Miscellaneous functions for the Steenrod algebra and its elements

AUTHORS:

  • John H. Palmieri (2008-07-30): initial version (as the file steenrod_algebra_element.py)
  • John H. Palmieri (2010-06-30): initial version of steenrod_misc.py. Implemented profile functions. Moved most of the methods for elements to the Element subclass of sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic.

The main functions here are

  • get_basis_name(). This function takes a string as input and attempts to interpret it as the name of a basis for the Steenrod algebra; it returns the canonical name attached to that basis. This allows for the use of synonyms when defining bases, while the resulting algebras will be identical.
  • normalize_profile(). This function returns the canonical (and hashable) description of any profile function. See sage.algebras.steenrod.steenrod_algebra and SteenrodAlgebra for information on profile functions.
  • functions named *_mono_to_string where * is a basis name (milnor_mono_to_string(), etc.). These convert tuples representing basis elements to strings, for _repr_ and _latex_ methods.
sage.algebras.steenrod.steenrod_algebra_misc.arnonA_long_mono_to_string(mono, latex=False, p=2)

Alternate string representation of element of Arnon’s A basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of non-negative integers (m,k) with \(m >= k\)
  • latex - boolean (optional, default False), if true, output LaTeX string

OUTPUT: string - concatenation of strings of the form Sq(2^m)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import arnonA_long_mono_to_string
sage: arnonA_long_mono_to_string(((1,2),(3,0)))
'Sq^{8} Sq^{4} Sq^{2} Sq^{1}'
sage: arnonA_long_mono_to_string(((1,2),(3,0)),latex=True)
'\text{Sq}^{8} \text{Sq}^{4} \text{Sq}^{2} \text{Sq}^{1}'

The empty tuple represents the unit element:

sage: arnonA_long_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.arnonA_mono_to_string(mono, latex=False, p=2)

String representation of element of Arnon’s A basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of non-negative integers (m,k) with \(m >= k\)
  • latex - boolean (optional, default False), if true, output LaTeX string

OUTPUT: string - concatenation of strings of the form X^{m}_{k} for each pair (m,k)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import arnonA_mono_to_string
sage: arnonA_mono_to_string(((1,2),(3,0)))
'X^{1}_{2} X^{3}_{0}'
sage: arnonA_mono_to_string(((1,2),(3,0)),latex=True)
'X^{1}_{2} X^{3}_{0}'

The empty tuple represents the unit element:

sage: arnonA_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.comm_long_mono_to_string(mono, latex=False, p=2)

Alternate string representation of element of a commutator basis.

Okay in low dimensions, but gets unwieldy as the dimension increases.

INPUT:

  • mono - tuple of pairs of integers (s,t) with \(s >= 0\), \(t > 0\)
  • latex - boolean (optional, default False), if true, output LaTeX string
  • p - positive prime number (optional, default 2).

OUTPUT: string - concatenation of strings of the form s_{2^s ... 2^(s+t-1)} for each pair (s,t)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import comm_long_mono_to_string
sage: comm_long_mono_to_string(((1,2),(0,3)))
's_{24} s_{124}'
sage: comm_long_mono_to_string(((1,2),(0,3)),latex=True)
's_{24} s_{124}'
sage: comm_long_mono_to_string(((1, 4), (((1,2), 1),((0,3), 2))), p=5)
'Q_{1} Q_{4} s_{5,25} s_{1,5,25}^2'
sage: comm_long_mono_to_string(((1, 4), (((1,2), 1),((0,3), 2))), latex=True, p=3)
'Q_{1} Q_{4} s_{3,9} s_{1,3,9}^{2}'

The empty tuple represents the unit element:

sage: comm_long_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.comm_mono_to_string(mono, latex=False, p=2)

String representation of element of a commutator basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of integers (s,t) with \(s >= 0\), \(t > 0\)
  • latex - boolean (optional, default False), if true, output LaTeX string
  • p - positive prime number (optional, default 2)

