Nonsymmetric Macdonald polynomials¶
AUTHORS:
- Anne Schilling and Nicolas M. Thiery (2013): initial version
ACKNOWLEDGEMENTS:
The initial version of this code (together with root_lattice_realization_algebras.Algebras
and hecke_algebra_representation.HeckeAlgebraRepresentation
) was written by
Anne Schilling and Nicolas M. Thiery during the ICERM Semester Program on “Automorphic Forms,
Combinatorial Representation Theory and Multiple Dirichlet Series” (January 28, 2013 - May 3, 2013)
with the help of Dan Bump, Ben Brubaker, Bogdan Ion, Dan Orr, Arun Ram, Siddhartha Sahi, and Mark Shimozono.
Special thanks go to Bogdan Ion and Mark Shimozono for their patient explanations and hand computations
to check the code.
-
class
sage.combinat.root_system.non_symmetric_macdonald_polynomials.
NonSymmetricMacdonaldPolynomials
(KL, q, q1, q2, normalized)¶ Bases:
sage.combinat.root_system.hecke_algebra_representation.CherednikOperatorsEigenvectors
Nonsymmetric Macdonald polynomials
INPUT:
KL
– an affine Cartan type or the group algebra of a realization of the affine weight latticeq
,q1
,q2
– parameters in the base ring of the group algebra (default:q
,q1
,q2
)normalized
– a boolean (default:True
) whether to normalize the result to have leading coefficient 1
This implementation covers all reduced affine root systems. The polynomials are constructed recursively by the application of intertwining operators.
Todo
- Non-reduced case (Koornwinder polynomials).
- Non-equal parameters for the affine Hecke algebra.
- Choice of convention (dominant/anti-dominant, ...).
- More uniform implementation of the
operator.
- Optimizations, in particular in the calculation of the eigenvalues for the recursion.
EXAMPLES:
We construct the family of nonsymmetric Macdonald polynomials in three variables in type
:
sage: E = NonSymmetricMacdonaldPolynomials(["A",2,1])
They are constructed as elements of the group algebra of the classical weight lattice
(or one of its realizations, such as the ambient space, which is used here) and indexed by elements of
:
sage: L0 = E.keys(); L0 Ambient space of the Root system of type ['A', 2]
Here is the nonsymmetric Macdonald polynomial with leading term
:
sage: E[L0([2,0,1])] ((-q*q1-q*q2)/(-q*q1-q2))*B[(1, 1, 1)] + ((-q1-q2)/(-q*q1-q2))*B[(2, 1, 0)] + B[(2, 0, 1)]
It can be seen as a polynomial (or in general a Laurent polynomial) by interpreting each term as an exponent vector. The parameter
is the exponential of the null (co)root, whereas
and
are the two eigenvalues of each generator
of the affine Hecke algebra (see the background section for details).
By setting
,
and using the
root_lattice_realization_algebras.Algebras.ElementMethods.expand()
method, we recover the nonsymmetric Macdonald polynomial as computed by [HHL06]‘s combinatorial formula:sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: E = NonSymmetricMacdonaldPolynomials(["A",2,1], q=q, q1=t, q2=-1) sage: vars = K['x0,x1,x2'].gens() sage: E[L0([2,0,1])].expand(vars) ((-t + 1)/(-q*t + 1))*x0^2*x1 + x0^2*x2 + ((-q*t + q)/(-q*t + 1))*x0*x1*x2 sage: from sage.combinat.sf.ns_macdonald import E sage: E([2,0,1]) ((-t + 1)/(-q*t + 1))*x0^2*x1 + x0^2*x2 + ((-q*t + q)/(-q*t + 1))*x0*x1*x2
Here is a type
nonsymmetric Macdonald polynomial:
sage: E = NonSymmetricMacdonaldPolynomials(["G",2,1]) sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: E[ omega[2]-omega[1] ] ((-q*q1^3*q2-q*q1^2*q2^2)/(q*q1^4-q2^4))*B[(0, 0, 0)] + B[(1, -1, 0)] + ((-q1*q2^3-q2^4)/(q*q1^4-q2^4))*B[(1, 0, -1)]
Many more examples are given after the background section.
