Group algebras of root lattice realizations

class sage.combinat.root_system.root_lattice_realization_algebras.Algebras(category, *args)

Bases: sage.categories.algebra_functor.AlgebrasCategory

The category of group algebras of root lattice realizations.

This includes typically weight rings (group algebras of weight lattices).

TESTS:

sage: for ct in CartanType.samples(crystallographic=True): # long time
....:     TestSuite(RootSystem(ct).root_lattice().algebra(QQ)).run()
class ElementMethods
acted_upon(w)

Implements the action of w on self.

INPUT:

  • w – an element of the Weyl group acting on the underlying weight lattice realization

EXAMPLES:

sage: L = RootSystem(["A",3]).ambient_space()
sage: W = L.weyl_group()
sage: M = L.algebra(QQ['q','t'])
sage: m = M.an_element(); m  # TODO: investigate why we don't get something more interesting
B[(2, 2, 3, 0)]
sage: m = (m+1)^2; m
B[(0, 0, 0, 0)] + 2*B[(2, 2, 3, 0)] + B[(4, 4, 6, 0)]
sage: w = W.an_element(); w.reduced_word()
[1, 2, 3]
sage: m.acted_upon(w)
B[(0, 0, 0, 0)] + 2*B[(0, 2, 2, 3)] + B[(0, 4, 4, 6)]
expand(alphabet)

Expand self into variables in the alphabet.

INPUT:

  • alphabet – a non empty list/tuple of (invertible) variables in a ring to expand in

EXAMPLES:

sage: L = RootSystem(["A",2]).ambient_lattice()
sage: KL = L.algebra(QQ)
sage: p = KL.an_element() + KL.sum_of_monomials(L.some_elements()); p
B[(1, 0, 0)] + B[(1, -1, 0)] + B[(1, 1, 0)] + 2*B[(2, 2, 3)] + B[(0, 1, -1)]
sage: F = LaurentPolynomialRing(QQ, 'x,y,z')
sage: p.expand(F.gens())
2*x^2*y^2*z^3 + x*y + x + y*z^-1 + x*y^-1

TESTS:

sage: type(p.expand(F.gens()))
<type 'sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair'>

sage: p = KL.zero()
sage: p.expand(F.gens())
0
sage: type(p.expand(F.gens()))
<type 'sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair'>
class Algebras.ParentMethods
T0_check_on_basis(q1, q2, convention='antidominant')

Return the \(T_0^\vee\) operator acting on the basis.

This implements the formula for \(T_{0'}\) in Section 6.12 of [Haiman06].

REFERENCES:

[Haiman06](1, 2)
  1. Haiman, Cherednik algebras, Macdonald polynomials and combinatorics, ICM 2006.

Warning

The current implementation probably returns just nonsense, if the convention is not “dominant”.

EXAMPLES:

sage: K = QQ['q1,q2'].fraction_field()
sage: q1,q2 = K.gens()

sage: L = RootSystem(["A",1,1]).ambient_space()
sage: L0 = L.classical()
sage: KL = L.algebra(K)
sage: some_weights = L.fundamental_weights()
sage: f = KL.T0_check_on_basis(q1,q2, convention="dominant")
sage: f(L0.zero())
(q1+q2)*B[(0, 0)] + q1*B[(1, -1)]

sage: L = RootSystem(["A",3,1]).ambient_space()
sage: L0 = L.classical()
sage: KL = L.algebra(K)
sage: some_weights = L0.fundamental_weights()
sage: f = KL.T0_check_on_basis(q1,q2, convention="dominant")
sage: f(L0.zero())       # not checked
(q1+q2)*B[(0, 0, 0, 0)] + q1^3/q2^2*B[(1, 0, 0, -1)]

The following results have not been checked:

sage: for x in some_weights:
....:     print x, ": ", f(x)
(1, 0, 0, 0) :  q1*B[(1, 0, 0, 0)]
(1, 1, 0, 0) :  q1*B[(1, 1, 0, 0)]
(1, 1, 1, 0) :  q1*B[(1, 1, 1, 0)]

Some examples for type \(B_2^{(1)}\) dual:

sage: L = RootSystem("B2~*").ambient_space()
sage: L0 = L.classical()
sage: e = L.basis()
sage: K = QQ['q,u'].fraction_field()
sage: q,u = K.gens()
sage: q1 = u
sage: q2 = -1/u
sage: KL = L.algebra(K)
sage: KL0 = KL.classical()
sage: f = KL.T0_check_on_basis(q1,q2, convention="dominant")
sage: T = KL.twisted_demazure_lusztig_operators(q1,q2, convention="dominant")

Direct calculation:

sage: T.Tw(0)(KL0.monomial(L0([0,0])))
((u^2-1)/u)*B[(0, 0)] + u^3*B[(1, 1)]
sage: KL.T0_check_on_basis(q1,q2, convention="dominant")(L0([0,0]))
((u^2-1)/u)*B[(0, 0)] + u^3*B[(1, 1)]

Step by step calculation, comparing by hand with Mark Shimozono:

sage: res = T.Tw(2)(KL0.monomial(L0([0,0]))); res
u*B[(0, 0)]
sage: res = res * KL0.monomial(L0([-1,1])); res
u*B[(-1, 1)]
sage: res = T.Tw_inverse(1)(res); res
(u^2-1)*B[(0, 0)] + u^2*B[(1, -1)]
sage: res = T.Tw_inverse(2)(res); res
((u^2-1)/u)*B[(0, 0)] + u^3*B[(1, 1)]
cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: A = RootSystem(["A",2,1]).ambient_space().algebra(QQ)
sage: A.cartan_type()
['A', 2, 1]
sage: A = RootSystem(["B",2]).weight_space().algebra(QQ)
sage: A.cartan_type()
['B', 2]
classical()

Return the group algebra of the corresponding classical lattice.