OUTPUT: string - concatenation of strings of the form c_{s,t} for each pair (s,t)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import comm_mono_to_string
sage: comm_mono_to_string(((1,2),(0,3)), p=2)
'c_{1,2} c_{0,3}'
sage: comm_mono_to_string(((1,2),(0,3)), latex=True, p=2)
'c_{1,2} c_{0,3}'
sage: comm_mono_to_string(((1, 4), (((1,2), 1),((0,3), 2))), p=5)
'Q_{1} Q_{4} c_{1,2} c_{0,3}^2'
sage: comm_mono_to_string(((1, 4), (((1,2), 1),((0,3), 2))), latex=True, p=5)
'Q_{1} Q_{4} c_{1,2} c_{0,3}^{2}'

The empty tuple represents the unit element:

sage: comm_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.convert_perm(m)

Convert tuple m of non-negative integers to a permutation in one-line form.

INPUT:

  • m - tuple of non-negative integers with no repetitions

OUTPUT: list - conversion of m to a permutation of the set 1,2,...,len(m)

If m=(3,7,4), then one can view m as representing the permutation of the set \((3,4,7)\) sending 3 to 3, 4 to 7, and 7 to 4. This function converts m to the list [1,3,2], which represents essentially the same permutation, but of the set \((1,2,3)\). This list can then be passed to Permutation, and its signature can be computed.

EXAMPLES:

sage: sage.algebras.steenrod.steenrod_algebra_misc.convert_perm((3,7,4))
[1, 3, 2]
sage: sage.algebras.steenrod.steenrod_algebra_misc.convert_perm((5,0,6,3))
[3, 1, 4, 2]
sage.algebras.steenrod.steenrod_algebra_misc.get_basis_name(basis, p)

Return canonical basis named by string basis at the prime p.

INPUT:

  • basis - string
  • p - positive prime number

OUTPUT:

  • basis_name - string

Specify the names of the implemented bases. The input is converted to lower-case, then processed to return the canonical name for the basis.

For the Milnor and Serre-Cartan bases, use the list of synonyms defined by the variables _steenrod_milnor_basis_names and _steenrod_serre_cartan_basis_names. Their canonical names are ‘milnor’ and ‘serre-cartan’, respectively.

For the other bases, use pattern-matching rather than a list of synonyms:

  • Search for ‘wood’ and ‘y’ or ‘wood’ and ‘z’ to get the Wood bases. Canonical names ‘woody’, ‘woodz’.
  • Search for ‘arnon’ and ‘c’ for the Arnon C basis. Canonical name: ‘arnonc’.
  • Search for ‘arnon’ (and no ‘c’) for the Arnon A basis. Also see if ‘long’ is present, for the long form of the basis. Canonical names: ‘arnona’, ‘arnona_long’.
  • Search for ‘wall’ for the Wall basis. Also see if ‘long’ is present. Canonical names: ‘wall’, ‘wall_long’.
  • Search for ‘pst’ for P^s_t bases, then search for the order type: ‘rlex’, ‘llex’, ‘deg’, ‘revz’. Canonical names: ‘pst_rlex’, ‘pst_llex’, ‘pst_deg’, ‘pst_revz’.
  • For commutator types, search for ‘comm’, an order type, and also check to see if ‘long’ is present. Canonical names: ‘comm_rlex’, ‘comm_llex’, ‘comm_deg’, ‘comm_revz’, ‘comm_rlex_long’, ‘comm_llex_long’, ‘comm_deg_long’, ‘comm_revz_long’.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import get_basis_name
sage: get_basis_name('adem', 2)
'serre-cartan'
sage: get_basis_name('milnor', 2)
'milnor'
sage: get_basis_name('MiLNoR', 5)
'milnor'
sage: get_basis_name('pst-llex', 2)
'pst_llex'
sage: get_basis_name('wood_abcdedfg_y', 2)
'woody'
sage: get_basis_name('wood', 2)
Traceback (most recent call last):
...
ValueError: wood is not a recognized basis at the prime 2.
sage: get_basis_name('arnon--hello--long', 2)
'arnona_long'
sage: get_basis_name('arnona_long', p=5)
Traceback (most recent call last):
...
ValueError: arnona_long is not a recognized basis at the prime 5.
sage: get_basis_name('NOT_A_BASIS', 2)
Traceback (most recent call last):
...
ValueError: not_a_basis is not a recognized basis at the prime 2.
sage.algebras.steenrod.steenrod_algebra_misc.is_valid_profile(profile, truncation_type, p=2)

