Iwahori-Hecke Algebras

AUTHORS:

  • Daniel Bump, Nicolas Thiery (2010): Initial version
  • Brant Jones, Travis Scrimshaw, Andrew Mathas (2013): Moved into the category framework and implemented the Kazhdan-Lusztig C and C^{\prime} bases
class sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra(W, q1, q2, base_ring)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

Returns the Iwahori-Hecke algebra of the Coxeter group W with the specified parameters.

INPUT:

  • W – a Coxeter group or Cartan type
  • q1 – a parameter

OPTIONAL ARGUMENTS:

  • q2 – (default -1) another parameter
  • base_ring – (default q1.parent()) a ring containing q1 and q2

The Iwahori-Hecke algebra [I64] is a deformation of the group algebra of a Weyl group or, more generally, a Coxeter group. These algebras are defined by generators and relations and they depend on a deformation parameter q. Taking q = 1, as in the following example, gives a ring isomorphic to the group algebra of the corresponding Coxeter group.

Let (W, S) be a Coxeter system and let R be a commutative ring containing elements q_1 and q_2. Then the Iwahori-Hecke algebra H = H_{q_1,q_2}(W,S) of (W,S) with parameters q_1 and q_2 is the unital associative algebra with generators \{T_s \mid s\in S\} and relations:

\begin{aligned}
    (T_s - q_1)(T_s - q_2) &= 0\\
    T_r T_s T_r \cdots &= T_s T_r T_s \cdots,
\end{aligned}

where the number of terms on either side of the second relations (the braid relations) is the order of rs in the Coxeter group W, for r,s \in S.

Iwahori-Hecke algebras are fundamental in many areas of mathematics, ranging from the representation theory of Lie groups and quantum groups, to knot theory and statistical mechanics. For more information see, for example, [KL79], [HKP], [J87] and Wikipedia article Iwahori-Hecke_algebra.

Bases

A reduced expression for an element w \in W is any minimal length word w = s_1 \cdots s_k, with s_i \in S. If w = s_1 \cdots s_k is a reduced expression for w then Matsumoto’s Monoid Lemma implies that T_w = T_{s_1} \cdots T_{s_k} depends on w and not on the choice of reduced expressions. Moreover, \{ T_w \mid w\in W \} is a basis for the Iwahori-Hecke algebra H and

T_s T_w = \begin{cases}
   T_{sw}, &                     \text{if } \ell(sw) = \ell(w)+1,\\
   (q_1+q_2)T_w -q_1q_2 T_{sw}, & \text{if } \ell(sw) = \ell(w)-1.
\end{cases}

The T-basis of H is implemented for any choice of parameters q_1 and q_2:

sage: R.<u,v> = LaurentPolynomialRing(ZZ,2)
sage: H = IwahoriHeckeAlgebra('A3', u,v)
sage: T = H.T()
sage: T[1]
T[1]
sage: T[1,2,1] + T[2]
T[1,2,1] + T[2]
sage: T[1] * T[1,2,1]
(u+v)*T[1,2,1] + (-u*v)*T[2,1]
sage: T[1]^-1
(-u^-1*v^-1)*T[1] + (v^-1+u^-1)

Working over the Laurent polynomial ring Z[q^{\pm 1/2}] Kazhdan and Lusztig proved that there exist two distinguished bases \{ C^{\prime}_w \mid w \in W \} and \{ C_w \mid w \in W \} of H which are uniquely determined by the properties that they are invariant under the bar involution on H and have triangular transitions matrices with polynomial entries of a certain form with the T-basis; see [KL79] for a precise statement.

It turns out that the Kazhdan-Lusztig bases can be defined (by specialization) in H whenever -q_1 q_2 is a square in the base ring. The Kazhdan-Lusztig bases are implemented inside H whenever -q_1 q_2 has a square root:

sage: H = IwahoriHeckeAlgebra('A3', u^2,-v^2)
sage: T=H.T(); Cp= H.Cp(); C=H.C()
sage: T(Cp[1])
(u^-1*v^-1)*T[1] + (u^-1*v)
sage: T(C[1])
(u^-1*v^-1)*T[1] + (-u*v^-1)
sage: Cp(C[1])
Cp[1] + (-u*v^-1-u^-1*v)
sage: elt = Cp[2]*Cp[3]+C[1]; elt
Cp[2,3] + Cp[1] + (-u*v^-1-u^-1*v)
sage: c = C(elt); c
C[2,3] + C[1] + (u*v^-1+u^-1*v)*C[2] + (u*v^-1+u^-1*v)*C[3] + (u^2*v^-2+2+u^-2*v^2)
sage: t = T(c); t
(u^-2*v^-2)*T[2,3] + (u^-1*v^-1)*T[1] + (u^-2)*T[2] + (u^-2)*T[3] + (-u*v^-1+u^-2*v^2)
sage: Cp(t)
Cp[2,3] + Cp[1] + (-u*v^-1-u^-1*v)
sage: Cp(c)
Cp[2,3] + Cp[1] + (-u*v^-1-u^-1*v)

The conversions to and from the Kazhdan-Lusztig bases are done behind the scenes whenever the Kazhdan-Lusztig bases are well-defined. Once a suitable Iwahori-Hecke algebra is defined they will work without further intervention.