EXAMPLES:

sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(QQ)
sage: KL.classical()
Group algebra of the Ambient space of the Root system of type ['A', 2] over Rational Field
demazure_lusztig_operator_on_basis(weight, i, q1, q2, convention='antidominant')

Return the result of applying the \(i\)-th Demazure-Lusztig operator on weight.

INPUT:

  • weight – an element \(\lambda\) of the weight lattice
  • i – an element of the index set
  • q1,q2 – two elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

See demazure_lusztig_operators() for the details.

EXAMPLES:

sage: L = RootSystem(["A",1]).ambient_space()
sage: K = QQ['q1,q2']
sage: q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: KL.demazure_lusztig_operator_on_basis(L((2,2)), 1, q1, q2)
q1*B[(2, 2)]
sage: KL.demazure_lusztig_operator_on_basis(L((3,0)), 1, q1, q2)
(q1+q2)*B[(1, 2)] + (q1+q2)*B[(2, 1)] + (q1+q2)*B[(3, 0)] + q1*B[(0, 3)]
sage: KL.demazure_lusztig_operator_on_basis(L((0,3)), 1, q1, q2)
(-q1-q2)*B[(1, 2)] + (-q1-q2)*B[(2, 1)] + (-q2)*B[(3, 0)]

At \(q_1=1\) and \(q_2=0\) we recover the action of the isobaric divided differences \(\pi_i\):

sage: KL.demazure_lusztig_operator_on_basis(L((2,2)), 1, 1, 0)
B[(2, 2)]
sage: KL.demazure_lusztig_operator_on_basis(L((3,0)), 1, 1, 0)
B[(1, 2)] + B[(2, 1)] + B[(3, 0)] + B[(0, 3)]
sage: KL.demazure_lusztig_operator_on_basis(L((0,3)), 1, 1, 0)
-B[(1, 2)] - B[(2, 1)]

Or \(1-\pi_i\) for bar=True:

sage: KL.demazure_lusztig_operator_on_basis(L((2,2)), 1, 1, 0, convention="bar")
0
sage: KL.demazure_lusztig_operator_on_basis(L((3,0)), 1, 1, 0, convention="bar")
-B[(1, 2)] - B[(2, 1)] - B[(0, 3)]
sage: KL.demazure_lusztig_operator_on_basis(L((0,3)), 1, 1, 0, convention="bar")
B[(1, 2)] + B[(2, 1)] + B[(0, 3)]

At \(q_1=1\) and \(q_2=-1\) we recover the action of the simple reflection \(s_i\):

sage: KL.demazure_lusztig_operator_on_basis(L((2,2)), 1, 1, -1)
B[(2, 2)]
sage: KL.demazure_lusztig_operator_on_basis(L((3,0)), 1, 1, -1)
B[(0, 3)]
sage: KL.demazure_lusztig_operator_on_basis(L((0,3)), 1, 1, -1)
B[(3, 0)]
demazure_lusztig_operator_on_classical_on_basis(weight, i, q, q1, q2, convention='antidominant')

Return the result of applying the \(i\)-th Demazure-Lusztig operator on the classical weight weight embedded at level 0.

INPUT:

  • weight – a classical weight \(\lambda\)
  • i – an element of the index set
  • q1,q2 – two elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

See demazure_lusztig_operators() for the details.

Todo

  • Optimize the code to only do the embedding/projection for T_0
  • Add an option to specify at which level one wants to work. Currently this is level 0.

EXAMPLES:

sage: L = RootSystem(["A",1,1]).ambient_space()
sage: L0 = L.classical()
sage: K = QQ['q,q1,q2']
sage: q, q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: KL0 = L0.algebra(K)

These operators coincide with the usual Demazure-Lusztig operators:

sage: KL.demazure_lusztig_operator_on_classical_on_basis(L0((2,2)), 1, q, q1, q2)
q1*B[(2, 2)]
sage: KL0.demazure_lusztig_operator_on_basis(L0((2,2)), 1, q1, q2)
q1*B[(2, 2)]

sage: KL.demazure_lusztig_operator_on_classical_on_basis(L0((3,0)), 1, q, q1, q2)
(q1+q2)*B[(1, 2)] + (q1+q2)*B[(2, 1)] + (q1+q2)*B[(3, 0)] + q1*B[(0, 3)]
sage: KL0.demazure_lusztig_operator_on_basis(L0((3,0)), 1, q1, q2)
(q1+q2)*B[(1, 2)] + (q1+q2)*B[(2, 1)] + (q1+q2)*B[(3, 0)] + q1*B[(0, 3)]

except that we now have an action of \(T_0\), which introduces some \(q\) s:

sage: KL.demazure_lusztig_operator_on_classical_on_basis(L0((2,2)), 0, q, q1, q2)
q1*B[(2, 2)]
sage: KL.demazure_lusztig_operator_on_classical_on_basis(L0((3,0)), 0, q, q1, q2)
(-q^2*q1-q^2*q2)*B[(1, 2)] + (-q*q1-q*q2)*B[(2, 1)] + (-q^3*q2)*B[(0, 3)]
demazure_lusztig_operators(q1, q2, convention='antidominant')

Return the Demazure-Lusztig operators acting on self.