True if profile, together with truncation_type, is a valid profile at the prime \(p\).

INPUT:

  • profile - when \(p=2\), a tuple or list of numbers; when \(p\) is odd, a pair of such lists
  • truncation_type - either 0 or \(\infty\)
  • \(p\) - prime number, optional, default 2

OUTPUT: True if the profile function is valid, False otherwise.

See the documentation for sage.algebras.steenrod.steenrod_algebra for descriptions of profile functions and how they correspond to sub-Hopf algebras of the Steenrod algebra. Briefly: at the prime 2, a profile function \(e\) is valid if it satisfies the condition

  • \(e(r) \geq \min( e(r-i) - i, e(i))\) for all \(0 < i < r\).

At odd primes, a pair of profile functions \(e\) and \(k\) are valid if they satisfy

  • \(e(r) \geq \min( e(r-i) - i, e(i))\) for all \(0 < i < r\).
  • if \(k(i+j) = 1\), then either \(e(i) \leq j\) or \(k(j) = 1\) for all \(i \geq 1\), \(j \geq 0\).

In this function, profile functions are lists or tuples, and truncation_type is appended as the last element of the list \(e\) before testing.

EXAMPLES:

\(p=2\):

sage: from sage.algebras.steenrod.steenrod_algebra_misc import is_valid_profile
sage: is_valid_profile([3,2,1], 0)
True
sage: is_valid_profile([3,2,1], Infinity)
True
sage: is_valid_profile([1,2,3], 0)
False
sage: is_valid_profile([6,2,0], Infinity)
False
sage: is_valid_profile([0,3], 0)
False
sage: is_valid_profile([0,0,4], 0)
False
sage: is_valid_profile([0,0,0,4,0], 0)
True

Odd primes:

sage: is_valid_profile(([0,0,0], [2,1,1,1,2,2]), 0, p=3)
True
sage: is_valid_profile(([1], [2,2]), 0, p=3)
True
sage: is_valid_profile(([1], [2]), 0, p=7)
False
sage: is_valid_profile(([1,2,1], []), 0, p=7)
True
sage.algebras.steenrod.steenrod_algebra_misc.milnor_mono_to_string(mono, latex=False, p=2)

String representation of element of the Milnor basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - if \(p=2\), tuple of non-negative integers (a,b,c,...); if \(p>2\), pair of tuples of non-negative integers ((e0, e1, e2, ...), (r1, r2, ...))
  • latex - boolean (optional, default False), if true, output LaTeX string
  • p - positive prime number (optional, default 2)

OUTPUT: rep - string

This returns a string like Sq(a,b,c,...) when p=2, or a string like Q_e0 Q_e1 Q_e2 ... P(r1, r2, ...) when p is odd.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import milnor_mono_to_string
sage: milnor_mono_to_string((1,2,3,4))
'Sq(1,2,3,4)'
sage: milnor_mono_to_string((1,2,3,4),latex=True)
'\text{Sq}(1,2,3,4)'
sage: milnor_mono_to_string(((1,0), (2,3,1)), p=3)
'Q_{1} Q_{0} P(2,3,1)'
sage: milnor_mono_to_string(((1,0), (2,3,1)), latex=True, p=3)
'Q_{1} Q_{0} \mathcal{P}(2,3,1)'

The empty tuple represents the unit element:

sage: milnor_mono_to_string(())
'1'
sage: milnor_mono_to_string((), p=5)
'1'
sage.algebras.steenrod.steenrod_algebra_misc.normalize_profile(profile, precision=None, truncation_type='auto', p=2)

Given a profile function and related data, return it in a standard form, suitable for hashing and caching as data defining a sub-Hopf algebra of the Steenrod algebra.