Background
The polynomial module
The nonsymmetric Macdonald polynomials are a distinguished basis of the “polynomial” module of the affine Hecke algebra. Given:
- a ground ring `K`, which contains the input parameters `q, q_1, q_2` - an affine root system, specified by a Cartan type `C` - a realization `L` of the weight lattice of type `C`
the polynomial module is the group algebra
of the classical weight lattice
with coefficients in
. It is isomorphic to the Laurent polynomial ring over
generated by the formal exponentials of any basis of
.
In our running example
is the ambient space of type
:
sage: K = QQ['q,q1,q2'].fraction_field() sage: q, q1, q2 = K.gens() sage: C = CartanType(["C",2,1]) sage: L = RootSystem(C).ambient_space(); L Ambient space of the Root system of type ['C', 2, 1] sage: L.simple_roots() Finite family {0: -2*e[0] + e['delta'], 1: e[0] - e[1], 2: 2*e[1]} sage: omega = L.fundamental_weights(); omega Finite family {0: e['deltacheck'], 1: e[0] + e['deltacheck'], 2: e[0] + e[1] + e['deltacheck']} sage: L0 = L.classical(); L0 Ambient space of the Root system of type ['C', 2] sage: KL0 = L0.algebra(K); KL0 Group algebra of the Ambient space of the Root system of type ['C', 2] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field
Affine Hecke algebra
The affine Hecke algebra is generated by elements
for
i
in the set of affine Dynkin nodes. They satisfy the same braid relations as the simple reflectionsof the affine Weyl group. The
satisfy the quadratic relation
where
and
are the input parameters. Some of the representation theory requires that
and
satisfy additional relations; typically one uses the specializations
and
or
and
). This can be achieved by constructing an appropriate field and passing
and
appropriately; see the examples. In principle, the parameter(s) could further depend on
i
; this is not yet implemented but the code has been designed in such a way that this feature is easy to add.Demazure-Lusztig operators
The
i
-th Demazure-Lusztig operator is an operator onwhich interpolates between the reflection
and the Demazure operator
(see
root_lattice_realization.RootLatticeRealization.Algebras.ParentMethods.demazure_lusztig_operators()
).:sage: KL = L.algebra(K); KL Group algebra of the Ambient space of the Root system of type ['C', 2, 1] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field sage: T = KL.demazure_lusztig_operators(q1, q2) sage: x = KL.monomial(omega[1]); x B[e[0] + e['deltacheck']] sage: T[2](x) q1*B[e[0] + e['deltacheck']] sage: T[1](x) (q1+q2)*B[e[0] + e['deltacheck']] + q1*B[e[1] + e['deltacheck']] sage: T[0](x) q1*B[e[0] + e['deltacheck']]
The affine Hecke algebra acts on
by letting the generators
act by the Demazure-Lusztig operators. The class
sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation
implements some simple generic features for representations of affine Hecke algebras defined by the action of their-generators.:
sage: T A representation of the (q1, q2)-Hecke algebra of type ['C', 2, 1] on Group algebra of the Ambient space of the Root system of type ['C', 2, 1] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field sage: type(T) <class 'sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation'> sage: T._test_relations() # long time (1.3s)
Here we construct the operator
from a signed reduced word:
sage: T.Tw([0,1,2],[1,-1,-1], q1^2) Generic endomorphism of Group algebra of the Ambient space of the Root system of type ['C', 2, 1] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field
(note the reversal of the word). Inverses are computed using the quadratic relation.