INPUT:

  • q1,q2 – two elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

If \(R\) is the parent weight ring, the Demazure-Lusztig operator \(T_i\) is the linear map \(R\rightarrow R\) obtained by interpolating between the isobaric divided difference operator \(\pi_i\) (see isobaric_divided_difference_on_basis()) and the simple reflection \(s_i\).

\[(q_1+q_2) \pi_i - q_2 s_i\]

The Demazure-Lusztig operators give the usual representation of the operator \(T_i\) of the (affine) Hecke algebra with eigenvalues \(q_1\) and \(q_2\) associated to the Weyl group.

Several variants are available to match with various conventions used in the literature:

  • “bar” replaces \(\pi_i\) in the formula above by \(\overline{\pi}_i = (1-\pi_i)\).
  • “dominant” conjugates the operator by \(x^\lambda \mapsto x^-\lambda\).

The names dominant and antidominant for the conventions were chosen with regards to the nonsymmetric Macdonald polynomials. The \(Y\) operators for the Macdonald polynomials in the “dominant” convention satisfy \(Y_\lambda = T_{t_{\lambda}}\) for \(\lambda\) dominant. This is also the convention used in [Haiman06]. For the “antidominant” convention, \(Y_\lambda = T_{t_{\lambda}}\) with \(\lambda\) antidominant.

REFERENCES:

[Lusztig1985]G. Lusztig, Equivariant K-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 337-342.
[Cherednik1995]I. Cherednik, Nonsymmetric Macdonald polynomials. IMRN 10, 483-515 (1995).

EXAMPLES:

sage: L = RootSystem(["A",1]).ambient_space()
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: T = KL.demazure_lusztig_operators(q1, q2)
sage: Tbar = KL.demazure_lusztig_operators(q1, q2, convention="bar")
sage: Tdominant = KL.demazure_lusztig_operators(q1, q2, convention="dominant")
sage: x = KL.monomial(L((3,0)))
sage: T[1](x)
(q1+q2)*B[(1, 2)] + (q1+q2)*B[(2, 1)] + (q1+q2)*B[(3, 0)] + q1*B[(0, 3)]
sage: Tbar[1](x)
(-q1-q2)*B[(1, 2)] + (-q1-q2)*B[(2, 1)] + (-q1-2*q2)*B[(0, 3)]
sage: Tbar[1](x) + T[1](x)
(q1+q2)*B[(3, 0)] + (-2*q2)*B[(0, 3)]
sage: Tdominant[1](x)
(-q1-q2)*B[(1, 2)] + (-q1-q2)*B[(2, 1)] + (-q2)*B[(0, 3)]

sage: Tdominant.Tw_inverse(1)(KL.monomial(-L.simple_root(1)))
((-q1-q2)/(q1*q2))*B[(0, 0)] - 1/q2*B[(1, -1)]

We repeat similar computation in the affine setting:

sage: L = RootSystem(["A",2,1]).ambient_space()
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: T = KL.demazure_lusztig_operators(q1, q2)
sage: Tbar = KL.demazure_lusztig_operators(q1, q2, convention="bar")
sage: Tdominant = KL.demazure_lusztig_operators(q1, q2, convention="dominant")
sage: e = L.basis()
sage: x = KL.monomial(3*e[0])
sage: T[1](x)
(q1+q2)*B[e[0] + 2*e[1]] + (q1+q2)*B[2*e[0] + e[1]] + (q1+q2)*B[3*e[0]] + q1*B[3*e[1]]
sage: Tbar[1](x)
(-q1-q2)*B[e[0] + 2*e[1]] + (-q1-q2)*B[2*e[0] + e[1]] + (-q1-2*q2)*B[3*e[1]]
sage: Tbar[1](x) + T[1](x)
(q1+q2)*B[3*e[0]] + (-2*q2)*B[3*e[1]]
sage: Tdominant[1](x)
(-q1-q2)*B[e[0] + 2*e[1]] + (-q1-q2)*B[2*e[0] + e[1]] + (-q2)*B[3*e[1]]
sage: Tdominant.Tw_inverse(1)(KL.monomial(-L.simple_root(1)))
((-q1-q2)/(q1*q2))*B[0] - 1/q2*B[e[0] - e[1]]

