \(p\) Milnor basis elements in dimension \(n\)

At the prime 2, the Milnor basis consists of symbols of the form \(\text{Sq}(m_1, m_2, ..., m_t)\), where each \(m_i\) is a nonnegative integer and if \(t>1\), then \(m_t \neq 0\). At odd primes, it consists of symbols of the form \(Q_{e_1} Q_{e_2} ... P(m_1, m_2, ..., m_t)\), where \(0 \leq e_1 < e_2 < ...\), each \(m_i\) is a nonnegative integer, and if \(t>1\), then \(m_t \neq 0\).

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import milnor_basis
sage: milnor_basis(7)
((0, 0, 1), (1, 2), (4, 1), (7,))
sage: milnor_basis(7, 2)
((0, 0, 1), (1, 2), (4, 1), (7,))
sage: milnor_basis(4, 2)
((1, 1), (4,))
sage: milnor_basis(4, 2, profile=[2,1])
((1, 1),)
sage: milnor_basis(4, 2, profile=(), truncation_type=0)
()
sage: milnor_basis(4, 2, profile=(), truncation_type=Infinity)
((1, 1), (4,))
sage: milnor_basis(9, 3)
(((1,), (1,)), ((0,), (2,)))
sage: milnor_basis(17, 3)
(((2,), ()), ((1,), (3,)), ((0,), (0, 1)), ((0,), (4,)))
sage: milnor_basis(48, p=5)
(((), (0, 1)), ((), (6,)))
sage: len(milnor_basis(100,3))
13
sage: len(milnor_basis(200,7))
0
sage: len(milnor_basis(240,7))
3
sage: len(milnor_basis(240,7, profile=((),()), truncation_type=Infinity))
3
sage: len(milnor_basis(240,7, profile=((),()), truncation_type=0))
0

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import milnor_basis
>>> milnor_basis(Integer(7))
((0, 0, 1), (1, 2), (4, 1), (7,))
>>> milnor_basis(Integer(7), Integer(2))
((0, 0, 1), (1, 2), (4, 1), (7,))
>>> milnor_basis(Integer(4), Integer(2))
((1, 1), (4,))
>>> milnor_basis(Integer(4), Integer(2), profile=[Integer(2),Integer(1)])
((1, 1),)
>>> milnor_basis(Integer(4), Integer(2), profile=(), truncation_type=Integer(0))
()
>>> milnor_basis(Integer(4), Integer(2), profile=(), truncation_type=Infinity)
((1, 1), (4,))
>>> milnor_basis(Integer(9), Integer(3))
(((1,), (1,)), ((0,), (2,)))
>>> milnor_basis(Integer(17), Integer(3))
(((2,), ()), ((1,), (3,)), ((0,), (0, 1)), ((0,), (4,)))
>>> milnor_basis(Integer(48), p=Integer(5))
(((), (0, 1)), ((), (6,)))
>>> len(milnor_basis(Integer(100),Integer(3)))
13
>>> len(milnor_basis(Integer(200),Integer(7)))
0
>>> len(milnor_basis(Integer(240),Integer(7)))
3
>>> len(milnor_basis(Integer(240),Integer(7), profile=((),()), truncation_type=Infinity))
3
>>> len(milnor_basis(Integer(240),Integer(7), profile=((),()), truncation_type=Integer(0)))
0
sage.algebras.steenrod.steenrod_algebra_bases.restricted_partitions(n, l, no_repeats=False)[source]

Iterator over ‘restricted’ partitions of \(n\): partitions with parts taken from list \(l\).

INPUT:

  • n – nonnegative integer

  • l – list of positive integers

  • no_repeats – boolean (default: False); if True, only return partitions with no repeated parts

OUTPUT: iterator of lists

One could also use Partitions(n, parts_in=l), but this function may be faster. Also, while Partitions(n, parts_in=l, max_slope=-1) should in theory return the partitions of \(n\) with parts in l with no repetitions, the max_slope=-1 argument is ignored, so it does not work. (At the moment, the no_repeats=True case is the only one used in the code.)