Cherednik operators
The affine Hecke algebra contains elements
indexed by the coroot lattice. Their action on
is implemented in Sage:
sage: Y = T.Y(); Y Lazy family (...)_{i in Coroot lattice of the Root system of type ['C', 2, 1]} sage: alphacheck = Y.keys().simple_roots() sage: Y1 = Y[alphacheck[1]] sage: Y1(x) ((q1^3+q1^2*q2)/(-q2^3))*B[-e[0] - 2*e[1] - e['delta'] + e['deltacheck']] + ((q1^3+2*q1^2*q2+q1*q2^2)/(-q2^3))*B[-e[0] - e['delta'] + e['deltacheck']] + ((q1^3+q1^2*q2)/(-q2^3))*B[e[0] - 2*e[1] - 2*e['delta'] + e['deltacheck']] + ((q1^3+q1^2*q2)/(-q2^3))*B[e[0] - 2*e[1] - e['delta'] + e['deltacheck']] + ((q1^3+2*q1^2*q2+q1*q2^2)/(-q2^3))*B[e[0] - 2*e['delta'] + e['deltacheck']] + ((q1^3+q1^2*q2)/(-q2^3))*B[e[0] - e['delta'] + e['deltacheck']] + ((q1^2+2*q1*q2+q2^2)/(-q1*q2))*B[e[0] + e['deltacheck']] + ((q1^3+q1^2*q2)/(-q2^3))*B[2*e[0] - e[1] - 2*e['delta'] + e['deltacheck']] + ((-q1^2-q1*q2)/(-q2^2))*B[2*e[0] - e[1] - e['delta'] + e['deltacheck']] + (q1^3/(-q2^3))*B[3*e[0] - 2*e[1] - 3*e['delta'] + e['deltacheck']] + ((q1^3+q1^2*q2)/(-q2^3))*B[3*e[0] - 3*e['delta'] + e['deltacheck']] + ((q1^3+2*q1^2*q2+q1*q2^2)/(-q2^3))*B[-e[1] - e['delta'] + e['deltacheck']] + ((-q1^2-2*q1*q2-q2^2)/(-q2^2))*B[-e[1] + e['deltacheck']] + ((q1+q2)/(-q2))*B[e[1] + e['deltacheck']]
The Cherednik operators span a Laurent polynomial ring inside the affine Hecke algebra; namely
is a group isomorphism from the classical root lattice (viewed additively) to the affine Hecke algebra (viewed multiplicatively). In practice,
is constructed by computing combinatorially its signed reduced word (and an overall scalar factor) using the periodic orientation of the alcove model in the coweight lattice (see
hecke_algebra_representation.HeckeAlgebraRepresentation.Y_lambdacheck()
):sage: Lcheck = L.root_system.coweight_lattice() sage: w = Lcheck.reduced_word_of_translation(Lcheck(alphacheck[1])); w [0, 2, 1, 0, 2, 1] sage: Lcheck.signs_of_alcovewalk(w) [1, -1, 1, -1, 1, 1]
Level zero representation of the affine Hecke algebra
The action of the affine Hecke algebra on
induces an action on
: the action of
on
for
a classical weight in
is obtained by embedding the weight at level zero in the affine weight lattice (see
weight_lattice_realizations.WeightLatticeRealizations.ParentMethods.embed_at_level()
) applying the Demazure-Lusztig operator there, and projecting frommapping the exponential of
to
(see
root_lattice_realization_algebras.Algebras.ParentMethods.q_project()
). This is implemented inroot_lattice_realization_algebras.Algebras.ParentMethods.demazure_lusztig_operators_on_classical()
:sage: T = KL.demazure_lusztig_operators_on_classical(q, q1,q2) sage: omega = L0.fundamental_weights() sage: x = KL0.monomial(omega[1]) sage: T[0](x) (-q*q2)*B[(-1, 0)]
For classical nodes these are the usual Demazure-Lusztig operators:
sage: T[1](x) (q1+q2)*B[(1, 0)] + q1*B[(0, 1)]
Nonsymmetric Macdonald polynomials
We can now finally define the nonsymmetric Macdonald polyomials. Because the Cherednik operators commute (and there is no radical), they can be simultaneously diagonalized; namely,
admits a
-basis of joint eigenvectors for the
. For
, the nonsymmetric Macdonald polynomial
is the unique eigenvector of the family of Cherednik operators
having
as leading term:
sage: E = NonSymmetricMacdonaldPolynomials(KL, q, q1, q2); E The family of the Macdonald polynomials of type ['C', 2, 1] with parameters q, q1, q2
Or for short:
sage: E = NonSymmetricMacdonaldPolynomials(C)
Recursive construction of the nonsymmetric Macdonald polynomials
The generators
of the affine Hecke algebra almost skew commute with the Cherednik operators. More precisely, one can deform them into the so-called intertwining operators:
(where
except for
in type
where
) which satisfy the following skew commutation relations:
If
, applying
on an eigenvector
produces a new eigenvector (essentially
) with a distinct eigenvalue. It follows that the eigenvectors indexed by an affine Weyl orbit of weights, may be recursively computed from a single weight in the orbit.
