uconf.models.barratt_eccles

Barratt–Eccles operad model on sequences of permutations.

Classes

BarrattEccles(n, base_ring)

Barratt--Eccles operad component in fixed arity.
class uconf.models.barratt_eccles.BarrattEccles(n, base_ring)[source]

Bases: CombinatorialFreeModule

Barratt–Eccles operad component in fixed arity.

Basis elements are tuples of permutations in S_n with no consecutive duplicates. Degenerate inputs map to zero, while malformed inputs raise.

Individual permutations follow Sage’s SymmetricGroup conventions: a list is one-line notation and a tuple is cycle notation. The two never coincide, so [1, 2] is the identity of S_2 while (1, 2) is the transposition.

EXAMPLES:

sage: from sage.all import QQ
sage: from uconf import BarrattEccles
sage: E2 = BarrattEccles(2, QQ)
sage: x = E2(([1, 2], [2, 1]))
sage: x.arity()
2
sage: E2.degree_on_basis(next(iter(x.support())))
1

The same element in cycle notation, with () for the identity:

sage: E2(((), (1, 2))) == x
True
connectivity: ClassVar[int] = 0

All components live in non-negative degrees.

__init__(n, base_ring)[source]

Initialize E_n over base_ring.

rho(data)[source]

Build the normalized element (id, sigma_1, sigma_1 sigma_2, ...).

Parameters:

data (tuple | list)

Return type:

Element

arity()[source]

Return the fixed arity of this operad component.

Return type:

int

static unit(base_ring)[source]

Return the operadic unit in arity 1.

Return type:

Element

static unit_key()[source]

Return the basis key of the unit element in arity 1.

Return type:

tuple

planar_basis_iter(d)[source]

Iterate over planar basis elements in degree d.

Parameters:

d (int)

Return type:

Iterator[Element]

basis_iter(d)[source]

Iterate over all basis elements in degree d.

Parameters:

d (int)

Return type:

Iterator[Element]

graded_basis[source]

/__w/uconf/uconf/src/uconf/models/barratt_eccles.py (starting at line 257) Return the Family of all basis elements in degree d.

Type:

File

graded_planar_basis[source]

/__w/uconf/uconf/src/uconf/models/barratt_eccles.py (starting at line 262) Return the Family of planar basis elements in degree d.

Type:

File

degree_on_basis(element)[source]

Return homological degree of a basis element.

Parameters:

element (tuple)

Return type:

int

static compose(x, i, y)[source]

Operadic composition x \circ_i y using shuffle/EZ lifting.

Parameters

x, y:

Barratt–Eccles elements over the same base ring.

i:

Input slot of x into which y is inserted.

Returns

BarrattEccles.Element

The composite in arity x.arity() + y.arity() - 1.

EXAMPLES:

sage: from sage.all import QQ
sage: from uconf import BarrattEccles
sage: x = BarrattEccles(2, QQ)(([1, 2],))
sage: y = BarrattEccles(2, QQ)(([1, 2], [2, 1]))
sage: z = BarrattEccles.compose(x, 1, y)
sage: z.arity()
3
sage: z.is_zero()
False
Parameters:
Return type:

BarrattEccles

BarrattEccles.Element[source]

alias of Element