One can obtain iterated operators by passing a reduced word or an element of the Weyl group:

sage: T[1,2](x)
(q1^2+2*q1*q2+q2^2)*B[e[0] + e[1] + e[2]] +
(q1^2+2*q1*q2+q2^2)*B[e[0] + 2*e[1]] +
(q1^2+q1*q2)*B[e[0] + 2*e[2]] + (q1^2+2*q1*q2+q2^2)*B[2*e[0] + e[1]] +
(q1^2+q1*q2)*B[2*e[0] + e[2]] + (q1^2+q1*q2)*B[3*e[0]] +
(q1^2+q1*q2)*B[e[1] + 2*e[2]] + (q1^2+q1*q2)*B[2*e[1] + e[2]] +
(q1^2+q1*q2)*B[3*e[1]] + q1^2*B[3*e[2]]

and use that to check, for example, the braid relations:

sage: T[1,2,1](x) - T[2,1,2](x)
0

The operators satisfy the relations of the affine Hecke algebra with parameters \(q_1\), \(q_2\):

sage: T._test_relations()
sage: Tdominant._test_relations()
sage: Tbar._test_relations() #-q2,q1+2*q2   # todo: not implemented: set the appropriate eigenvalues!

And the \(\bar{T}\) are basically the inverses of the \(T\) s:

sage: Tinv = KL.demazure_lusztig_operators(2/q1+1/q2,-1/q1,convention="bar")
sage: [Tinv[1](T[1](x))-x for x in KL.some_elements()]
[0, 0, 0, 0, 0, 0, 0]

We check that \(\Lambda_1-\Lambda_0\) is an eigenvector for the \(Y\) s in affine type:

sage: K = QQ['q,q1,q2'].fraction_field()
sage: q,q1,q2=K.gens()
sage: L = RootSystem(["A",2,1]).ambient_space()
sage: L0 = L.classical()
sage: Lambda = L.fundamental_weights()
sage: alphacheck = L0.simple_coroots()
sage: KL = L.algebra(K)
sage: T = KL.demazure_lusztig_operators(q1, q2, convention="dominant")
sage: Y = T.Y()
sage: alphacheck = Y.keys().alpha() # alpha of coroot lattice is alphacheck
sage: alphacheck
Finite family {0: alphacheck[0], 1: alphacheck[1], 2: alphacheck[2]}
sage: x = KL.monomial(Lambda[1]-Lambda[0]); x
B[e[0]]

In fact it is not exactly an eigenvector, but the extra ‘delta` term is to be interpreted as a \(q\) parameter:

sage: Y[alphacheck[0]](KL.one())
q2^2/q1^2*B[0]
sage: Y[alphacheck[1]](x)
((-q2^2)/(-q1^2))*B[e[0] - e['delta']]
sage: Y[alphacheck[2]](x)
(q1/(-q2))*B[e[0]]
sage: KL.q_project(Y[alphacheck[1]](x),q)
((-q2^2)/(-q*q1^2))*B[(1, 0, 0)]

sage: KL.q_project(x, q)
B[(1, 0, 0)]
sage: KL.q_project(Y[alphacheck[0]](x),q)
((-q*q1)/q2)*B[(1, 0, 0)]
sage: KL.q_project(Y[alphacheck[1]](x),q)
((-q2^2)/(-q*q1^2))*B[(1, 0, 0)]
sage: KL.q_project(Y[alphacheck[2]](x),q)
(q1/(-q2))*B[(1, 0, 0)]

We now check systematically that the Demazure-Lusztig operators satisfy the relations of the Iwahori-Hecke algebra:

sage: K = QQ['q1,q2']
sage: q1, q2 = K.gens()
sage: for cartan_type in CartanType.samples(crystallographic=True): # long time 12s
....:    L = RootSystem(cartan_type).root_lattice()
....:    KL = L.algebra(K)
....:    T = KL.demazure_lusztig_operators(q1,q2)
....:    T._test_relations()

sage: for cartan_type in CartanType.samples(crystallographic=True): # long time 12s
....:    L = RootSystem(cartan_type).weight_lattice()
....:    KL = L.algebra(K)
....:    T = KL.demazure_lusztig_operators(q1,q2)
....:    T._test_relations()

Recall that the Demazure-Lusztig operators are only defined when all monomials belong to the weight lattice. Thus, in the group algebra of the ambient space, we need to specify explicitly the elements on which to run the tests:

sage: for cartan_type in CartanType.samples(crystallographic=True): # long time 12s
....:    L = RootSystem(cartan_type).ambient_space()
....:    KL = L.algebra(K)
....:    weight_lattice = RootSystem(cartan_type).weight_lattice(extended=L.is_extended())
....:    elements = [ KL.monomial(L(weight)) for weight in weight_lattice.some_elements() ]
....:    T = KL.demazure_lusztig_operators(q1,q2)
....:    T._test_relations(elements=elements)
demazure_lusztig_operators_on_classical(q, q1, q2, convention='antidominant')

Return the Demazure-Lusztig operators acting at level 1 on self.classical().

INPUT:

  • q,q1,q2 – three elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

Let \(KL\) be the group algebra of an affine weight lattice realization \(L\). The Demazure-Lusztig operators for \(KL\) act on the group algebra of the corresponding classical weight lattice by embedding it at level 1, and projecting back.

