Group algebras of root lattice realizations#

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

Bases: AlgebrasCategory

The category of group algebras of root lattice realizations.

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

class ElementMethods#

Bases: object

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()                                                # needs sage.libs.gap
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()                              # needs sage.libs.gap
[1, 2, 3]
sage: m.acted_upon(w)                                                   # needs sage.libs.gap
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
class ParentMethods#

Bases: object

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)

M. 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()                            # needs sage.graphs
sage: f = KL.T0_check_on_basis(q1,q2, convention="dominant")            # needs sage.graphs
sage: f(L0.zero())                                                      # needs sage.graphs
(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")            # needs sage.graphs
sage: f(L0.zero())       # not checked                                  # needs sage.graphs
(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:                                            # needs sage.graphs
....:     print("{} : {}".format(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")            # needs sage.graphs
sage: T = KL.twisted_demazure_lusztig_operators(q1,q2, convention="dominant")

Direct calculation:

sage: T.Tw(0)(KL0.monomial(L0([0,0])))                                  # needs sage.graphs
((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]))     # needs sage.graphs
((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()
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)),     # needs sage.graphs
....:                                                    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)),     # needs sage.graphs
....:                                                    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)),     # needs sage.graphs
....:                                                    0, q, q1, q2)
q1*B[(2, 2)]
sage: KL.demazure_lusztig_operator_on_classical_on_basis(L0((3,0)),     # needs sage.graphs
....:                                                    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.

See also

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()]                # needs sage.graphs
[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()                                  # needs sage.graphs
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                         # needs sage.graphs
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: # needs sage.graphs
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: # needs sage.graphs
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.

See also

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)         # needs sage.graphs

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)                                                           # needs sage.graphs
(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)                                                           # needs sage.graphs
(-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: # needs sage.graphs
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: # needs sage.graphs
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: # needs sage.graphs
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())          # needs sage.libs.gap
sage: pi = KL.demazure_operators()
sage: pi0 = pi[w0]                                                      # needs sage.libs.gap
sage: pi0(KL.monomial(L((2,1))))                                        # needs sage.libs.gap
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())                       # needs sage.libs.gap
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()                                    # needs sage.combinat
sage: s[2,1].expand(3, P.variable_names())                              # needs sage.combinat
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():                      # needs sage.combinat sage.libs.gap
....:     assert (s.monomial(p).expand(3, P.variable_names())
....:              == 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)]

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

See also

demazure_operators()

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)                    # needs sage.graphs
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 \(\delta\).

See also

q_project_on_basis()

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]]...]
sage: A.some_elements()                                                 # needs sage.graphs
[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[Lambda[1]], B[Lambda[2]]]
sage: A.some_elements()                                                 # needs sage.graphs
[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")

See also

twisted_demazure_lusztig_operators()

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(                    # needs sage.graphs
....:     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()

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 hopefully 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()                                           # needs sage.graphs
Traceback (most recent call last):
... tester.assertTrue(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,                  # needs sage.graphs
....:                                          convention="dominant")
sage: def T0c(*l0): return T0_check_on_basis(L0(l0))

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