For example, with the “standard parameters”, so that (T_r-q^2)(T_r+1) = 0:

sage: R.<q> = LaurentPolynomialRing(ZZ)
sage: H = IwahoriHeckeAlgebra('A3', q^2)
sage: T=H.T(); Cp=H.Cp(); C=H.C()
sage: C(T[1])
q*C[1] + q^2
sage: elt = Cp(T[1,2,1]); elt
q^3*Cp[1,2,1] + (-q^2)*Cp[1,2] + (-q^2)*Cp[2,1] + q*Cp[1] + q*Cp[2] + (-1)
sage: C(elt)
q^3*C[1,2,1] + q^4*C[1,2] + q^4*C[2,1] + q^5*C[1] + q^5*C[2] + q^6

With the “normalized presentation”, so that (T_r-q)(T_r+q^{-1}) = 0:

sage: R.<q> = LaurentPolynomialRing(ZZ)
sage: H = IwahoriHeckeAlgebra('A3', q, -q^-1)
sage: T=H.T(); Cp=H.Cp(); C=H.C()
sage: C(T[1])
C[1] + q
sage: elt = Cp(T[1,2,1]); elt
Cp[1,2,1] + (-q^-1)*Cp[1,2] + (-q^-1)*Cp[2,1] + (q^-2)*Cp[1] + (q^-2)*Cp[2] + (-q^-3)
sage: C(elt)
C[1,2,1] + q*C[1,2] + q*C[2,1] + q^2*C[1] + q^2*C[2] + q^3

In the group algebra, so that (T_r-1)(T_r+1) = 0:

sage: H = IwahoriHeckeAlgebra('A3', 1)
sage: T=H.T(); Cp=H.Cp(); C=H.C()
sage: C(T[1])
C[1] + 1
sage: Cp(T[1,2,1])
Cp[1,2,1] - Cp[1,2] - Cp[2,1] + Cp[1] + Cp[2] - 1
sage: C(_)
C[1,2,1] + C[1,2] + C[2,1] + C[1] + C[2] + 1

On the other hand, if the Kazhdan-Lusztig bases are not well-defined (when -q_1 q_2 is not a square), attempting to use the Kazhdan-Lusztig bases triggers an error:

sage: R.<q>=LaurentPolynomialRing(ZZ)
sage: H = IwahoriHeckeAlgebra('A3', q)
sage: C=H.C()
Traceback (most recent call last):
...
ValueError: The Kazhdan_Lusztig bases are defined only when -q_1*q_2 is a square

We give an example in affine type:

sage: R.<v> = LaurentPolynomialRing(ZZ)
sage: H = IwahoriHeckeAlgebra(['A',2,1], v^2)
sage: T=H.T(); Cp=H.Cp(); C=H.C()
sage: C(T[1,0,2])
v^3*C[1,0,2] + v^4*C[1,0] + v^4*C[0,2] + v^4*C[1,2]
 + v^5*C[0] + v^5*C[2] + v^5*C[1] + v^6
sage: Cp(T[1,0,2])
v^3*Cp[1,0,2] + (-v^2)*Cp[1,0] + (-v^2)*Cp[0,2] + (-v^2)*Cp[1,2]
 + v*Cp[0] + v*Cp[2] + v*Cp[1] + (-1)
sage: T(C[1,0,2])
(v^-3)*T[1,0,2] + (-v^-1)*T[1,0] + (-v^-1)*T[0,2] + (-v^-1)*T[1,2]
 + v*T[0] + v*T[2] + v*T[1] + (-v^3)
sage: T(Cp[1,0,2])
(v^-3)*T[1,0,2] + (v^-3)*T[1,0] + (v^-3)*T[0,2] + (v^-3)*T[1,2]
 + (v^-3)*T[0] + (v^-3)*T[2] + (v^-3)*T[1] + (v^-3)

REFERENCES:

[I64](1, 2) N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I, 10 (1964), 215–236 (1964). MathSciNet MR0165016
[HKP]T. J. Haines, R. E. Kottwitz, A. Prasad, Iwahori-Hecke Algebras, J. Ramanujan Math. Soc., 25 (2010), 113–145. Arxiv 0309168v3 MathSciNet MR2642451
[J87]V. Jones, Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2) 126 (1987), no. 2, 335–388. doi:10.2307/1971403 MathSciNet MR0908150

EXAMPLES:

We start by creating a Iwahori-Hecke algebra together with the three bases for these algebras that are currently supported:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()

It is also possible to define these three bases quickly using the inject_shorthands() method.

Next we create our generators for the T-basis and do some basic computations and conversions between the bases:

sage: T1,T2,T3 = T.algebra_generators()
sage: T1 == T[1]
True
sage: T1*T2 == T[1,2]
True
sage: T1 + T2
T[1] + T[2]
sage: T1*T1
(v^2-1)*T[1] + v^2
sage: (T1 + T2)*T3 + T1*T1 - (v + v^-1)*T2
T[3,1] + T[2,3] + (v^2-1)*T[1] + (-v-v^-1)*T[2] + v^2
sage: Cp(T1)
v*Cp[1] + (-1)
sage: Cp((v^1 - 1)*T1*T2 - T3)
(v^3-v^2)*Cp[1,2] + (-v^2+v)*Cp[1] + (-v^2+v)*Cp[2] + (-v)*Cp[3] + v
sage: C(T1)
v*C[1] + v^2
sage: p = C(T2*T3 - v*T1); p
v^2*C[2,3] + (-v^2)*C[1] + v^3*C[2] + v^3*C[3] + (v^4-v^3)
sage: Cp(p)
v^2*Cp[2,3] + (-v^2)*Cp[1] + (-v)*Cp[2] + (-v)*Cp[3] + (v+1)
sage: Cp(T2*T3 - v*T1)
v^2*Cp[2,3] + (-v^2)*Cp[1] + (-v)*Cp[2] + (-v)*Cp[3] + (v+1)

