Hochschild lattices

Hochschild lattices were defined in [Cha2020] as posets and shown to be congruence uniform lattices in [Com2021].

The name comes from the Hochschild polytopes introduced by Sanebdlidze in [San2009] for the study of free loop spaces in algebraic topology. The Hasse diagram of the Hochschild lattice is an orientation of the 1-skeleton of the Hochschild polytope.

For \(n \geq 1\), the cardinality of the Hochschild lattice \(H_n\) is \(2^{n - 2} \times (n + 3)\), starting with \(2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, \ldots\).

The underlying set of \(H_n\) consists of some words in the alphabet \((0,1,2)\), whose precise description can be found in [Com2021].

sage.combinat.posets.hochschild_lattice.hochschild_fan(n)[source]

Return Saneblidze’s fan for the Hochschild polytope.

The dual polytope is obtained from a standard simplex by a sequence of truncations.

See also

hochschild_simplicial_complex(), hochschild_lattice()

EXAMPLES:

sage: # needs sage.geometry.polyhedron
sage: from sage.combinat.posets.hochschild_lattice import hochschild_fan
sage: F = hochschild_fan(4); F
Rational polyhedral fan in 4-d lattice N
sage: F.f_vector()
(1, 11, 39, 56, 28)

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> from sage.combinat.posets.hochschild_lattice import hochschild_fan
>>> F = hochschild_fan(Integer(4)); F
Rational polyhedral fan in 4-d lattice N
>>> F.f_vector()
(1, 11, 39, 56, 28)
sage.combinat.posets.hochschild_lattice.hochschild_lattice(n)[source]

Return the Hochschild lattice \(H_n\).

INPUT:

  • \(n \geq 1\) – an integer

The cardinality of \(H_n\) is \(2^{n - 2} \times (n + 3)\).

See also

hochschild_simplicial_complex(), hochschild_fan()

EXAMPLES:

sage: P = posets.HochschildLattice(5); P
Finite lattice containing 64 elements
sage: P.degree_polynomial()
x^5 + 9*x^4*y + 22*x^3*y^2 + 22*x^2*y^3 + 9*x*y^4 + y^5

>>> from sage.all import *
>>> P = posets.HochschildLattice(Integer(5)); P
Finite lattice containing 64 elements
>>> P.degree_polynomial()
x^5 + 9*x^4*y + 22*x^3*y^2 + 22*x^2*y^3 + 9*x*y^4 + y^5
sage.combinat.posets.hochschild_lattice.hochschild_simplicial_complex(n)[source]

Return a simplicial complex related to the Hochschild lattice \(H_n\).

This is a pure spherical simplicial complex, whose flip graph is isomorphic to the Hasse diagram of \(H_n\).

See also

hochschild_fan(), hochschild_lattice()

EXAMPLES:

sage: # needs sage.geometry.polyhedron
sage: C = simplicial_complexes.HochschildSphere(3); C
Simplicial complex with 8 vertices and 12 facets
sage: H = C.flip_graph()
sage: P = posets.HochschildLattice(3)
sage: H.is_isomorphic(P.hasse_diagram().to_undirected())
True

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> C = simplicial_complexes.HochschildSphere(Integer(3)); C
Simplicial complex with 8 vertices and 12 facets
>>> H = C.flip_graph()
>>> P = posets.HochschildLattice(Integer(3))
>>> H.is_isomorphic(P.hasse_diagram().to_undirected())
True