EXAMPLES:

sage: L = RootSystem(["A",1,1]).ambient_space()
sage: K = QQ['q,q1,q2'].fraction_field()
sage: q, q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: KL0 = KL.classical()
sage: L0 = KL0.basis().keys()
sage: T = KL.demazure_lusztig_operators_on_classical(q, q1, q2)

sage: x = KL0.monomial(L0((3,0))); x
B[(3, 0)]

For \(T_1,\dots\) we recover the usual Demazure-Lusztig operators:

sage: T[1](x)
(q1+q2)*B[(1, 2)] + (q1+q2)*B[(2, 1)] + (q1+q2)*B[(3, 0)] + q1*B[(0, 3)]

For \(T_0\), we can note that, in the projection, \(\delta\) is mapped to \(q\):

sage: T[0](x)
(-q^2*q1-q^2*q2)*B[(1, 2)] + (-q*q1-q*q2)*B[(2, 1)] + (-q^3*q2)*B[(0, 3)]

Note that there is no translation part, and in particular 1 is an eigenvector for all \(T_i\)‘s:

sage: T[0](KL0.one())
q1*B[(0, 0)]
sage: T[1](KL0.one())
q1*B[(0, 0)]

sage: Y = T.Y()
sage: alphacheck=Y.keys().simple_roots()
sage: Y[alphacheck[0]](KL0.one())
((-q2)/(q*q1))*B[(0, 0)]

Matching with Ion Bogdan’s hand calculations from 3/15/2013:

sage: L = RootSystem(["A",1,1]).weight_space(extended=True)
sage: K = QQ['q,u'].fraction_field()
sage: q, u = K.gens()
sage: KL = L.algebra(K)
sage: KL0 = KL.classical()
sage: L0 = KL0.basis().keys()
sage: omega = L0.fundamental_weights()
sage: T = KL.demazure_lusztig_operators_on_classical(q, u, -1/u, convention="dominant")
sage: Y = T.Y()
sage: alphacheck = Y.keys().simple_roots()

sage: Ydelta = Y[Y.keys().null_root()]
sage: Ydelta.word, Ydelta.signs, Ydelta.scalar
((), (), 1/q)

sage: Y1 = Y[alphacheck[1]]
sage: Y1.word, Y1.signs, Y1.scalar # This is T_0 T_1 (T_1 acts first, then T_0); Ion gets T_1 T_0
((1, 0), (1, 1), 1)

sage: Y0 = Y[alphacheck[0]]
sage: Y0.word, Y0.signs, Y0.scalar # This is 1/q T_1^-1 T_0^-1
((0, 1), (-1, -1), 1/q)

Note that the following computations use the “dominant” convention:

sage: T0 = T.Tw(0)
sage: T0(KL0.monomial(omega[1]))
q*u*B[-Lambda[1]] + ((u^2-1)/u)*B[Lambda[1]]
sage: T0(KL0.monomial(2*omega[1]))
((q*u^2-q)/u)*B[0] + q^2*u*B[-2*Lambda[1]] + ((u^2-1)/u)*B[2*Lambda[1]]

sage: T0(KL0.monomial(-omega[1]))
1/(q*u)*B[Lambda[1]]
sage: T0(KL0.monomial(-2*omega[1]))
((-u^2+1)/(q*u))*B[0] + 1/(q^2*u)*B[2*Lambda[1]]
demazure_operators()

Return the Demazure operators acting on self.

The \(i\)-th Demazure operator is defined by:

\[\pi_i = \frac{ 1 - e^{-\alpha_i}s_i }{ 1-e^{-\alpha_i} }\]

It acts on \(e^\lambda\), for \(\lambda\) a weight, by:

\[\pi_i e^\lambda = \frac{e^\lambda - e^{-\alpha_i+s_i\lambda}}{1-e^{-\alpha_i}}\]

This matches with Lascoux’ definition [Lascoux2003] of \(\pi_i\), and with the \(i\)-th Demazure operator of [Kumar1987], which also works for general Kac-Moody types.

REFERENCES:

[Kumar1987]S. Kumar, Demazure character formula in arbitrary Kac-Moody setting, Invent. Math. 89 (1987), no. 2, 395-423.

EXAMPLES:

We compute some Schur functions, as images of dominant monomials under the action of the maximal isobaric divided difference \(\Delta_{w_0}\):

sage: L = RootSystem(["A",2]).ambient_lattice()
sage: KL = L.algebra(QQ)
sage: w0 = tuple(L.weyl_group().long_element().reduced_word())
sage: pi = KL.demazure_operators()
sage: pi0 = pi[w0]
sage: pi0(KL.monomial(L((2,1))))
2*B[(1, 1, 1)] + B[(1, 2, 0)] + B[(1, 0, 2)] + B[(2, 1, 0)] + B[(2, 0, 1)] + B[(0, 1, 2)] + B[(0, 2, 1)]

Let us make the result into an actual polynomial:

sage: P = QQ['x,y,z']
sage: pi0(KL.monomial(L((2,1)))).expand(P.gens())
x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2

This is indeed a Schur function:

sage: s = SymmetricFunctions(QQ).s()
sage: s[2,1].expand(3, P.gens())
x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2