In the case at hand, there is a little complication: namely, the simple reflections
acting at level 0 do not act transitively on classical weights; in fact the orbits for the classical Weyl group and for the affine Weyl group are the same. Thus, one can construct the nonsymmetric Macdonald polynomials for all weights from those for the classical dominant weights, but one is lacking a creation operator to construct the nonsymmetric Macdonald polynomials for dominant weights.
Twisted Demazure-Lusztig operators
To compensate for this, one needs to consider another affinization of the action of the classical Demazure-Lusztig operators
, which gives rise to the double affine Hecke algebra. Following Cherednik, one adds another operator
implemented in:
root_lattice_realization_algebras.Algebras.ParentMethods.T0_check_on_basis()
. See also:root_lattice_realization_algebras.Algebras.ParentMethods.twisted_demazure_lusztig_operators()
.Depending on the type (untwisted or not), this is a representation of the affine Hecke algebra for another affinization of the classical Cartan type. The corresponding action of the affine Weyl group – which is used to compute the recursion on
– occurs in the corresponding weight lattice realization:
sage: E.L() Ambient space of the Root system of type ['C', 2, 1] sage: E.L_prime() Coambient space of the Root system of type ['B', 2, 1] sage: E.L_prime().classical() Ambient space of the Root system of type ['C', 2]
See
L_prime()
andcartan_type.CartanType_affine.other_affinization()
.REFERENCES:
[HHL06] (1, 2, 3, 4, 5) J. Haglund, M. Haiman and N. Loehr, A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math. 130, No. 2 (2008), 359-383. [LNSSS12] C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, preprint arXiv.1211.2042 [math.QA] More examples
We show how to create the nonsymmetric Macdonald polynomials in two different ways and check that they are the same:
sage: K = QQ['q,u'].fraction_field() sage: q, u = K.gens() sage: E = NonSymmetricMacdonaldPolynomials(['D',3,1], q, u, -1/u) sage: omega = E.keys().fundamental_weights() sage: E[omega[1]+omega[3]] ((-q*u^2+q)/(-q*u^4+1))*B[(1/2, -1/2, 1/2)] + ((-q*u^2+q)/(-q*u^4+1))*B[(1/2, 1/2, -1/2)] + B[(3/2, 1/2, 1/2)] sage: KL = RootSystem(["D",3,1]).ambient_space().algebra(K) sage: P = NonSymmetricMacdonaldPolynomials(KL, q, u, -1/u) sage: E[omega[1]+omega[3]] == P[omega[1]+omega[3]] True sage: E[E.keys()((0,1,-1))] ((-q*u^2+q)/(-q*u^2+1))*B[(0, 0, 0)] + ((-u^2+1)/(-q*u^2+1))*B[(1, 1, 0)] + ((-u^2+1)/(-q*u^2+1))*B[(1, 0, -1)] + B[(0, 1, -1)]
In type
, there is also a combinatorial implementation of the nonsymmetric Macdonald polynomials in terms of augmented diagram fillings as in [HHL06]. See