In addition to explicitly creating generators, we have two shortcuts to basis elements. The first is by using elements of the underlying Coxeter group, the other is by using reduced words:

sage: s1,s2,s3 = H.coxeter_group().gens()
sage: T[s1*s2*s1*s3] == T[1,2,1,3]
True
sage: T[1,2,1,3] == T1*T2*T1*T3
True

TESTS:

We check the defining properties of the bases:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()
sage: T(Cp[1])
(v^-1)*T[1] + (v^-1)
sage: T(C[1])
(v^-1)*T[1] + (-v)
sage: C(Cp[1])
C[1] + (v+v^-1)
sage: Cp(C[1])
Cp[1] + (-v-v^-1)
sage: all(C[x] == C[x].bar() for x in W) # long time
True
sage: all(Cp[x] == Cp[x].bar() for x in W) # long time
True
sage: all(T(C[x]).bar() == T(C[x]) for x in W) # long time
True
sage: all(T(Cp[x]).bar() == T(Cp[x]) for x in W) # long time
True
sage: KL = KazhdanLusztigPolynomial(W, v)
sage: term = lambda x,y: (-1)^y.length() * v^(-2*y.length()) * KL.P(y, x).substitute(v=v^-2)*T[y]
sage: all(T(C[x]) == (-v)^x.length()*sum(term(x,y) for y in W) for x in W) # long time
True
sage: all(T(Cp[x]) == v^-x.length()*sum(KL.P(y,x).substitute(v=v^2)*T[y] for y in W) for x in W) # long time
True

We check conversion between the bases for type B_2 as well as some of the defining properties:

sage: H = IwahoriHeckeAlgebra(['B',2], v**2)
sage: W = H.coxeter_group()
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()
sage: all(T[x] == T(C(T[x])) for x in W) # long time
True
sage: all(T[x] == T(Cp(T[x])) for x in W) # long time
True
sage: all(C[x] == C(T(C[x])) for x in W) # long time
True
sage: all(C[x] == C(Cp(C[x])) for x in W) # long time
True
sage: all(Cp[x] == Cp(T(Cp[x])) for x in W) # long time
True
sage: all(Cp[x] == Cp(C(Cp[x])) for x in W) # long time
True
sage: all(T(C[x]).bar() == T(C[x]) for x in W) # long time
True
sage: all(T(Cp[x]).bar() == T(Cp[x]) for x in W) # long time
True
sage: KL = KazhdanLusztigPolynomial(W, v)
sage: term = lambda x,y: (-1)^y.length() * v^(-2*y.length()) * KL.P(y, x).substitute(v=v^-2)*T[y]
sage: all(T(C[x]) == (-v)^x.length()*sum(term(x,y) for y in W) for x in W) # long time
True
sage: all(T(Cp[x]) == v^-x.length()*sum(KL.P(y,x).substitute(v=v^2)*T[y] for y in W) for x in W) # long time
True

Todo

Implement multi-parameter Iwahori-Hecke algebras together with their Kazhdan-Lusztig bases. That is, Iwahori-Hecke algebras with (possibly) different parameters for each conjugacy class of simple reflections in the underlying Coxeter group.

Todo

When given “generic parameters” we should return the generic Iwahori-Hecke algebra with these parameters and allow the user to work inside this algebra rather than doing calculations behind the scenes in a copy of the generic Iwahori-Hecke algebra. The main problem is that it is not clear how to recognise when the parameters are “generic”.

class C(IHAlgebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra._KLHeckeBasis

The Kazhdan-Lusztig C-basis of Iwahori-Hecke algebra.

Assuming the standard quadratic relations of (T_r-q)(T_r+1)=0, for every element w in the Coxeter group, there is a unique element C_w in the Iwahori-Hecke algebra which is uniquely determined by the two properties:

\begin{aligned}
    \overline{C_w} &= C_w \\
    C_w &= (-1)^{\ell(w)} q^{\ell(w)/2}
        \sum_{v \leq w} (-q)^{-\ell(v)}\overline{P_{v,w}(q)} T_v
\end{aligned}

where \leq is the Bruhat order on the underlying Coxeter group and P_{v,w}(q)\in\ZZ[q,q^{-1}] are polynomials in \ZZ[q] such that P_{w,w}(q) = 1 and if v < w then \deg P_{v,w}(q) \leq \frac{1}{2}(\ell(w) - \ell(v) - 1).

More generally, if the quadratic relations are of the form (T_s-q_1)(T_s-q_2)=0` and \sqrt{-q_1q_2} exists then for a simple reflection s then the corresponding Kazhdan-Lusztig basis element is:

C_s = (-q_1 q_2)^{1/2} (1 - (-q_1 q_2)^{-1/2} T_s).

This is related to the C^{\prime} Kazhdan-Lusztig basis by C_i =
-\alpha(C_i^{\prime}) where \alpha is the \ZZ-linear Hecke involution defined by q^{1/2} \mapsto q^{-1/2} and \alpha(T_i) =
-(q_1 q_2)^{-1/2} T_i.

See [KL79] for more details.