INPUT:

  • profile - a profile function in form specified below
  • precision - integer or None, optional, default None
  • truncation_type - 0 or \(\infty\) or ‘auto’, optional, default ‘auto’
  • \(p\) - prime, optional, default 2

OUTPUT: a triple profile, precision, truncation_type, in standard form as described below.

The “standard form” is as follows: profile should be a tuple of integers (or \(\infty\)) with no trailing zeroes when \(p=2\), or a pair of such when \(p\) is odd. precision should be a positive integer. truncation_type should be 0 or \(\infty\). Furthermore, this must be a valid profile, as determined by the funtion is_valid_profile(). See also the documentation for the module sage.algebras.steenrod.steenrod_algebra for information about profile functions.

For the inputs: when \(p=2\), profile should be a valid profile function, and it may be entered in any of the following forms:

  • a list or tuple, e.g., [3,2,1,1]
  • a function from positive integers to non-negative integers (and \(\infty\)), e.g., lambda n: n+2. This corresponds to the list [3, 4, 5, ...].
  • None or Infinity - use this for the profile function for the whole Steenrod algebra. This corresponds to the list [Infinity, Infinity, Infinity, ...]

To make this hashable, it gets turned into a tuple. In the first case it is clear how to do this; also in this case, precision is set to be one more than the length of this tuple. In the second case, construct a tuple of length one less than precision (default value 100). In the last case, the empty tuple is returned and precision is set to 1.

Once a sub-Hopf algebra of the Steenrod algebra has been defined using such a profile function, if the code requires any remaining terms (say, terms after the 100th), then they are given by truncation_type if that is 0 or \(\infty\). If truncation_type is ‘auto’, then in the case of a tuple, it gets set to 0, while for the other cases it gets set to \(\infty\).

See the examples below.

When \(p\) is odd, profile is a pair of “functions”, so it may have the following forms:

  • a pair of lists or tuples, the second of which takes values in the set \(\{1,2\}\), e.g., ([3,2,1,1], [1,1,2,2,1]).
  • a pair of functions, one (called \(e\)) from positive integers to non-negative integers (and \(\infty\)), one (called \(k\)) from non-negative integers to the set \(\{1,2\}\), e.g., (lambda n: n+2, lambda n: 1). This corresponds to the pair ([3, 4, 5, ...], [1, 1, 1, ...]).
  • None or Infinity - use this for the profile function for the whole Steenrod algebra. This corresponds to the pair ([Infinity, Infinity, Infinity, ...], [2, 2, 2, ...]).

You can also mix and match the first two, passing a pair with first entry a list and second entry a function, for instance. The values of precision and truncation_type are determined by the first entry.

EXAMPLES:

\(p=2\):

sage: from sage.algebras.steenrod.steenrod_algebra_misc import normalize_profile
sage: normalize_profile([1,2,1,0,0])
((1, 2, 1), 0)

The full mod 2 Steenrod algebra:

sage: normalize_profile(Infinity)
((), +Infinity)
sage: normalize_profile(None)
((), +Infinity)
sage: normalize_profile(lambda n: Infinity)
((), +Infinity)

The precision argument has no effect when the first argument is a list or tuple:

sage: normalize_profile([1,2,1,0,0], precision=12)
((1, 2, 1), 0)