Let us check this systematically on Schur functions of degree 6:

sage: for p in Partitions(6, max_length=3).list():
....:     assert s.monomial(p).expand(3, P.gens()) == pi0(KL.monomial(L(tuple(p)))).expand(P.gens())

We check systematically that these operators satisfy the Iwahori-Hecke algebra relations:

sage: for cartan_type in CartanType.samples(crystallographic=True): # long time 12s
....:     L = RootSystem(cartan_type).weight_lattice()
....:     KL = L.algebra(QQ)
....:     T = KL.demazure_operators()
....:     T._test_relations()

sage: L = RootSystem(['A',1,1]).weight_lattice()
sage: KL = L.algebra(QQ)
sage: T = KL.demazure_operators()
sage: T._test_relations()

Warning

The Demazure operators are only defined if all the elements in the support have integral scalar products with the coroots (basically, they are in the weight lattice). Otherwise an error is raised:

sage: L = RootSystem(CartanType(["G",2]).dual()).ambient_space()
sage: KL = L.algebra(QQ)
sage: pi = KL.demazure_operators()
sage: pi[1](KL.monomial(L([0,0,1])))
Traceback (most recent call last):
...
ValueError: the weight does not have an integral scalar product with the coroot
divided_difference_on_basis(weight, i)

Return the result of applying the \(i\)-th divided difference on weight.

INPUT:

  • weight – a weight
  • i – an element of the index set

Todo

type free definition (Viviane’s definition uses that we are in the ambient space)

EXAMPLES:

sage: L = RootSystem(["A",1]).ambient_space()
sage: KL = L.algebra(QQ)
sage: KL.divided_difference_on_basis(L((2,2)), 1) # todo: not implemented
0
sage: KL.divided_difference_on_basis(L((3,0)), 1) # todo: not implemented
B[(2, 0)] + B[(1, 1)] + B[(0, 2)]
sage: KL.divided_difference_on_basis(L((0,3)), 1) # todo: not implemented
-B[(2, 0)] - B[(1, 1)] - B[(0, 2)]

In type \(A\) and in the ambient lattice, we recover the usual action of divided differences polynomials:

sage: x,y = QQ['x,y'].gens()
sage: d = lambda p: (p - p(y,x)) / (x-y)
sage: d(x^2*y^2)
0
sage: d(x^3)
x^2 + x*y + y^2
sage: d(y^3)
-x^2 - x*y - y^2
from_polynomial(p)

Construct an element of self from a polynomial \(p\).

INPUT:

  • p – a polynomial

EXAMPLES:

sage: L = RootSystem(["A",2]).ambient_lattice()
sage: KL = L.algebra(QQ)
sage: x,y,z = QQ['x,y,z'].gens()
sage: KL.from_polynomial(x)
B[(1, 0, 0)]
sage: KL.from_polynomial(x^2*y + 2*y - z)
B[(2, 1, 0)] + 2*B[(0, 1, 0)] - B[(0, 0, 1)]

TESTS:

sage: KL.from_polynomial(x).leading_support().parent() is L
True
sage: KL.from_polynomial(x-x)
0
sage: KL.from_polynomial(x-x).parent() is KL
True

Todo

make this work for Laurent polynomials too

isobaric_divided_difference_on_basis(weight, i)

Return the result of applying the \(i\)-th isobaric divided difference on weight.

INPUT:

  • weight – a weight
  • i – an element of the index set

EXAMPLES:

sage: L = RootSystem(["A",1]).ambient_space()
sage: KL = L.algebra(QQ)
sage: KL.isobaric_divided_difference_on_basis(L((2,2)), 1)
B[(2, 2)]
sage: KL.isobaric_divided_difference_on_basis(L((3,0)), 1)
B[(1, 2)] + B[(2, 1)] + B[(3, 0)] + B[(0, 3)]
sage: KL.isobaric_divided_difference_on_basis(L((0,3)), 1)
-B[(1, 2)] - B[(2, 1)]

In type \(A\) and in the ambient lattice, we recover the usual action of divided differences on polynomials:

sage: x,y = QQ['x,y'].gens()
sage: d = lambda p: (x*p - (x*p)(y,x)) / (x-y)
sage: d(x^2*y^2)
x^2*y^2
sage: d(x^3)
x^3 + x^2*y + x*y^2 + y^3
sage: d(y^3)
-x^2*y - x*y^2

REFERENCES:

[Lascoux2003]Alain Lascoux, Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, 99, 2003.
q_project(x, q)

Implement the \(q\)-projection morphism from self to the group algebra of the classical space.

INPUT:

  • x – an element of the group algebra of self
  • q – an element of the ground ring

This is an algebra morphism mapping \(\delta\) to \(q\) and \(X^b\) to its classical counterpart for the other elements \(b\) of the basis of the realization.