sage.combinat.sf.ns_macdonald.E()
. First we check that these polynomials are indeed eigenvectors of the Cherednik operators:sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: q1 = t; q2 = -1 sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(K) sage: KL0 = KL.classical() sage: E = NonSymmetricMacdonaldPolynomials(KL,q, q1, q2) sage: omega = E.keys().fundamental_weights() sage: w = omega[1] sage: import sage.combinat.sf.ns_macdonald as NS sage: p = NS.E([1,0,0]); p x0 sage: pp = KL0.from_polynomial(p) sage: E.eigenvalues(KL0.from_polynomial(p)) [t, (-1)/(-q*t^2), t] sage: def eig(l): return E.eigenvalues(KL0.from_polynomial(NS.E(l))) sage: eig([1,0,0]) [t, (-1)/(-q*t^2), t] sage: eig([2,0,0]) [q*t, (-1)/(-q^2*t^2), t] sage: eig([3,0,0]) [q^2*t, (-1)/(-q^3*t^2), t] sage: eig([2,0,4]) [(-1)/(-q^3*t), 1/(q^2*t), q^4*t^2]
Next we check explicitly that they agree with the current implementation:
sage: K = QQ['q','t'].fraction_field() sage: q,t = K.gens() sage: KL = RootSystem(["A",1,1]).ambient_lattice().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t, -1) sage: L0 = E.keys() sage: KL0 = KL.classical() sage: P = K['x0,x1'] sage: def EE(weight): return E[L0(weight)].expand(P.gens()) sage: import sage.combinat.sf.ns_macdonald as NS sage: EE([0,0]) 1 sage: NS.E([0,0]) 1 sage: EE([1,0]) x0 sage: NS.E([1,0]) x0 sage: EE([0,1]) ((-t + 1)/(-q*t + 1))*x0 + x1 sage: NS.E([0,1]) ((-t + 1)/(-q*t + 1))*x0 + x1 sage: NS.E([2,0]) x0^2 + ((-q*t + q)/(-q*t + 1))*x0*x1 sage: EE([2,0]) x0^2 + ((-q*t + q)/(-q*t + 1))*x0*x1
The same, directly in the ambient lattice with several shifts:
sage: E[L0([2,0])] ((-q*t+q)/(-q*t+1))*B[(1, 1)] + B[(2, 0)] sage: E[L0([1,-1])] ((-q*t+q)/(-q*t+1))*B[(0, 0)] + B[(1, -1)] sage: E[L0([0,-2])] ((-q*t+q)/(-q*t+1))*B[(-1, -1)] + B[(0, -2)]
Systematic checks with Sage’s implementation of [HHL06]:
sage: assert all(EE([x,y]) == NS.E([x,y]) for d in range(5) for x,y in IntegerVectors(d,2))
With the current implementation, we can compute nonsymmetric Macdonald polynomials for any type, for example for type
:
sage: K=QQ['q,u'].fraction_field() sage: q, u = K.gens() sage: KL = RootSystem(["E",6,1]).weight_space(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q,u,-1/u) sage: L0 = E.keys() sage: E[L0.fundamental_weight(1).weyl_action([2,4,3,2,1])] ((-u^2+1)/(-q*u^16+1))*B[-Lambda[1] + Lambda[3]] + ((-u^2+1)/(-q*u^16+1))*B[Lambda[1]] + B[-Lambda[2] + Lambda[5]] + ((-u^2+1)/(-q*u^16+1))*B[Lambda[2] - Lambda[4] + Lambda[5]] + ((-u^2+1)/(-q*u^16+1))*B[-Lambda[3] + Lambda[4]] sage: E[L0.fundamental_weight(2).weyl_action([2,5,3,4,2])] # long time (6s) ((-q^2*u^20+q^2*u^18+q*u^2-q)/(-q^2*u^32+2*q*u^16-1))*B[0] + B[Lambda[1] - Lambda[3] + Lambda[4] - Lambda[5] + Lambda[6]] + ((-u^2+1)/(-q*u^16+1))*B[Lambda[1] - Lambda[3] + Lambda[5]] + ((-q*u^20+q*u^18+u^2-1)/(-q^2*u^32+2*q*u^16-1))*B[-Lambda[2] + Lambda[4]] + ((-q*u^20+q*u^18+u^2-1)/(-q^2*u^32+2*q*u^16-1))*B[Lambda[2]] + ((u^4-2*u^2+1)/(q^2*u^32-2*q*u^16+1))*B[Lambda[3] - Lambda[4] + Lambda[5]] + ((-u^2+1)/(-q*u^16+1))*B[Lambda[3] - Lambda[5] + Lambda[6]] sage: E[L0.fundamental_weight(1)+L0.fundamental_weight(6)] # long time (13s) ((q^2*u^10-q^2*u^8-q^2*u^2+q^2)/(q^2*u^26-q*u^16-q*u^10+1))*B[0] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[1] - Lambda[2] + Lambda[6]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[1] + Lambda[2] - Lambda[4] + Lambda[6]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[1] - Lambda[3] + Lambda[4] - Lambda[5] + Lambda[6]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[1] - Lambda[3] + Lambda[5]] + B[Lambda[1] + Lambda[6]] + ((-q*u^2+q)/(-q*u^10+1))*B[-Lambda[2] + Lambda[4]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[2]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[3] - Lambda[4] + Lambda[5]] + ((-q*u^2+q)/(-q*u^10+1))*B[Lambda[3] - Lambda[5] + Lambda[6]]
We test various other types:
sage: K=QQ['q,u'].fraction_field() sage: q, u = K.gens() sage: KL = RootSystem(["A",5,2]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL, q, u, -1/u) sage: L0 = E.keys() sage: E[L0.fundamental_weight(2)] ((-q*u^2+q)/(-q*u^8+1))*B[(0, 0, 0)] + B[(1, 1, 0)] sage: E[L0((0,-1,1))] # long time (1.5s) ((-q^2*u^10+q^2*u^8-q*u^6+q*u^4+q*u^2+u^2-q-1)/(-q^3*u^12+q^2*u^8+q*u^4-1))*B[(0, 0, 0)] + ((-u^2+1)/(-q*u^4+1))*B[(1, -1, 0)] + ((u^6-u^4-u^2+1)/(q^3*u^12-q^2*u^8-q*u^4+1))*B[(1, 1, 0)] + ((u^4-2*u^2+1)/(q^3*u^12-q^2*u^8-q*u^4+1))*B[(1, 0, -1)] + ((q^2*u^12-q^2*u^10-u^2+1)/(q^3*u^12-q^2*u^8-q*u^4+1))*B[(1, 0, 1)] + B[(0, -1, 1)] + ((-u^2+1)/(-q^2*u^8+1))*B[(0, 1, -1)] + ((-u^2+1)/(-q^2*u^8+1))*B[(0, 1, 1)] sage: K=QQ['q,u'].fraction_field() sage: q, u = K.gens() sage: KL = RootSystem(["E",6,2]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q,u,-1/u) sage: L0 = E.keys() sage: E[L0.fundamental_weight(4)] # long time (5s) ((-q^3*u^20+q^3*u^18+q^2*u^2-q^2)/(-q^3*u^28+q^2*u^22+q*u^6-1))*B[(0, 0, 0, 0)] + ((-q*u^2+q)/(-q*u^6+1))*B[(1/2, 1/2, -1/2, -1/2)] + ((-q*u^2+q)/(-q*u^6+1))*B[(1/2, 1/2, -1/2, 1/2)] + ((-q*u^2+q)/(-q*u^6+1))*B[(1/2, 1/2, 1/2, -1/2)] + ((-q*u^2+q)/(-q*u^6+1))*B[(1/2, 1/2, 1/2, 1/2)] + ((q*u^2-q)/(q*u^6-1))*B[(1, 0, 0, 0)] + B[(1, 1, 0, 0)] + ((-q*u^2+q)/(-q*u^6+1))*B[(0, 1, 0, 0)] sage: E[L0((1,-1,0,0))] # long time (23s) ((q^3*u^18-q^3*u^16+q*u^4-q^2*u^2-2*q*u^2+q^2+q)/(q^3*u^18-q^2*u^12-q*u^6+1))*B[(0, 