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra('A5', v**2)
sage: W = H.coxeter_group()
sage: s1,s2,s3,s4,s5 = W.simple_reflections()
sage: T = H.T()
sage: C = H.C()
sage: T(s1)**2
(v^2-1)*T[1] + v^2
sage: T(C(s1))
(v^-1)*T[1] + (-v)
sage: T(C(s1)*C(s2)*C(s1))
(v^-3)*T[1,2,1] + (-v^-1)*T[1,2] + (-v^-1)*T[2,1] + (v+v^-1)*T[1] + v*T[2] + (-v^3-v)
sage: R.<v> = LaurentPolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: s1,s2,s3 = W.simple_reflections()
sage: C = H.C()
sage: C(s1*s2*s1)
C[1,2,1]
sage: C(s1)**2
(-v-v^-1)*C[1]
sage: C(s1)*C(s2)*C(s1)
C[1,2,1] + C[1]

TESTS:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()
sage: all(C(T(C[x])) == C[x] for x in W) # long time
True
sage: all(C(Cp(C[x])) == C[x] for x in W) # long time
True

Check the defining property between C and C^{\prime}:

sage: T(C[1])
(v^-1)*T[1] + (-v)
sage: -T(Cp[1]).hash_involution()
(v^-1)*T[1] + (-v)
sage: T(Cp[1] + Cp[2]).hash_involution()
(-v^-1)*T[1] + (-v^-1)*T[2] + 2*v
sage: -T(C[1] + C[2])
(-v^-1)*T[1] + (-v^-1)*T[2] + 2*v
sage: Cp(-C[1].hash_involution())
Cp[1]
sage: Cp(-C[1,2,3].hash_involution())
Cp[1,2,3]
sage: Cp(C[1,2,1,3].hash_involution())
Cp[1,2,3,1]
sage: all(C((-1)**x.length()*Cp[x].hash_involution()) == C[x] for x in W) # long time
True
hash_involution_on_basis(w)

Return the effect of applying the hash involution to the basis element self[w].

This function is not intended to be called directly. Instead, use hash_involution().

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: C=H.C()
sage: s=H.coxeter_group().simple_reflection(1)
sage: C.hash_involution_on_basis(s)
(-1)*C[1] + (-v-v^-1)
sage: C[s].hash_involution()
(-1)*C[1] + (-v-v^-1)
class IwahoriHeckeAlgebra.Cp(IHAlgebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra._KLHeckeBasis

The C^{\prime} Kazhdan-Lusztig basis of Iwahori-Hecke algebra.

Assuming the standard quadratic relations of (T_r-q)(T_r+1)=0, for every element w in the Coxeter group, there is a unique element C^{\prime}_w in the Iwahori-Hecke algebra which is uniquely determined by the two properties:

\begin{aligned}
    \overline{ C^{\prime}_w } &= C^{\prime}_w\\
    C^{\prime}_w &= q^{-\ell(w)/2}
        \sum_{v \leq w} P_{v,w}(q) T_v
\end{aligned}

where \leq is the Bruhat order on the underlying Coxeter group and P_{v,w}(q) \in \ZZ[q,q^{-1}] are polynomials in \ZZ[q] such that P_{w,w}(q) = 1 and if v < w then \deg P_{v,w}(q) \leq
\frac{1}{2}(\ell(w)-\ell(v)-1).

More generally, if the quadratic relations are of the form (T_s-q_1)(T_s-q_2)=0` and \sqrt{-q_1q_2} exists then for a simple reflection s then the corresponding Kazhdan-Lusztig basis element is:

C^{\prime}_s = (-q_1 q_2)^{-1/2} (T_s + 1).

See [KL79] for more details.

EXAMPLES:

sage: R = LaurentPolynomialRing(QQ, 'v')
sage: v = R.gen(0)
sage: H = IwahoriHeckeAlgebra('A5', v**2)
sage: W = H.coxeter_group()
sage: s1,s2,s3,s4,s5 = W.simple_reflections()
sage: T = H.T()
sage: Cp = H.Cp()
sage: T(s1)**2
(v^2-1)*T[1] + v^2
sage: T(Cp(s1))
(v^-1)*T[1] + (v^-1)
sage: T(Cp(s1)*Cp(s2)*Cp(s1))
(v^-3)*T[1,2,1] + (v^-3)*T[1,2] + (v^-3)*T[2,1] + (v^-1+v^-3)*T[1] + (v^-3)*T[2] + (v^-1+v^-3)
sage: R = LaurentPolynomialRing(QQ, 'v')
sage: v = R.gen(0)
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: s1,s2,s3 = W.simple_reflections()
sage: Cp = H.Cp()
sage: Cp(s1*s2*s1)
Cp[1,2,1]
sage: Cp(s1)**2
(v+v^-1)*Cp[1]
sage: Cp(s1)*Cp(s2)*Cp(s1)
Cp[1,2,1] + Cp[1]
sage: Cp(s1)*Cp(s2)*Cp(s3)*Cp(s1)*Cp(s2) # long time
Cp[1,2,3,1,2] + Cp[1,2,1] + Cp[3,1,2]

TESTS:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()
sage: all(Cp(T(Cp[x])) == Cp[x] for x in W) # long time
True
sage: all(Cp(C(Cp[x])) == Cp[x] for x in W) # long time
True
hash_involution_on_basis(w)

Return the effect of applying the hash involution to the basis element self[w].

This function is not intended to be called directly. Instead, use hash_involution().

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: Cp=H.Cp()
sage: s=H.coxeter_group().simple_reflection(1)
sage: Cp.hash_involution_on_basis(s)
(-1)*Cp[1] + (v+v^-1)
sage: Cp[s].hash_involution()
(-1)*Cp[1] + (v+v^-1)
class IwahoriHeckeAlgebra.T(algebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra._Basis

The standard basis of Iwahori-Hecke algebra.

