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# 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:

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

  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

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

  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
      -(1-v^2)*T[1] + v^2
      sage: (T1 + T2)*T3 + T1*T1 - (v + v^-1)*T2
      T[3,1] + T[2,3] - (1-v^2)*T[1] - (v^-1+v)*T[2] + v^2
      sage: Cp(T1)
      v*Cp[1] - 1
      sage: Cp((v^1 - 1)*T1*T2 - T3)
      -(v^2-v^3)*Cp[1,2] + (v-v^2)*Cp[1] + (v-v^2)*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^3-v^4)
      sage: Cp(p)
      v^2*Cp[2,3] - v^2*Cp[1] - v*Cp[2] - v*Cp[3] + (1+v)
      sage: Cp(T2*T3 - v*T1)
      v^2*Cp[2,3] - v^2*Cp[1] - v*Cp[2] - v*Cp[3] + (1+v)

  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^-1+v)
      sage: Cp(C[1])
      Cp[1] - (v^-1+v)
      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:

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

    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
        -(1-v^2)*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^-1+v)*T[1] + v*T[2] - (v+v^3)

        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^-1+v)*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)
          -C[1] - (v^-1+v)
          sage: C[s].hash_involution()
          -C[1] - (v^-1+v)

  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:

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

    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
        -(1-v^2)*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^-3+v^-1)*T[1] + (v^-3)*T[2] + (v^-3+v^-1)

        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^-1+v)*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)
          -Cp[1] + (v^-1+v)
          sage: Cp[s].hash_involution()
          -Cp[1] + (v^-1+v)

  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()
        2*s[1,2,3,2,1] + 3*s[1,2,3,1] + s[1,2,3] + 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(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

    class `Element`(M, x)

      Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

      A class for elements of an Iwahori-Hecke algebra in the \(T\) basis.

      TESTS:

          sage: R.<q> = QQ[]
          sage: H = IwahoriHeckeAlgebra("B3",q).T()
          sage: T1,T2,T3 = H.algebra_generators()
          sage: T1+2*T2*T3
          2*T[2,3] + T[1]
          sage: T1*T1
          (q-1)*T[1] + q

          sage: R.<q1,q2> = QQ[]
          sage: H = IwahoriHeckeAlgebra("A2", q1, q2=q2).T(prefix="x")
          sage: sum(H.algebra_generators())^2
          x[1,2] + x[2,1] + (q1+q2)*x[1] + (q1+q2)*x[2] + (-2*q1*q2)

          sage: H = IwahoriHeckeAlgebra("A2", q1, q2=q2).T(prefix="t")
          sage: t1,t2 = H.algebra_generators()
          sage: (t1-t2)^3
          (q1^2-q1*q2+q2^2)*t[1] + (-q1^2+q1*q2-q2^2)*t[2]

          sage: R.<q> = QQ[]
          sage: H = IwahoriHeckeAlgebra("G2", q).T()
          sage: [T1, T2] = H.algebra_generators()
          sage: T1*T2*T1*T2*T1*T2 == T2*T1*T2*T1*T2*T1
          True
          sage: T1*T2*T1 == T2*T1*T2
          False

          sage: H = IwahoriHeckeAlgebra("A2", 1).T()
          sage: [T1,T2] = H.algebra_generators()
          sage: T1+T2
          T[1] + T[2]

          sage: -(T1+T2)
          -T[1] - T[2]
          sage: 1-T1
          -T[1] + 1

          sage: T1.parent()
          Iwahori-Hecke algebra of type A2 in 1,-1 over Integer Ring in the T-basis

      `inverse`()

        Return the inverse if `self` is a basis element.

        An element is a basis element if it is \(T_w\) where \(w\) is in the Weyl group. The base ring must be a field or Laurent polynomial ring. Other elements of the ring have inverses but the inverse method is only implemented for the basis elements.

        EXAMPLES:

            sage: R.<q> = LaurentPolynomialRing(QQ)
            sage: H = IwahoriHeckeAlgebra("A2", q).T()
            sage: [T1,T2] = H.algebra_generators()
            sage: x = (T1*T2).inverse(); x
            (q^-2)*T[2,1] + (q^-2-q^-1)*T[1] + (q^-2-q^-1)*T[2] + (q^-2-2*q^-1+1)
            sage: x*T1*T2
            1

        TESTS:

        We check some alternative forms of input for inverting an element:

            sage: R.<q> = LaurentPolynomialRing(QQ)
            sage: H = IwahoriHeckeAlgebra("A2", q).T()
            sage: T1,T2 = H.algebra_generators()
            sage: ~(T1*T2)
            (q^-2)*T[2,1] + (q^-2-q^-1)*T[1] + (q^-2-q^-1)*T[2] + (q^-2-2*q^-1+1)
            sage: (T1*T2)^(-1)
            (q^-2)*T[2,1] + (q^-2-q^-1)*T[1] + (q^-2-q^-1)*T[2] + (q^-2-2*q^-1+1)

    `IwahoriHeckeAlgebra.T.``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^-6-v^-4)*T[1,2]
           + (v^-6-v^-4)*T[3,1]
           + (v^-6-v^-4)*T[2,3]
           + (v^-6-2*v^-4+v^-2)*T[1]
           + (v^-6-2*v^-4+v^-2)*T[2]
           + (v^-6-2*v^-4+v^-2)*T[3]
           + (v^-6-3*v^-4+3*v^-2-1)
          sage: b.bar()
          T[1,2,3]

    `IwahoriHeckeAlgebra.T.``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^-11+2*v^-8-v^-7)*T[1,2,3,2] + (v^-4)*T[1,2]
          sage: h.hash_involution().hash_involution() == h
          True

    `IwahoriHeckeAlgebra.T.``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)

    `IwahoriHeckeAlgebra.T.``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] + (q^-1-1), (q^-1)*V[2] + (q^-1-1)]
          sage: V1*W1, W2*V2
          (1, 1)

    `IwahoriHeckeAlgebra.T.``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]]

    `IwahoriHeckeAlgebra.T.``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

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

      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]]

    `IwahoriHeckeAlgebra.T.``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]]

    `IwahoriHeckeAlgebra.T.``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+v^3)*C[1] + (v+v^3)*C[2] + (1+2*v^2+v^4)
          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

    `IwahoriHeckeAlgebra.T.``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.integer.Integer'>]

  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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