0, 0, 0)] + ((-q^3*u^18+q^3*u^16+q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(1/2, -1/2, -1/2, -1/2)] + ((-q^3*u^18+q^3*u^16+q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(1/2, -1/2, -1/2, 1/2)] + ((q^3*u^18-q^3*u^16-q*u^2+q)/(q^3*u^18-q^2*u^12-q*u^6+1))*B[(1/2, -1/2, 1/2, -1/2)] + ((q^3*u^18-q^3*u^16-q*u^2+q)/(q^3*u^18-q^2*u^12-q*u^6+1))*B[(1/2, -1/2, 1/2, 1/2)] + ((q*u^8-q*u^6-q*u^2+q)/(q^3*u^18-q^2*u^12-q*u^6+1))*B[(1/2, 1/2, -1/2, -1/2)] + ((q*u^8-q*u^6-q*u^2+q)/(q^3*u^18-q^2*u^12-q*u^6+1))*B[(1/2, 1/2, -1/2, 1/2)] + ((-q*u^8+q*u^6+q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(1/2, 1/2, 1/2, -1/2)] + ((-q*u^8+q*u^6+q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(1/2, 1/2, 1/2, 1/2)] + ((-q^2*u^18+q^2*u^16-q*u^8+q*u^6+q*u^2+u^2-q-1)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(1, 0, 0, 0)] + B[(1, -1, 0, 0)] + ((-u^2+1)/(-q^2*u^12+1))*B[(1, 1, 0, 0)] + ((-u^2+1)/(-q^2*u^12+1))*B[(1, 0, -1, 0)] + ((u^2-1)/(q^2*u^12-1))*B[(1, 0, 1, 0)] + ((-u^2+1)/(-q^2*u^12+1))*B[(1, 0, 0, -1)] + ((-u^2+1)/(-q^2*u^12+1))*B[(1, 0, 0, 1)] + ((-q*u^2+q)/(-q*u^6+1))*B[(0, -1, 0, 0)] + ((-q*u^4+2*q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(0, 1, 0, 0)] + ((-q*u^4+2*q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(0, 0, -1, 0)] + ((-q*u^4+2*q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(0, 0, 1, 0)] + ((-q*u^4+2*q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(0, 0, 0, -1)] + ((-q*u^4+2*q*u^2-q)/(-q^3*u^18+q^2*u^12+q*u^6-1))*B[(0, 0, 0, 1)]
Next we test a twisted type (checked against Maple computation by Bogdan Ion for
and
):
sage: E = NonSymmetricMacdonaldPolynomials(["A",5,2]) sage: omega = E.keys() sage: E[omega[1]] B[(1, 0, 0)] sage: E[-omega[1]] B[(-1, 0, 0)] + ((-q*q1^6-q*q1^5*q2-q1*q2^5-q2^6)/(-q^3*q1^6-q^2*q1^5*q2-q*q1*q2^5-q2^6))*B[(1, 0, 0)] + ((-q1-q2)/(-q*q1-q2))*B[(0, -1, 0)] + ((q1+q2)/(q*q1+q2))*B[(0, 1, 0)] + ((-q1-q2)/(-q*q1-q2))*B[(0, 0, -1)] + ((-q1-q2)/(-q*q1-q2))*B[(0, 0, 1)] sage: E[omega[2]] ((-q1*q2^3-q2^4)/(q*q1^4-q2^4))*B[(1, 0, 0)] + B[(0, 1, 0)] sage: E[-omega[2]] ((q^2*q1^7+q^2*q1^6*q2-q1*q2^6-q2^7)/(q^3*q1^7-q^2*q1^5*q2^2+q*q1^2*q2^5-q2^7))*B[(1, 0, 0)] + B[(0, -1, 0)] + ((q*q1^5*q2^2+q*q1^4*q2^3-q1*q2^6-q2^7)/(q^3*q1^7-q^2*q1^5*q2^2+q*q1^2*q2^5-q2^7))*B[(0, 1, 0)] + ((-q1*q2-q2^2)/(q*q1^2-q2^2))*B[(0, 0, -1)] + ((q1*q2+q2^2)/(-q*q1^2+q2^2))*B[(0, 0, 1)] sage: E[-omega[1]-omega[2]] ((-q^3*q1^6-q^3*q1^5*q2-2*q^2*q1^6-3*q^2*q1^5*q2+q^2*q1^4*q2^2+2*q^2*q1^3*q2^3+q*q1^5*q2+2*q*q1^4*q2^2-q*q1^3*q2^3-2*q*q1^2*q2^4+q*q1*q2^5+q*q2^6-q1^3*q2^3-q1^2*q2^4+2*q1*q2^5+2*q2^6)/(-q^4*q1^6-q^3*q1^5*q2+q^3*q1^4*q2^2-q*q1^2*q2^4+q*q1*q2^5+q2^6))*B[(0, 