For every simple reflection s_i of the Coxeter group, there is a corresponding generator T_i of Iwahori-Hecke algebra. These are subject to the relations:

(T_i - q_1) (T_i - q_2) = 0

together with the braid relations:

T_i T_j T_i \cdots = T_j T_i T_j \cdots,

where the number of terms on each of the two sides is the order of s_i s_j in the Coxeter group.

Weyl group elements form a basis of Iwahori-Hecke algebra H with the property that if w_1 and w_2 are Coxeter group elements such that \ell(w_1 w_2) = \ell(w_1) + \ell(w_2) then T_{w_1 w_2} = T_{w_1} T_{w_2}.

With the default value q_2 = -1 and with q_1 = q the generating relation may be written T_i^2 = (q-1) \cdot T_i + q \cdot 1 as in [I64].

EXAMPLES:

sage: H = IwahoriHeckeAlgebra("A3", 1)
sage: T = H.T()
sage: T1,T2,T3 = T.algebra_generators()
sage: T1*T2*T3*T1*T2*T1 == T3*T2*T1*T3*T2*T3
True
sage: w0 = T(H.coxeter_group().long_element())
sage: w0
T[1,2,3,1,2,1]
sage: T = H.T(prefix="s")
sage: T.an_element()
s[1,2,3,1,2,1] + 2*s[1,2,3,1,2] + 3*s[1,2,3,2,1] + s[1,2,3]

TESTS:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: T = H.T()
sage: C = H.C()
sage: Cp = H.Cp()
sage: all(T(C(T[x])) == T[x] for x in W) # long time
True
sage: all(T(Cp(T[x])) == T[x] for x in W) # long time
True

We check a property of the bar involution and R-polynomials:

sage: KL = KazhdanLusztigPolynomial(W, v)
sage: all(T[x].bar() == sum(v^(-2*y.length()) * KL.R(y, x).substitute(v=v^-2) * T[y] for y in W) for x in W) # long time
True
Element

alias of T.Element

bar_on_basis(w)

Return the bar involution of T_w, which is T^{-1}_{w^-1}.

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: W = H.coxeter_group()
sage: s1,s2,s3 = W.simple_reflections()
sage: T = H.T()
sage: b = T.bar_on_basis(s1*s2*s3); b
(v^-6)*T[1,2,3]
  + (-v^-4+v^-6)*T[1,2]
  + (-v^-4+v^-6)*T[3,1]
  + (-v^-4+v^-6)*T[2,3]
  + (v^-2-2*v^-4+v^-6)*T[1]
  + (v^-2-2*v^-4+v^-6)*T[2]
  + (v^-2-2*v^-4+v^-6)*T[3]
  + (-1+3*v^-2-3*v^-4+v^-6)
sage: b.bar()
T[1,2,3]
hash_involution_on_basis(w)

Return the hash involution on the basis element self[w].

The hash involution \alpha is a \ZZ-algebra involution of the Iwahori-Hecke algebra determined by q^{1/2} \mapsto q^{-1/2}, and T_w \mapsto -1^{\ell(w)}
(q_1 q_2)^{-\ell(w)} T_w, for w an element of the corresponding Coxeter group.

This map is defined in [KL79] and it is used to change between the C and C^{\prime} bases because \alpha(C_w) = (-1)^{\ell(w)}C^{\prime}_w.

This function is not intended to be called directly. Instead, use hash_involution().

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
sage: H = IwahoriHeckeAlgebra('A3', v**2)
sage: T=H.T()
sage: s=H.coxeter_group().simple_reflection(1)
sage: T.hash_involution_on_basis(s)
(-v^-2)*T[1]
sage: T[s].hash_involution()
(-v^-2)*T[1]
sage: h = T[1]*T[2] + (v^3 - v^-1 + 2)*T[3,1,2,3]
sage: h.hash_involution()
(-v^-7+2*v^-8+v^-11)*T[1,2,3,2] + (v^-4)*T[1,2]
sage: h.hash_involution().hash_involution() == h
True
inverse_generator(i)

Return the inverse of the i-th generator, if it exists.

This method is only available if the Iwahori-Hecke algebra parameters q1 and q2 are both invertible. In this case, the algebra generators are also invertible and this method returns the inverse of self.algebra_generator(i).

EXAMPLES:

sage: P.<q1, q2>=QQ[]
sage: F = Frac(P)
sage: H = IwahoriHeckeAlgebra("A2", q1, q2=q2, base_ring=F).T()
sage: H.base_ring()
Fraction Field of Multivariate Polynomial Ring in q1, q2 over Rational Field
sage: H.inverse_generator(1)
-1/(q1*q2)*T[1] + ((q1+q2)/(q1*q2))
sage: H = IwahoriHeckeAlgebra("A2", q1, base_ring=F).T()
sage: H.inverse_generator(2)
-(1/(-q1))*T[2] + ((q1-1)/(-q1))
sage: P1.<r1, r2> = LaurentPolynomialRing(QQ)
sage: H1 = IwahoriHeckeAlgebra("B2", r1, q2=r2, base_ring=P1).T()
sage: H1.base_ring()
Multivariate Laurent Polynomial Ring in r1, r2 over Rational Field
sage: H1.inverse_generator(2)
(-r1^-1*r2^-1)*T[2] + (r2^-1+r1^-1)
sage: H2 = IwahoriHeckeAlgebra("C2", r1, base_ring=P1).T()
sage: H2.inverse_generator(2)
(r1^-1)*T[2] + (-1+r1^-1)
inverse_generators()

Return the inverses of all the generators, if they exist.

