Blob Algebras

AUTHORS:

  • Travis Scrimshaw (2020-05-16): Initial version

class sage.combinat.blob_algebra.BlobAlgebra(k, q1, q2, q3, base_ring, prefix)[source]

Bases: CombinatorialFreeModule

The blob algebra.

The blob algebra (also known as the Temperley-Lieb algebra of type \(B\) in [ILZ2018], but is a quotient of the Temperley-Lieb algebra of type \(B\) defined in [Graham1985]) is a diagram-type algebra introduced in [MS1994] whose basis consists of Temperley-Lieb diagrams, noncrossing perfect matchings, that may contain blobs on strands that can be deformed so that the blob touches the left side (which we can think of as a frozen pole).

The form we give here has 3 parameters, the natural one from the Temperley-Lieb algebra, one for the idempotent relation, and one for a loop with a blob.

INPUT:

  • k – the order

  • q1 – the loop parameter

  • q2 – the idempotent parameter

  • q3 – the blob loop parameter

EXAMPLES:

sage: R.<q,r,s> = ZZ[]
sage: B4 = algebras.Blob(4, q, r, s)
sage: B = sorted(B4.basis())
sage: B[14]
B({{-4, -3}}, {{-2, -1}, {1, 2}, {3, 4}})
sage: B[40]
B({{3, 4}}, {{-4, -3}, {-2, -1}, {1, 2}})
sage: B[14] * B[40]
q*r*s*B({}, {{-4, -3}, {-2, -1}, {1, 2}, {3, 4}})

>>> from sage.all import *
>>> R = ZZ['q, r, s']; (q, r, s,) = R._first_ngens(3)
>>> B4 = algebras.Blob(Integer(4), q, r, s)
>>> B = sorted(B4.basis())
>>> B[Integer(14)]
B({{-4, -3}}, {{-2, -1}, {1, 2}, {3, 4}})
>>> B[Integer(40)]
B({{3, 4}}, {{-4, -3}, {-2, -1}, {1, 2}})
>>> B[Integer(14)] * B[Integer(40)]
q*r*s*B({}, {{-4, -3}, {-2, -1}, {1, 2}, {3, 4}})

REFERENCES:

one_basis()[source]

Return the index of the basis element \(1\).

EXAMPLES:

sage: R.<q,r,s> = ZZ[]
sage: B4 = algebras.Blob(4, q, r, s)
sage: B4.one_basis()
({}, {{-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}})

>>> from sage.all import *
>>> R = ZZ['q, r, s']; (q, r, s,) = R._first_ngens(3)
>>> B4 = algebras.Blob(Integer(4), q, r, s)
>>> B4.one_basis()
({}, {{-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}})
order()[source]

Return the order of self.

The order of a partition algebra is defined as half of the number of nodes in the diagrams.

EXAMPLES:

sage: R.<q,r,s> = ZZ[]
sage: B4 = algebras.Blob(4, q, r, s)
sage: B4.order()
4

>>> from sage.all import *
>>> R = ZZ['q, r, s']; (q, r, s,) = R._first_ngens(3)
>>> B4 = algebras.Blob(Integer(4), q, r, s)
>>> B4.order()
4
product_on_basis(top, bot)[source]

Return the product of the basis elements indexed by top and bot.

EXAMPLES:

sage: R.<q,r,s> = ZZ[]
sage: B4 = algebras.Blob(4, q, r, s)
sage: B = B4.basis()
sage: BD = sorted(B.keys())
sage: BD[14]
({{-4, -3}}, {{-2, -1}, {1, 2}, {3, 4}})
sage: BD[40]
({{3, 4}}, {{-4, -3}, {-2, -1}, {1, 2}})
sage: B4.product_on_basis(BD[14], BD[40])
q*r*s*B({}, {{-4, -3}, {-2, -1}, {1, 2}, {3, 4}})
sage: all(len((x*y).support()) == 1 for x in B for y in B)
True

>>> from sage.all import *
>>> R = ZZ['q, r, s']; (q, r, s,) = R._first_ngens(3)
>>> B4 = algebras.Blob(Integer(4), q, r, s)
>>> B = B4.basis()
>>> BD = sorted(B.keys())
>>> BD[Integer(14)]
({{-4, -3}}, {{-2, -1}, {1, 2}, {3, 4}})
>>> BD[Integer(40)]
({{3, 4}}, {{-4, -3}, {-2, -1}, {1, 2}})
>>> B4.product_on_basis(BD[Integer(14)], BD[Integer(40)])
q*r*s*B({}, {{-4, -3}, {-2, -1}, {1, 2}, {3, 4}})
>>> all(len((x*y).support()) == Integer(1) for x in B for y in B)
True
class sage.combinat.blob_algebra.BlobDiagram(parent, marked, unmarked)[source]

Bases: Element

A blob diagram.

A blob diagram consists of a perfect matching of the set \(\{1, \ldots, n\} \sqcup \{-1, \ldots, -n\}\) such that the result is a noncrossing matching (a Temperley-Lieb diagram), divided into two sets of pairs: one for the pairs with blobs and one for those without. The blobed pairs must either be either the leftmost propagating strand or to the left of it and not nested.

temperley_lieb_diagram()[source]

Return the Temperley-Lieb diagram corresponding to self.

EXAMPLES:

sage: from sage.combinat.blob_algebra import BlobDiagrams
sage: BD4 = BlobDiagrams(4)
sage: B = BD4([[1,-3]], [[2,-4], [3,4], [-1,-2]])
sage: B.temperley_lieb_diagram()
{{-4, 2}, {-3, 1}, {-2, -1}, {3, 4}}

>>> from sage.all import *
>>> from sage.combinat.blob_algebra import BlobDiagrams
>>> BD4 = BlobDiagrams(Integer(4))
>>> B = BD4([[Integer(1),-Integer(3)]], [[Integer(2),-Integer(4)], [Integer(3),Integer(4)], [-Integer(1),-Integer(2)]])
>>> B.temperley_lieb_diagram()
{{-4, 2}, {-3, 1}, {-2, -1}, {3, 4}}
class sage.combinat.blob_algebra.BlobDiagrams(n)[source]

Bases: Parent, UniqueRepresentation

The set of all blob diagrams.

Element[source]

alias of BlobDiagram

base_set()[source]

Return the base set of self.

EXAMPLES:

sage: from sage.combinat.blob_algebra import BlobDiagrams
sage: BD4 = BlobDiagrams(4)
sage: sorted(BD4.base_set())
[-4, -3, -2, -1, 1, 2, 3, 4]

>>> from sage.all import *
>>> from sage.combinat.blob_algebra import BlobDiagrams
>>> BD4 = BlobDiagrams(Integer(4))
>>> sorted(BD4.base_set())
[-4, -3, -2, -1, 1, 2, 3, 4]
cardinality()[source]

Return the cardinality of self.

EXAMPLES:

sage: from sage.combinat.blob_algebra import BlobDiagrams
sage: BD4 = BlobDiagrams(4)
sage: BD4.cardinality()
70

>>> from sage.all import *
>>> from sage.combinat.blob_algebra import BlobDiagrams
>>> BD4 = BlobDiagrams(Integer(4))
>>> BD4.cardinality()
70
order()[source]

Return the order of self.

EXAMPLES:

sage: from sage.combinat.blob_algebra import BlobDiagrams
sage: BD4 = BlobDiagrams(4)
sage: BD4.order()
4

>>> from sage.all import *
>>> from sage.combinat.blob_algebra import BlobDiagrams
>>> BD4 = BlobDiagrams(Integer(4))
>>> BD4.order()
4