Todo

This should be re-implemented in a non-recursive way.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import restricted_partitions
sage: list(restricted_partitions(10, [7,5,1]))
[[7, 1, 1, 1], [5, 5], [5, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
sage: list(restricted_partitions(10, [6,5,4,3,2,1], no_repeats=True))
[[6, 4], [6, 3, 1], [5, 4, 1], [5, 3, 2], [4, 3, 2, 1]]
sage: list(restricted_partitions(10, [6,4,2]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
sage: list(restricted_partitions(10, [6,4,2], no_repeats=True))
[[6, 4]]

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import restricted_partitions
>>> list(restricted_partitions(Integer(10), [Integer(7),Integer(5),Integer(1)]))
[[7, 1, 1, 1], [5, 5], [5, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)], no_repeats=True))
[[6, 4], [6, 3, 1], [5, 4, 1], [5, 3, 2], [4, 3, 2, 1]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2)]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2)], no_repeats=True))
[[6, 4]]

l may have repeated elements. If no_repeats is False, this has no effect. If no_repeats is True, and if the repeated elements appear consecutively in l, then each element may be used only as many times as it appears in l:

sage: list(restricted_partitions(10, [6,4,2,2], no_repeats=True))
[[6, 4], [6, 2, 2]]
sage: list(restricted_partitions(10, [6,4,2,2,2], no_repeats=True))
[[6, 4], [6, 2, 2], [4, 2, 2, 2]]

>>> from sage.all import *
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2),Integer(2)], no_repeats=True))
[[6, 4], [6, 2, 2]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2),Integer(2),Integer(2)], no_repeats=True))
[[6, 4], [6, 2, 2], [4, 2, 2, 2]]

(If the repeated elements don’t appear consecutively, the results are likely meaningless, containing several partitions more than once, for example.)

In the following examples, no_repeats is False:

sage: list(restricted_partitions(10, [6,4,2]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
sage: list(restricted_partitions(10, [6,4,2,2,2]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
sage: list(restricted_partitions(10, [6,4,4,4,2,2,2,2,2,2]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]

>>> from sage.all import *
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2)]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(2),Integer(2),Integer(2)]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
>>> list(restricted_partitions(Integer(10), [Integer(6),Integer(4),Integer(4),Integer(4),Integer(2),Integer(2),Integer(2),Integer(2),Integer(2),Integer(2)]))
[[6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]]
sage.algebras.steenrod.steenrod_algebra_bases.serre_cartan_basis(n, p=2, bound=1, **kwds)[source]

Serre-Cartan basis in dimension \(n\).

INPUT:

  • n – nonnegative integer

  • bound – positive integer (optional)

  • prime – positive prime number (default: 2)

OUTPUT: tuple of mod \(p\) Serre-Cartan basis elements in dimension \(n\)

The Serre-Cartan basis consists of ‘admissible monomials in the Steenrod squares’. Thus at the prime 2, it consists of monomials \(\text{Sq}^{m_1} \text{Sq}^{m_2} ... \text{Sq}^{m_t}\) with \(m_i \geq 2m_{i+1}\) for each \(i\). At odd primes, it consists of monomials \(\beta^{e_0} P^{s_1} \beta^{e_1} P^{s_2} ... P^{s_k} \beta^{e_k}\) with each \(e_i\) either 0 or 1, \(s_i \geq p s_{i+1} + e_i\) for all \(i\), and \(s_k \geq 1\).

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_bases import serre_cartan_basis
sage: serre_cartan_basis(7)
((7,), (6, 1), (4, 2, 1), (5, 2))
sage: serre_cartan_basis(13,3)
((1, 3, 0), (0, 3, 1))
sage: serre_cartan_basis(50,5)
((1, 5, 0, 1, 1), (1, 6, 1))

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import serre_cartan_basis
>>> serre_cartan_basis(Integer(7))
((7,), (6, 1), (4, 2, 1), (5, 2))
>>> serre_cartan_basis(Integer(13),Integer(3))
((1, 3, 0), (0, 3, 1))
>>> serre_cartan_basis(Integer(50),Integer(5))
((1, 5, 0, 1, 1), (1, 6, 1))

If optional argument bound is present, include only those monomials whose last term is at least bound (when \(p=2\)), or those for which \(s_k - e_k \geq bound\) (when \(p\) is odd).

sage: serre_cartan_basis(7, bound=2)
((7,), (5, 2))
sage: serre_cartan_basis(13, 3, bound=3)
((1, 3, 0),)

>>> from sage.all import *
>>> serre_cartan_basis(Integer(7), bound=Integer(2))
((7,), (5, 2))
>>> serre_cartan_basis(Integer(13), Integer(3), bound=Integer(3))
((1, 3, 0),)
sage.algebras.steenrod.steenrod_algebra_bases.steenrod_algebra_basis(basis='milnor', p=2, **kwds)[source]

Basis for the Steenrod algebra in degree \(n\).