This method is only available if q1 and q2 are invertible. In that case, the algebra generators are also invertible.

EXAMPLES:

sage: P.<q> = PolynomialRing(QQ)
sage: F = Frac(P)
sage: H = IwahoriHeckeAlgebra("A2", q, base_ring=F).T()
sage: T1,T2 = H.algebra_generators()
sage: U1,U2 = H.inverse_generators()
sage: U1*T1,T1*U1
(1, 1)
sage: P1.<q> = LaurentPolynomialRing(QQ)
sage: H1 = IwahoriHeckeAlgebra("A2", q, base_ring=P1).T(prefix="V")
sage: V1,V2 = H1.algebra_generators()
sage: W1,W2 = H1.inverse_generators()
sage: [W1,W2]
[(q^-1)*V[1] + (-1+q^-1), (q^-1)*V[2] + (-1+q^-1)]
sage: V1*W1, W2*V2
(1, 1)
product_by_generator(x, i, side='right')

Return T_i \cdot x, where T_i is the i-th generator. This is coded individually for use in x._mul_().

EXAMPLES:

sage: R.<q> = QQ[]; H = IwahoriHeckeAlgebra("A2", q).T()
sage: T1, T2 = H.algebra_generators()
sage: [H.product_by_generator(x, 1) for x in [T1,T2]]
[(q-1)*T[1] + q, T[2,1]]
sage: [H.product_by_generator(x, 1, side = "left") for x in [T1,T2]]
[(q-1)*T[1] + q, T[1,2]]
product_by_generator_on_basis(w, i, side='right')

Return the product T_w T_i (resp. T_i T_w) if side is 'right' (resp. 'left').

If the quadratic relation is (T_i-u)(T_i-v) = 0, then we have

T_w T_i = \begin{cases}
T_{ws_i} & \text{if } \ell(ws_i) = \ell(w) + 1, \\
(u+v) T_{ws_i} - uv T_w & \text{if } \ell(w s_i) = \ell(w)-1.
\end{cases}

The left action is similar.

INPUT:

  • w – an element of the Coxeter group
  • i – an element of the index set
  • side'right' (default) or 'left'

EXAMPLES:

sage: R.<q> = QQ[]; H = IwahoriHeckeAlgebra("A2", q)
sage: T = H.T()
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: [T.product_by_generator_on_basis(w, 1) for w in [s1,s2,s1*s2]]
[(q-1)*T[1] + q, T[2,1], T[1,2,1]]
sage: [T.product_by_generator_on_basis(w, 1, side="left") for w in [s1,s2,s1*s2]]
[(q-1)*T[1] + q, T[1,2], (q-1)*T[1,2] + q*T[2]]
product_on_basis(w1, w2)

Return T_{w_1} T_{w_2}, where w_1 and w_2 are words in the Coxeter group.

EXAMPLES:

sage: R.<q> = QQ[]; H = IwahoriHeckeAlgebra("A2", q)
sage: T = H.T()
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: [T.product_on_basis(s1,x) for x in [s1,s2]]
[(q-1)*T[1] + q, T[1,2]]
to_C_basis(w)

Return T_w as a linear combination of C-basis elements.

EXAMPLES:

sage: R = LaurentPolynomialRing(QQ, 'v')
sage: v = R.gen(0)
sage: H = IwahoriHeckeAlgebra('A2', v**2)
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: C = H.C()
sage: T.to_C_basis(s1)
v*T[1] + v^2
sage: C(T(s1))
v*C[1] + v^2
sage: C(v^-1*T(s1) - v)
C[1]
sage: C(T(s1*s2)+T(s1)+T(s2)+1)
v^2*C[1,2] + (v^3+v)*C[1] + (v^3+v)*C[2] + (v^4+2*v^2+1)
sage: C(T(s1*s2*s1))
v^3*C[1,2,1] + v^4*C[1,2] + v^4*C[2,1] + v^5*C[1] + v^5*C[2] + v^6
to_Cp_basis(w)

Return T_w as a linear combination of C^{\prime}-basis elements.

EXAMPLES:

sage: R.<v> = LaurentPolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra('A2', v**2)
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: Cp = H.Cp()
sage: T.to_Cp_basis(s1)
v*Cp[1] + (-1)
sage: Cp(T(s1))
v*Cp[1] + (-1)
sage: Cp(T(s1)+1)
v*Cp[1]
sage: Cp(T(s1*s2)+T(s1)+T(s2)+1)
v^2*Cp[1,2]
sage: Cp(T(s1*s2*s1))
v^3*Cp[1,2,1] + (-v^2)*Cp[1,2] + (-v^2)*Cp[2,1] + v*Cp[1] + v*Cp[2] + (-1)
IwahoriHeckeAlgebra.a_realization()

Return a particular realization of self (the T-basis).

EXAMPLES:

sage: H = IwahoriHeckeAlgebra("B2", 1)
sage: H.a_realization()
Iwahori-Hecke algebra of type B2 in 1,-1 over Integer Ring in the T-basis
IwahoriHeckeAlgebra.cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: IwahoriHeckeAlgebra("D4", 1).cartan_type()
['D', 4]
IwahoriHeckeAlgebra.coxeter_group()

Return the Coxeter group of self.

EXAMPLES:

sage: IwahoriHeckeAlgebra("B2", 1).coxeter_group()
Weyl Group of type ['B', 2] (as a matrix group acting on the ambient space)
IwahoriHeckeAlgebra.q1()

Return the parameter q_1 of self.