If the first argument is a function, then construct a list of length one less than precision, by plugging in the numbers 1, 2, ..., precision - 1:

sage: normalize_profile(lambda n: 4-n, precision=4)
((3, 2, 1), +Infinity)
sage: normalize_profile(lambda n: 4-n, precision=4, truncation_type=0)
((3, 2, 1), 0)

Negative numbers in profile functions are turned into zeroes:

sage: normalize_profile(lambda n: 4-n, precision=6)
((3, 2, 1, 0, 0), +Infinity)

If it doesn’t give a valid profile, an error is raised:

sage: normalize_profile(lambda n: 3, precision=4, truncation_type=0)
Traceback (most recent call last):
...
ValueError: Invalid profile
sage: normalize_profile(lambda n: 3, precision=4, truncation_type = Infinity)
((3, 3, 3), +Infinity)

When \(p\) is odd, the behavior is similar:

sage: normalize_profile(([2,1], [2,2,2]), p=13)
(((2, 1), (2, 2, 2)), 0)

The full mod \(p\) Steenrod algebra:

sage: normalize_profile(None, p=7)
(((), ()), +Infinity)
sage: normalize_profile(Infinity, p=11)
(((), ()), +Infinity)
sage: normalize_profile((lambda n: Infinity, lambda n: 2), p=17)
(((), ()), +Infinity)

Note that as at the prime 2, the precision argument has no effect on a list or tuple in either entry of profile. If truncation_type is ‘auto’, then it gets converted to either 0 or +Infinity depending on the first entry of profile:

sage: normalize_profile(([2,1], [2,2,2]), precision=84, p=13)
(((2, 1), (2, 2, 2)), 0)
sage: normalize_profile((lambda n: 0, lambda n: 2), precision=4, p=11)
(((0, 0, 0), ()), +Infinity)
sage: normalize_profile((lambda n: 0, (1,1,1,1,1,1,1)), precision=4, p=11)
(((0, 0, 0), (1, 1, 1, 1, 1, 1, 1)), +Infinity)
sage: normalize_profile(((4,3,2,1), lambda n: 2), precision=6, p=11)
(((4, 3, 2, 1), (2, 2, 2, 2, 2)), 0)
sage: normalize_profile(((4,3,2,1), lambda n: 1), precision=3, p=11, truncation_type=Infinity)
(((4, 3, 2, 1), (1, 1)), +Infinity)

As at the prime 2, negative numbers in the first component are converted to zeroes. Numbers in the second component must be either 1 and 2, or else an error is raised:

sage: normalize_profile((lambda n: -n, lambda n: 1), precision=4, p=11)
(((0, 0, 0), (1, 1, 1)), +Infinity)
sage: normalize_profile([[0,0,0], [1,2,3,2,1]], p=11)
Traceback (most recent call last):
...
ValueError: Invalid profile
sage.algebras.steenrod.steenrod_algebra_misc.pst_mono_to_string(mono, latex=False, p=2)

String representation of element of a \(P^s_t\)-basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of integers (s,t) with \(s >= 0\), \(t > 0\)
  • latex - boolean (optional, default False), if true, output LaTeX string
  • p - positive prime number (optional, default 2).

OUTPUT: string - concatenation of strings of the form P^{s}_{t} for each pair (s,t)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import pst_mono_to_string
sage: pst_mono_to_string(((1,2),(0,3)), p=2)
'P^{1}_{2} P^{0}_{3}'
sage: pst_mono_to_string(((1,2),(0,3)),latex=True, p=2)
'P^{1}_{2} P^{0}_{3}'
sage: pst_mono_to_string(((1,4), (((1,2), 1),((0,3), 2))), p=3)
'Q_{1} Q_{4} P^{1}_{2} (P^{0}_{3})^2'
sage: pst_mono_to_string(((1,4), (((1,2), 1),((0,3), 2))), latex=True, p=3)
'Q_{1} Q_{4} P^{1}_{2} (P^{0}_{3})^{2}'