EXAMPLES:

sage: K = QQ['q'].fraction_field()
sage: q = K.gen()
sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(K)
sage: L = KL.basis().keys()
sage: e = L.basis()
sage: x = KL.an_element() + KL.monomial(4*e[1] + 3*e[2] + e['deltacheck'] - 2*e['delta']); x
B[2*e[0] + 2*e[1] + 3*e[2]] + B[4*e[1] + 3*e[2] - 2*e['delta'] + e['deltacheck']]
sage: KL.q_project(x, q)
B[(2, 2, 3)] + 1/q^2*B[(0, 4, 3)]

sage: KL = RootSystem(["BC",3,2]).ambient_space().algebra(K)
sage: L = KL.basis().keys()
sage: e = L.basis()
sage: x = KL.an_element() + KL.monomial(4*e[1] + 3*e[2] + e['deltacheck'] - 2*e['delta']); x
B[2*e[0] + 2*e[1] + 3*e[2]] + B[4*e[1] + 3*e[2] - 2*e['delta'] + e['deltacheck']]
sage: KL.q_project(x, q)
B[(2, 2, 3)] + 1/q^2*B[(0, 4, 3)]

Warning

Recall that the null root, usually denoted \(\delta\), is in fact a[0]\delta in Sage’s notation, in order to avoid half integer coefficients (this only makes a difference in type BC). Similarly, what’s usually denoted \(q\) is in fact q^a[0] in Sage’s notations, to avoid manipulating square roots:

sage: KL.q_project(KL.monomial(L.null_root()),q)
q^2*B[(0, 0, 0)]
q_project_on_basis(l, q)

Return the monomial \(c * cl(l)\) in the group algebra of the classical lattice.

INPUT:

  • l – an element of the root lattice realization
  • q – an element of the ground ring

Here, \(cl(l)\) is the projection of \(l\) in the classical lattice, and \(c\) is the coefficient of \(l\) in \(\delta\).

EXAMPLES:

sage: K = QQ['q'].fraction_field()
sage: q = K.gen()
sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(K)
sage: L = KL.basis().keys()
sage: e = L.basis()
sage: KL.q_project_on_basis( 4*e[1] + 3*e[2] + e['deltacheck'] - 2*e['delta'], q)
1/q^2*B[(0, 4, 3)]
some_elements()

Return some elements of the algebra self.

EXAMPLES:

sage: A = RootSystem(["A",2,1]).ambient_space().algebra(QQ)
sage: A.some_elements()
[B[2*e[0] + 2*e[1] + 3*e[2]],
B[-e[0] + e[2] + e['delta']],
B[e[0] - e[1]],
B[e[1] - e[2]],
B[e['deltacheck']],
B[e[0] + e['deltacheck']],
B[e[0] + e[1] + e['deltacheck']]]

sage: A = RootSystem(["B",2]).weight_space().algebra(QQ)
sage: A.some_elements()
[B[2*Lambda[1] + 2*Lambda[2]],
B[2*Lambda[1] - 2*Lambda[2]],
B[-Lambda[1] + 2*Lambda[2]],
B[Lambda[1]],
B[Lambda[2]]]
twisted_demazure_lusztig_operator_on_basis(weight, i, q1, q2, convention='antidominant')

Return the twisted Demazure-Lusztig operator acting on the basis.

INPUT:

  • weight – an element \(\lambda\) of the weight lattice
  • i – an element of the index set
  • q1,q2 – two elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

EXAMPLES:

sage: L = RootSystem(["A",3,1]).ambient_space()
sage: e = L.basis()
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: Lambda = L.classical().fundamental_weights()
sage: KL.twisted_demazure_lusztig_operator_on_basis(Lambda[1]+2*Lambda[2], 1, q1, q2, convention="dominant")
(-q2)*B[(2, 3, 0, 0)]
sage: KL.twisted_demazure_lusztig_operator_on_basis(Lambda[1]+2*Lambda[2], 2, q1, q2, convention="dominant")
(-q1-q2)*B[(3, 1, 1, 0)] + (-q2)*B[(3, 0, 2, 0)]
sage: KL.twisted_demazure_lusztig_operator_on_basis(Lambda[1]+2*Lambda[2], 3, q1, q2, convention="dominant")
q1*B[(3, 2, 0, 0)]
sage: KL.twisted_demazure_lusztig_operator_on_basis(Lambda[1]+2*Lambda[2], 0, q1, q2, convention="dominant")
((q1*q2+q2^2)/q1)*B[(1, 2, 1, 1)] + ((q1*q2+q2^2)/q1)*B[(1, 2, 2, 0)] + q2^2/q1*B[(1, 2, 0, 2)]
+ ((q1^2+2*q1*q2+q2^2)/q1)*B[(2, 1, 1, 1)] + ((q1^2+2*q1*q2+q2^2)/q1)*B[(2, 1, 2, 0)]
+ ((q1*q2+q2^2)/q1)*B[(2, 1, 0, 2)] + ((q1^2+2*q1*q2+q2^2)/q1)*B[(2, 2, 1, 0)] + ((q1*q2+q2^2)/q1)*B[(2, 2, 0, 1)]
twisted_demazure_lusztig_operators(q1, q2, convention='antidominant')

Return the twisted Demazure-Lusztig operators acting on self.