EXAMPLES:

sage: H = IwahoriHeckeAlgebra("B2", 1)
sage: H.q1()
1
IwahoriHeckeAlgebra.q2()

Return the parameter q_2 of self.

EXAMPLES:

sage: H = IwahoriHeckeAlgebra("B2", 1)
sage: H.q2()
-1
sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebraT(W, q1, q2=-1, base_ring=None, prefix='T')

TESTS:

sage: H = IwahoriHeckeAlgebraT("A2", 1)
doctest:...: DeprecationWarning: this class is deprecated. Use IwahoriHeckeAlgebra().T instead
See http://trac.sagemath.org/14261 for details.
class sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard(W)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra

This is a class which is used behind the scenes by IwahoriHeckeAlgebra to compute the Kazhdan-Lusztig bases. It is not meant to be used directly. It implements the slightly idiosyncratic (but convenient) Iwahori-Hecke algebra with two parameters which is defined over the Laurent polynomial ring \ZZ[u,u^{-1},v,v^{-1}] in two variables and has quadratic relations:

(T_r - u)(T_r + v^2/u) = 0.

The point of these relations is that the product of the two parameters is v^2 which is a square in \ZZ[u,u^{-1},v,v^{-1}]. Consequently, the Kazhdan-Lusztig bases are defined for this algebra.

More generally, if we have a Iwahori-Hecke algebra with two parameters which has quadratic relations of the form:

(T_r - q_1)(T_r - q_2) = 0

where -q_1 q_2 is a square then the Kazhdan-Lusztig bases are well-defined for this algebra. Moreover, these bases be computed by specialization from the generic Iwahori-Hecke algebra using the specialization which sends u \mapsto q_1 and v \mapsto \sqrt{-q_1 q_2}, so that v^2 / u \mapsto -q_2.

For example, if q_1 = q = Q^2 and q_2 = -1 then u \mapsto q and v \mapsto \sqrt{q} = Q; this is the standard presentation of the Iwahori-Hecke algebra with (T_r - q)(T_r + 1) = 0. On the other hand, when q_1 = q and q_2 = -q^{-1} then u \mapsto q and v \mapsto 1. This is the normalized presentation with (T_r - v)(T_r + v^{-1}) = 0.

Warning

This class uses non-standard parameters for the Iwahori-Hecke algebra and are related to the standard parameters by an outer automorphism that is non-trivial on the T-basis.

class C(IHAlgebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra.C

The Kazhdan-Lusztig C-basis for the generic Iwahori-Hecke algebra.

to_T_basis(w)

Return C_w as a linear combination of T-basis elements.

EXAMPLES:

sage: H = sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard("A3")
sage: s1,s2,s3 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: C = H.C()
sage: C.to_T_basis(s1)
(v^-1)*T[1] + (-u*v^-1)
sage: C.to_T_basis(s1*s2)
(v^-2)*T[1,2] + (-u*v^-2)*T[1] + (-u*v^-2)*T[2] + (u^2*v^-2)
sage: C.to_T_basis(s1*s2*s1)
(v^-3)*T[1,2,1] + (-u*v^-3)*T[1,2] + (-u*v^-3)*T[2,1]
 + (u^2*v^-3)*T[1] + (u^2*v^-3)*T[2] + (-u^3*v^-3)
sage: T(C(s1*s2*s1))
(v^-3)*T[1,2,1] + (-u*v^-3)*T[1,2] + (-u*v^-3)*T[2,1]
 + (u^2*v^-3)*T[1] + (u^2*v^-3)*T[2] + (-u^3*v^-3)
sage: T(C(s2*s1*s3*s2))
(v^-4)*T[2,3,1,2] + (-u*v^-4)*T[1,2,1] + (-u*v^-4)*T[3,1,2]
 + (-u*v^-4)*T[2,3,1] + (-u*v^-4)*T[2,3,2] + (u^2*v^-4)*T[1,2]
 + (u^2*v^-4)*T[2,1] + (u^2*v^-4)*T[3,1] + (u^2*v^-4)*T[2,3]
 + (u^2*v^-4)*T[3,2] + (-u^3*v^-4)*T[1]
 + (-u^3*v^-4-u*v^-2)*T[2] + (-u^3*v^-4)*T[3]
 + (u^4*v^-4+u^2*v^-2)
class IwahoriHeckeAlgebra_nonstandard.Cp(IHAlgebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra.Cp

The Kazhdan-Lusztig C^{\prime}-basis for the generic Iwahori-Hecke algebra.

to_T_basis(w)

Return C^{\prime}_w as a linear combination of T-basis elements.

EXAMPLES:

sage: H = sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard("A3")
sage: s1,s2,s3 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: Cp = H.Cp()
sage: Cp.to_T_basis(s1)
(v^-1)*T[1] + (u^-1*v)
sage: Cp.to_T_basis(s1*s2)
(v^-2)*T[1,2] + (u^-1)*T[1] + (u^-1)*T[2] + (u^-2*v^2)
sage: Cp.to_T_basis(s1*s2*s1)
(v^-3)*T[1,2,1] + (u^-1*v^-1)*T[1,2] + (u^-1*v^-1)*T[2,1]
 + (u^-2*v)*T[1] + (u^-2*v)*T[2] + (u^-3*v^3)
sage: T(Cp(s1*s2*s1))
(v^-3)*T[1,2,1] + (u^-1*v^-1)*T[1,2] + (u^-1*v^-1)*T[2,1]
 + (u^-2*v)*T[1] + (u^-2*v)*T[2] + (u^-3*v^3)
sage: T(Cp(s2*s1*s3*s2))
(v^-4)*T[2,3,1,2] + (u^-1*v^-2)*T[1,2,1] + (u^-1*v^-2)*T[3,1,2]
 + (u^-1*v^-2)*T[2,3,1] + (u^-1*v^-2)*T[2,3,2] + (u^-2)*T[1,2]
 + (u^-2)*T[2,1] + (u^-2)*T[3,1] + (u^-2)*T[2,3]
 + (u^-2)*T[3,2] + (u^-3*v^2)*T[1] + (u^-1+u^-3*v^2)*T[2]
 + (u^-3*v^2)*T[3] + (u^-2*v^2+u^-4*v^4)
class IwahoriHeckeAlgebra_nonstandard.T(algebra, prefix=None)