The empty tuple represents the unit element:

sage: pst_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.serre_cartan_mono_to_string(mono, latex=False, p=2)

String representation of element of the Serre-Cartan basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of positive integers (a,b,c,...) when \(p=2\), or tuple (e0, n1, e1, n2, ...) when \(p>2\), where each ei is 0 or 1, and each ni is positive
  • latex - boolean (optional, default False), if true, output LaTeX string
  • p - positive prime number (optional, default 2)

OUTPUT: rep - string

This returns a string like Sq^{a} Sq^{b} Sq^{c} ... when \(p=2\), or a string like \beta^{e0} P^{n1} \beta^{e1} P^{n2} ... when \(p\) is odd.

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import serre_cartan_mono_to_string
sage: serre_cartan_mono_to_string((1,2,3,4))
'Sq^{1} Sq^{2} Sq^{3} Sq^{4}'
sage: serre_cartan_mono_to_string((1,2,3,4),latex=True)
'\\text{Sq}^{1} \\text{Sq}^{2} \\text{Sq}^{3} \\text{Sq}^{4}'
sage: serre_cartan_mono_to_string((0,5,1,1,0), p=3)
'P^{5} beta P^{1}'
sage: serre_cartan_mono_to_string((0,5,1,1,0), p=3, latex=True)
'\\mathcal{P}^{5} \\beta \\mathcal{P}^{1}'

The empty tuple represents the unit element 1:

sage: serre_cartan_mono_to_string(())
'1'
sage: serre_cartan_mono_to_string((), p=7)
'1'
sage.algebras.steenrod.steenrod_algebra_misc.wall_long_mono_to_string(mono, latex=False)

Alternate string representation of element of Wall’s basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of non-negative integers (m,k) with \(m >= k\)
  • latex - boolean (optional, default False), if true, output LaTeX string

OUTPUT: string - concatenation of strings of the form Sq^(2^m)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import wall_long_mono_to_string
sage: wall_long_mono_to_string(((1,2),(3,0)))
'Sq^{1} Sq^{2} Sq^{4} Sq^{8}'
sage: wall_long_mono_to_string(((1,2),(3,0)),latex=True)
'\text{Sq}^{1} \text{Sq}^{2} \text{Sq}^{4} \text{Sq}^{8}'

The empty tuple represents the unit element:

sage: wall_long_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.wall_mono_to_string(mono, latex=False)

String representation of element of Wall’s basis.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of non-negative integers (m,k) with \(m >= k\)
  • latex - boolean (optional, default False), if true, output LaTeX string

OUTPUT: string - concatenation of strings Q^{m}_{k} for each pair (m,k)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import wall_mono_to_string
sage: wall_mono_to_string(((1,2),(3,0)))
'Q^{1}_{2} Q^{3}_{0}'
sage: wall_mono_to_string(((1,2),(3,0)),latex=True)
'Q^{1}_{2} Q^{3}_{0}'

The empty tuple represents the unit element:

sage: wall_mono_to_string(())
'1'
sage.algebras.steenrod.steenrod_algebra_misc.wood_mono_to_string(mono, latex=False)

String representation of element of Wood’s Y and Z bases.

This is used by the _repr_ and _latex_ methods.

INPUT:

  • mono - tuple of pairs of non-negative integers (s,t)
  • latex - boolean (optional, default False), if true, output LaTeX string

OUTPUT: string - concatenation of strings of the form Sq^{2^s (2^{t+1}-1)} for each pair (s,t)

EXAMPLES:

sage: from sage.algebras.steenrod.steenrod_algebra_misc import wood_mono_to_string
sage: wood_mono_to_string(((1,2),(3,0)))
'Sq^{14} Sq^{8}'
sage: wood_mono_to_string(((1,2),(3,0)),latex=True)
'\text{Sq}^{14} \text{Sq}^{8}'

The empty tuple represents the unit element:

sage: wood_mono_to_string(())
'1'

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