INPUT:

  • n – nonnegative integer

  • basis – string, which basis to use (default: 'milnor')

  • p – positive prime number (default: 2)

  • profile – profile function (default: None); this is just passed on to the functions milnor_basis() and pst_basis()

  • truncation_type – truncation type, either 0 or Infinity (default: Infinity if no profile function is specified, 0 otherwise). This is just passed on to the function milnor_basis().

  • generic – boolean (default: None)

OUTPUT:

Tuple of objects representing basis elements for the Steenrod algebra in dimension n.

The choices for the string basis are as follows; see the documentation for 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,))

>>> from sage.all import *
>>> from sage.algebras.steenrod.steenrod_algebra_bases import steenrod_algebra_basis
>>> steenrod_algebra_basis(Integer(7), 'milnor') # indirect doctest
((0, 0, 1), (1, 2), (4, 1), (7,))
>>> steenrod_algebra_basis(Integer(5))   # milnor basis is the default
((2, 1), (5,))

Bases in negative dimensions are empty:

sage: steenrod_algebra_basis(-2, 'wall')
()

>>> from sage.all import *
>>> steenrod_algebra_basis(-Integer(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,)))

>>> from sage.all import *
>>> steenrod_algebra_basis(Integer(9), 'milnor', p=Integer(3))
(((1,), (1,)), ((0,), (2,)))
>>> steenrod_algebra_basis(Integer(9), 'milnor', Integer(3))
(((1,), (1,)), ((0,), (2,)))
>>> steenrod_algebra_basis(Integer(17), 'milnor', Integer(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 sage.all import *
>>> steenrod_algebra_basis(Integer(7), 'admissible')
((7,), (6, 1), (4, 2, 1), (5, 2))
>>> steenrod_algebra_basis(Integer(13), 'admissible', p=Integer(3))
((1, 3, 0), (0, 3, 1))
>>> steenrod_algebra_basis(Integer(5), 'wall')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
>>> steenrod_algebra_basis(Integer(5), 'wall_long')
(((2, 2), (0, 0)), ((1, 1), (1, 0)))
>>> steenrod_algebra_basis(Integer(5), 'pst-rlex')
(((0, 1), (2, 1)), ((1, 1), (0, 2)))
sage.algebras.steenrod.steenrod_algebra_bases.steenrod_basis_error_check(dim, p, **kwds)[source]

This performs crude error checking.

INPUT:

  • dim – nonnegative integer

  • p – positive prime number

OUTPUT: none

This checks to see if the different bases have the same length, and if the change-of-basis matrices are invertible. If something goes wrong, an error message is printed.

This function checks at the prime \(p\) as the dimension goes up from 0 to dim.

If you set the Sage verbosity level to a positive integer (using set_verbose(n)), then some extra messages will be printed.

EXAMPLES:

sage: # long time
sage: from sage.algebras.steenrod.steenrod_algebra_bases import steenrod_basis_error_check
sage: steenrod_basis_error_check(15, 2)
sage: steenrod_basis_error_check(15, 2, generic=True)
sage: steenrod_basis_error_check(40, 3)
sage: steenrod_basis_error_check(80, 5)

>>> from sage.all import *
>>> # long time
>>> from sage.algebras.steenrod.steenrod_algebra_bases import steenrod_basis_error_check
>>> steenrod_basis_error_check(Integer(15), Integer(2))
>>> steenrod_basis_error_check(Integer(15), Integer(2), generic=True)
>>> steenrod_basis_error_check(Integer(40), Integer(3))
>>> steenrod_basis_error_check(Integer(80), Integer(5))
sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(n, p=2, reverse=True)[source]

Decreasing list of degrees of the \(\xi_i\)’s, starting in degree \(n\).

INPUT:

  • n – integer

  • p – prime number (default: 2)

  • reverse – boolean (default: True)

OUTPUT: list of integers

When \(p=2\): decreasing list of the degrees of the \(\xi_i\)’s with degree at most \(n\).

At odd primes: decreasing list of these degrees, each divided by \(2(p-1)\).

If reverse is False, then return an increasing list rather than a decreasing one.

EXAMPLES:

sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17)
[15, 7, 3, 1]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17, reverse=False)
[1, 3, 7, 15]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(17, p=3)
[13, 4, 1]
sage: sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(400, p=17)
[307, 18, 1]

>>> from sage.all import *
>>> sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(Integer(17))
[15, 7, 3, 1]
>>> sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(Integer(17), reverse=False)
[1, 3, 7, 15]
>>> sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(Integer(17), p=Integer(3))
[13, 4, 1]
>>> sage.algebras.steenrod.steenrod_algebra_bases.xi_degrees(Integer(400), p=Integer(17))
[307, 18, 1]