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra.T

The T-basis for the generic Iwahori-Hecke algebra.

to_C_basis(w)

Return T_w as a linear combination of C-basis elements.

To compute this we piggy back off the C^{\prime}-basis conversion using the observation that the hash involution sends T_w to (-q_1 q_1)^{\ell(w)} T_w and C_w to (-1)^{\ell(w)} C^{\prime}_w. Therefore, if

T_w = \sum_v a_{vw} C^{\prime}_v

then

T_w = (-q_1 q_2)^{\ell(w)} \Big( \sum_v a_{vw} C^{\prime}_v
                                 \Big)^\#
    = \sum_v (-1)^{\ell(v)} \overline{a_{vw}} C_v

Note that we cannot just apply hash_involution() here because this involution always returns the answer with respect to the same basis.

EXAMPLES:

sage: H = sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard("A2")
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: C = H.C()
sage: T.to_C_basis(s1)
v*T[1] + u
sage: C(T(s1))
v*C[1] + u
sage: C(T( C[1] ))
C[1]
sage: C(T(s1*s2)+T(s1)+T(s2)+1)
v^2*C[1,2] + (u*v+v)*C[1] + (u*v+v)*C[2] + (u^2+2*u+1)
sage: C(T(s1*s2*s1))
v^3*C[1,2,1] + u*v^2*C[1,2] + u*v^2*C[2,1] + u^2*v*C[1] + u^2*v*C[2] + u^3
to_Cp_basis(w)

Return T_w as a linear combination of C^{\prime}-basis elements.

EXAMPLES:

sage: H = sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard("A2")
sage: s1,s2 = H.coxeter_group().simple_reflections()
sage: T = H.T()
sage: Cp = H.Cp()
sage: T.to_Cp_basis(s1)
v*Cp[1] + (-u^-1*v^2)
sage: Cp(T(s1))
v*Cp[1] + (-u^-1*v^2)
sage: Cp(T(s1)+1)
v*Cp[1] + (-u^-1*v^2+1)
sage: Cp(T(s1*s2)+T(s1)+T(s2)+1)
v^2*Cp[1,2] + (-u^-1*v^3+v)*Cp[1] + (-u^-1*v^3+v)*Cp[2] + (u^-2*v^4-2*u^-1*v^2+1)
sage: Cp(T(s1*s2*s1))
v^3*Cp[1,2,1] + (-u^-1*v^4)*Cp[1,2] + (-u^-1*v^4)*Cp[2,1]
 + (u^-2*v^5)*Cp[1] + (u^-2*v^5)*Cp[2] + (-u^-3*v^6)
sage.algebras.iwahori_hecke_algebra.index_cmp(x, y)

Compare two term indices x and y by Bruhat order, then by word length, and then by the generic comparison.

EXAMPLES:

sage: from sage.algebras.iwahori_hecke_algebra import index_cmp
sage: W = WeylGroup(['A',2,1])
sage: x = W.from_reduced_word([0,1])
sage: y = W.from_reduced_word([0,2,1])
sage: x.bruhat_le(y)
True
sage: index_cmp(x, y)
1
sage.algebras.iwahori_hecke_algebra.normalized_laurent_polynomial(R, p)

Returns a normalized version of the (Laurent polynomial) p in the ring R.

Various ring operations in sage return an element of the field of fractions of the parent ring even though the element is “known” to belong to the base ring. This function is a hack to recover from this. This occurs somewhat haphazardly with Laurent polynomial rings:

sage: R.<q>=LaurentPolynomialRing(ZZ)
sage: [type(c) for c in (q**-1).coefficients()]
[<type 'sage.rings.rational.Rational'>]

It also happens in any ring when dividing by units:

sage: type ( 3/1 )
<type 'sage.rings.rational.Rational'>
sage: type ( -1/-1 )
<type 'sage.rings.rational.Rational'>

This function is a variation on a suggested workaround of Nils Bruin.

EXAMPLES:

sage: from sage.algebras.iwahori_hecke_algebra import normalized_laurent_polynomial
sage: type ( normalized_laurent_polynomial(ZZ, 3/1) )
<type 'sage.rings.integer.Integer'>
sage: R.<q>=LaurentPolynomialRing(ZZ)
sage: [type(c) for c in normalized_laurent_polynomial(R, q**-1).coefficients()]
[<type 'sage.rings.integer.Integer'>]
sage: R.<u,v>=LaurentPolynomialRing(ZZ,2)
sage: p=normalized_laurent_polynomial(R, 2*u**-1*v**-1+u*v)
sage: ui=normalized_laurent_polynomial(R, u^-1)
sage: vi=normalized_laurent_polynomial(R, v^-1)
sage: p(ui,vi)
2*u*v + u^-1*v^-1
sage: q= u+v+ui
sage: q(ui,vi)
u + v^-1 + u^-1

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