Yokonuma-Hecke Algebras¶
AUTHORS:
- Travis Scrimshaw (2015-11): initial version
-
class
sage.algebras.yokonuma_hecke_algebra.
YokonumaHeckeAlgebra
(d, n, q, R)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
The Yokonuma-Hecke algebra
.
Let
be a commutative ring and
be a unit in
. The Yokonuma-Hecke algebra
is the associative, unital
-algebra generated by
and subject to the relations:
for all
,
,
,
, and
,
where
is the simple transposition
, along with the quadratic relation
Thus the Yokonuma-Hecke algebra can be considered a quotient of the framed braid group
, where
is the classical braid group on
strands, by the quadratic relations. Moreover, all of the algebra generators are invertible. In particular, we have
When we specialize
, we obtain the group algebra of the complex reflection group
. Moreover for
, the Yokonuma-Hecke algebra is equal to the :class`Iwahori-Hecke <IwahoriHeckeAlgebra>` of type
.
INPUT:
d
– the maximum power ofn
– the number of generatorsq
– (optional) an invertible element in a commutative ring; the default isR
– (optional) a commutative ring containingq
; the default is the parent of
EXAMPLES:
We construct
and do some computations:
sage: Y = algebras.YokonumaHecke(4, 3) sage: g1, g2, t1, t2, t3 = Y.algebra_generators() sage: g1 * g2 g[1,2] sage: t1 * g1 t1*g[1] sage: g2 * t2 t3*g[2] sage: g2 * t3 t2*g[2] sage: (g2 + t1) * (g1 + t2*t3) g[2,1] + t2*t3*g[2] + t1*g[1] + t1*t2*t3 sage: g1 * g1 1 - (1/4*q^-1-1/4*q)*g[1] - (1/4*q^-1-1/4*q)*t1*t2^3*g[1] - (1/4*q^-1-1/4*q)*t1^2*t2^2*g[1] - (1/4*q^-1-1/4*q)*t1^3*t2*g[1] sage: g2 * g1 * t1 t3*g[2,1]
We construct the elements
and show that they are idempotents:
sage: e1 = Y.e(1); e1 1/4 + 1/4*t1*t2^3 + 1/4*t1^2*t2^2 + 1/4*t1^3*t2 sage: e1 * e1 == e1 True sage: e2 = Y.e(2); e2 1/4 + 1/4*t2*t3^3 + 1/4*t2^2*t3^2 + 1/4*t2^3*t3 sage: e2 * e2 == e2 True
REFERENCES:
[CL13] Maria Chlouveraki and Sofia Lambropoulou. The Yokonuma-Hecke algebras and the HOMFLYPT polynomial. (2015) Arxiv 1204.1871v4. [CPdA14] Maria Chlouveraki and Loic Poulain d’Andecy. Representation theory of the Yokonuma-Hecke algebra. (2014) Arxiv 1302.6225v2. [ERH15] Jorge Espanoza and Steen Ryom-Hansen. Cell structures for the Yokonuma-Hecke algebra and the algebra of braids and ties. (2015) Arxiv 1506.00715. [JPdA15] N. Jacon and L. Poulain d’Andecy. An isomorphism theorem for Yokonuma-Hecke algebras and applications to link invariants. (2015) Arxiv 1501.06389v3. -
class
Element
(M, x)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModuleElement
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s
__call__()
method.TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c']) sage: B = F.basis() sage: f = B['a'] + 3*B['c']; f B['a'] + 3*B['c'] sage: f == loads(dumps(f)) True
-
inverse
()¶ Return the inverse if
self
is a basis element.EXAMPLES:
sage: Y = algebras.YokonumaHecke(3, 3) sage: t = prod(Y.t()); t t1*t2*t3 sage: ~t t1^2*t2^2*t3^2 sage: [3*~(t*g) for g in Y.g()] [(q^-1+q)*t2*t3^2 + (q^-1+q)*t1*t3^2 + (q^-1+q)*t1^2*t2^2*t3^2 + 3*t1^2*t2^2*t3^2*g[1], (q^-1+q)*t1^2*t3 + (q^-1+q)*t1^2*t2 + (q^-1+q)*t1^2*t2^2*t3^2 + 3*t1^2*t2^2*t3^2*g[2]]
-
-
YokonumaHeckeAlgebra.
algebra_generators
()¶ Return the algebra generators of
self
.EXAMPLES:
sage: Y = algebras.YokonumaHecke(5, 3) sage: dict(Y.algebra_generators()) {'g1': g[1], 'g2': g[2], 't1': t1, 't2': t2, 't3': t3}
-
YokonumaHeckeAlgebra.
e
(i)¶ Return the element
.
EXAMPLES:
sage: Y = algebras.YokonumaHecke(4, 3) sage: Y.e(1) 1/4 + 1/4*t1*t2^3 + 1/4*t1^2*t2^2 + 1/4*t1^3*t2 sage: Y.e(2) 1/4 + 1/4*t2*t3^3 + 1/4*t2^2*t3^2 + 1/4*t2^3*t3
-
YokonumaHeckeAlgebra.
g
(i=None)¶ Return the generator(s)
.
INPUT:
i
– (default:None
) the generatoror if
None
, then the list of all generators
EXAMPLES:
sage: Y = algebras.YokonumaHecke(8, 3) sage: Y.g(1) g[1] sage: Y.g() [g[1], g[2]]
-
YokonumaHeckeAlgebra.
gens
()¶ Return the generators of
self
.EXAMPLES:
sage: Y = algebras.YokonumaHecke(5, 3) sage: Y.gens() (g[1], g[2], t1, t2, t3)
-
YokonumaHeckeAlgebra.
inverse_g
(i)¶ Return the inverse of the generator
.
From the quadratic relation, we have
EXAMPLES:
sage: Y = algebras.YokonumaHecke(2, 4) sage: [2*Y.inverse_g(i) for i in range(1, 4)] [(q^-1+q) + 2*g[1] + (q^-1+q)*t1*t2, (q^-1+q) + 2*g[2] + (q^-1+q)*t2*t3, (q^-1+q) + 2*g[3] + (q^-1+q)*t3*t4]
-
YokonumaHeckeAlgebra.
one_basis
()¶ Return the index of the basis element of
.
EXAMPLES:
sage: Y = algebras.YokonumaHecke(5, 3) sage: Y.one_basis() ((0, 0, 0), [1, 2, 3])
-
YokonumaHeckeAlgebra.
product_on_basis
(m1, m2)¶ Return the product of the basis elements indexed by
m1
andm2
.EXAMPLES:
sage: Y = algebras.YokonumaHecke(4, 3) sage: m = ((1, 0, 2), Permutations(3)([2,1,3])) sage: 4 * Y.product_on_basis(m, m) -(q^-1-q)*t2^2*g[1] + 4*t1*t2 - (q^-1-q)*t1*t2*g[1] - (q^-1-q)*t1^2*g[1] - (q^-1-q)*t1^3*t2^3*g[1]
Check that we apply the permutation correctly on
:
sage: Y = algebras.YokonumaHecke(4, 3) sage: g1, g2, t1, t2, t3 = Y.algebra_generators() sage: g21 = g2 * g1 sage: g21 * t1 t3*g[2,1]
-
YokonumaHeckeAlgebra.
t
(i=None)¶ Return the generator(s)
.
INPUT:
i
– (default:None
) the generatoror if
None
, then the list of all generators
EXAMPLES:
sage: Y = algebras.YokonumaHecke(8, 3) sage: Y.t(2) t2 sage: Y.t() [t1, t2, t3]