INPUT:

  • q1,q2 – two elements of the ground ring
  • convention – “antidominant”, “bar”, or “dominant” (default: “antidominant”)

Warning

  • the code is currently only tested for \(q_1q_2=-1\)
  • only the “dominant” convention is functional for \(i=0\)

For \(T_1,\ldots,T_n\), these operators are the usual Demazure-Lusztig operators. On the other hand, the operator \(T_0\) is twisted:

sage: L = RootSystem(["A",3,1]).ambient_space()
sage: e = L.basis()
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KL = L.algebra(K)
sage: T = KL.twisted_demazure_lusztig_operators(q1, q2, convention="dominant")
sage: T._test_relations()

TESTS:

The following computations were checked with Mark Shimozono for type \(A_1^{(1)}\):

sage: L = RootSystem(["A",1,1]).ambient_space()
sage: e = L.basis()
sage: K = QQ['q1,q2'].fraction_field()
sage: q1,q2 = K.gens()
sage: KL = L.algebra(K)
sage: T = KL.twisted_demazure_lusztig_operators(q1, q2, convention="dominant")
sage: T._test_relations()
sage: L0 = L.classical()
sage: alpha = L0.simple_roots()
sage: T.Ti_on_basis(L0.zero(), 1)
q1*B[(0, 0)]
sage: T.Ti_inverse_on_basis(L0.zero(), 1)
1/q1*B[(0, 0)]
sage: T.Ti_on_basis(alpha[1], 1)
(-q1-q2)*B[(0, 0)] + (-q2)*B[(-1, 1)]
sage: T.Ti_inverse_on_basis(alpha[1], 1)
((q1+q2)/(q1*q2))*B[(0, 0)] + 1/q1*B[(-1, 1)] + ((q1+q2)/(q1*q2))*B[(1, -1)]
sage: T.Ti_on_basis(L0.zero(), 0)
(q1+q2)*B[(0, 0)] + q1*B[(1, -1)]

The next computations were checked with Mark Shimozono for type \(A_2^{(1)}\):

sage: L = RootSystem(["A",2,1]).ambient_space()
sage: e = L.basis()
sage: K = QQ['u'].fraction_field()
sage: u = K.gen()
sage: KL = L.algebra(K)
sage: T = KL.twisted_demazure_lusztig_operators(u, -~u, convention="dominant")
sage: T._test_relations()
sage: L0 = L.classical()
sage: KL0 = L0.algebra(K)
sage: alpha = L0.simple_roots()

sage: phi = L0.highest_root(); phi
(1, 0, -1)
sage: phi.to_simple_root(reduced_word=True)
(2, (1,))
sage: res = T.Ti_on_basis(L0([1,0,1]), 1); res
1/u*B[(0, 1, 1)]
sage: res = res * KL0.monomial(-alpha[2]); res
1/u*B[(0, 0, 2)]
sage: res = T.Tw_inverse(2)(res); res
((u^2-1)/u^2)*B[(0, 1, 1)] + B[(0, 2, 0)]
sage: res = T.Tw_inverse(1)(res); res
((u^2-1)/u)*B[(1, 1, 0)] + ((u^2-1)/u)*B[(1, 0, 1)] + u*B[(2, 0, 0)]

Todo

Choose a good set of Cartan Type to run on. Rank >4 is too big. But \(C_1\) and \(B_1\) are boring.

We now check systematically that those operators satisfy the relations of the Iwahori-Hecke algebra:

sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: for cartan_type in CartanType.samples(affine=True, crystallographic=True): # long time 12s
....:     if cartan_type.rank() > 4: continue
....:     if cartan_type.type() == 'BC': continue
....:     KL = RootSystem(cartan_type).weight_lattice().algebra(K)
....:     T = KL.twisted_demazure_lusztig_operators(q1, q2, convention="dominant")
....:     T._test_relations()

Todo

Investigate why \(T_0^\vee\) currently does not satisfy the quadratic relation in type \(BC\). This should hopefuly be fixed when \(T_0^\vee\) will have a more uniform implementation:

sage: cartan_type = CartanType(["BC",1,2])
sage: KL = RootSystem(cartan_type).weight_lattice().algebra(K)
sage: T = KL.twisted_demazure_lusztig_operators(q1,q2, convention="dominant")
sage: T._test_relations()
... tester.assert_(Ti(Ti(x,i,-q2),i,-q1).is_zero()) ...
AssertionError: False is not true

Comparison with T0:

sage: L = RootSystem(["A",2,1]).ambient_space()
sage: e = L.basis()
sage: K = QQ['t,q'].fraction_field()
sage: t,q = K.gens()
sage: q1 = t
sage: q2 = -1
sage: KL = L.algebra(K)
sage: L0 = L.classical()
sage: T = KL.demazure_lusztig_operators(q1,q2, convention="dominant")
sage: def T0(*l0): return KL.q_project(T[0].on_basis()(L.embed_at_level(L0(l0), 1)), q)
sage: T0_check_on_basis = KL.T0_check_on_basis(q1, q2, convention="dominant")
sage: def T0c(*l0): return T0_check_on_basis(L0(l0))

sage: T0(0,0,1)                                 # not double checked
((-t+1)/q)*B[(1, 0, 0)] + 1/q^2*B[(2, 0, -1)]
sage: T0c(0,0,1)
(t^2-t)*B[(1, 0, 0)] + (t^2-t)*B[(1, 1, -1)] + t^2*B[(2, 0, -1)] + (t-1)*B[(0, 0, 1)]

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