Barratt-Eccles operad

Barratt Eccles elements

class clesto.barratt_eccles.barratt_eccles.BarrattEccles_element(data=None, torsion=None)

Elements in the Barratt-Eccles operad

Examples

>>> x = BarrattEccles_element()
>>> print(x)
0
>>> y = BarrattEccles_element({((1,3,2), (2,1,3)): -1})
>>> print(y)
- ((1,3,2),(2,1,3))

References

[BF]: C. Berger, and B. Fresse. “Combinatorial operad actions on cochains.” Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 137. No. 1. Cambridge University Press, 2004.

property arity

Arity of self

Defined as None if self is not homogeneous. The arity of a basis element agrees with arity of any of the symmetric group elements

>>> x = BarrattEccles_element({((1,3,2), (2,3,1)): 1})
>>> x.arity
3

property degree

Degree of self

Defined as None if self is not homogeneous. The degree of a basis surjection agrees with the cardinality of the tuple minus one.

>>> x = BarrattEccles_element({((1,3,2), (2,3,1)): 1})
>>> x.degree
1

property complexity

boundary()

Boundary of self.

>>> x = BarrattEccles_element({((1,3,2), (2,3,1), (1,2,3)): 1})
>>> print(x.boundary())
((2,3,1),(1,2,3)) - ((1,3,2),(1,2,3)) + ((1,3,2),(2,3,1))

__rmul__(other)

Left action: other * self

Left multiplication by a symmetric group element or an integer.

>>> x = BarrattEccles_element({((1,3,2), (2,3,1)): 1})
>>> print(-x)
- ((1,3,2),(2,3,1))
>>> rho = SymmetricRing_element({(2,3,1): 1})
>>> print(rho * x)
((2,1,3),(3,1,2))

orbit(representation='trivial')

Returns the preferred representative in the orbit of self

The preferred representative in the orbit of a basis element is one whose first symmetric group element if the identity.

The representation used can be either ‘trivial’ or ‘sign’.

>>> x = BarrattEccles_element({((1,3,2), (1,2,3)): 1})
>>> print(x.orbit())
((1,2,3),(1,3,2))
>>> print(x.orbit('sign'))
- ((1,2,3),(1,3,2))

compose(other, position)

Operadic compositions: self o_position other

We think of other being inserted into self the pair is ordered: self tensor other.

>>> x = BarrattEccles_element({((1, 2), (2, 1)): 1})
>>> print(x.compose(x, 1))
- ((1,2,3),(2,1,3),(3,2,1)) + ((1,2,3),(3,1,2),(3,2,1))

table_reduction()

Table reduction of self

>>> b = BarrattEccles_element({((1,2,3,4), (1,4,3,2), (1,2,4,3)): 1})
>>> print(b.table_reduction())
(1,2,4,2,4,3) + (1,2,4,3,2,3)

diagonal(r=1)

Alexander Whitney diagonal

Defined on basis elements by sum_i [pi_0,…pi_i] tensor [pi_i,…,pi_d]

class clesto.barratt_eccles.barratt_eccles.BarrattEccles

Class producing Barratt-Eccles elements of special interest.

static steenrod_product(arity, degree, torsion=None)

Returns a representative of the requested Steenrod product

Constructed recursively by mapping the minimal resolution W(r) of Z[S_r] to E(r). We use the chain homotopy equivalence of Surj(r) and Z defined using the chain contraction (i, p, s) relating Surj(r-1) and Surj(r).