0, 0)] + B[(-1, -1, 0)] + ((q*q1^4+q*q1^3*q2+q1*q2^3+q2^4)/(q^3*q1^4+q^2*q1^3*q2+q*q1*q2^3+q2^4))*B[(-1, 1, 0)] + ((q1+q2)/(q*q1+q2))*B[(-1, 0, -1)] + ((-q1-q2)/(-q*q1-q2))*B[(-1, 0, 1)] + ((q*q1^4+q*q1^3*q2+q1*q2^3+q2^4)/(q^3*q1^4+q^2*q1^3*q2+q*q1*q2^3+q2^4))*B[(1, -1, 0)] + ((-q^2*q1^6-q^2*q1^5*q2-q*q1^5*q2+q*q1^3*q2^3+q1^5*q2+q1^4*q2^2-q1^3*q2^3-q1^2*q2^4+q1*q2^5+q2^6)/(-q^4*q1^6-q^3*q1^5*q2+q^3*q1^4*q2^2-q*q1^2*q2^4+q*q1*q2^5+q2^6))*B[(1, 1, 0)] + ((-q*q1^4-2*q*q1^3*q2-q*q1^2*q2^2+q1^3*q2+q1^2*q2^2-q1*q2^3-q2^4)/(-q^3*q1^4-q^2*q1^3*q2-q*q1*q2^3-q2^4))*B[(1, 0, -1)] + ((-q*q1^4-2*q*q1^3*q2-q*q1^2*q2^2+q1^3*q2+q1^2*q2^2-q1*q2^3-q2^4)/(-q^3*q1^4-q^2*q1^3*q2-q*q1*q2^3-q2^4))*B[(1, 0, 1)] + ((q1+q2)/(q*q1+q2))*B[(0, -1, -1)] + ((-q1-q2)/(-q*q1-q2))*B[(0, -1, 1)] + ((q*q1^4+2*q*q1^3*q2+q*q1^2*q2^2-q1^3*q2-q1^2*q2^2+q1*q2^3+q2^4)/(q^3*q1^4+q^2*q1^3*q2+q*q1*q2^3+q2^4))*B[(0, 1, -1)] + ((q*q1^4+2*q*q1^3*q2+q*q1^2*q2^2-q1^3*q2-q1^2*q2^2+q1*q2^3+q2^4)/(q^3*q1^4+q^2*q1^3*q2+q*q1*q2^3+q2^4))*B[(0, 1, 1)] sage: E[omega[1]-omega[2]] ((q^3*q1^7+q^3*q1^6*q2-q*q1*q2^6-q*q2^7)/(q^3*q1^7-q^2*q1^5*q2^2+q*q1^2*q2^5-q2^7))*B[(0, 0, 0)] + B[(1, -1, 0)] + ((q*q1^5*q2^2+q*q1^4*q2^3-q1*q2^6-q2^7)/(q^3*q1^7-q^2*q1^5*q2^2+q*q1^2*q2^5-q2^7))*B[(1, 1, 0)] + ((-q1*q2-q2^2)/(q*q1^2-q2^2))*B[(1, 0, -1)] + ((q1*q2+q2^2)/(-q*q1^2+q2^2))*B[(1, 0, 1)] sage: E[omega[3]] ((-q1*q2^2-q2^3)/(-q*q1^3-q2^3))*B[(1, 0, 0)] + ((-q1*q2^2-q2^3)/(-q*q1^3-q2^3))*B[(0, 1, 0)] + B[(0, 0, 1)] sage: E[-omega[3]] ((q*q1^4*q2+q*q1^3*q2^2-q1*q2^4-q2^5)/(-q^2*q1^5-q2^5))*B[(1, 0, 0)] + ((q*q1^4*q2+q*q1^3*q2^2-q1*q2^4-q2^5)/(-q^2*q1^5-q2^5))*B[(0, 1, 0)] + B[(0, 0, -1)] + ((-q1*q2^4-q2^5)/(-q^2*q1^5-q2^5))*B[(0, 0, 1)]
Comparison with the energy function of crystals
Next we test that the nonsymmetric Macdonald polynomials at
match with the one-dimensional configuration sums involving Kirillov-Reshetikhin crystals for various types. See [LNSSS12]:
sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: KL = RootSystem(["A",5,2]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL, q, t, -1) sage: omega = E.keys().fundamental_weights() sage: E[-omega[1]].map_coefficients(lambda x:x.subs(t=0)) B[(-1, 0, 0)] + B[(1, 0, 0)] + B[(0, -1, 0)] + B[(0, 1, 0)] + B[(0, 0, -1)] + B[(0, 0, 1)] sage: E[-omega[2]].map_coefficients(lambda x:x.subs(t=0)) # long time (3s) (q+2)*B[(0, 0, 0)] + B[(-1, -1, 0)] + B[(-1, 1, 0)] + B[(-1, 0, -1)] + B[(-1, 0, 1)] + B[(1, -1, 0)] + B[(1, 1, 0)] + B[(1, 0, -1)] + B[(1, 0, 1)] + B[(0, -1, -1)] + B[(0, -1, 1)] + B[(0, 1, -1)] + B[(0, 1, 1)]
sage: KL = RootSystem(["C",3,1]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: E[-omega[2]].map_coefficients(lambda x:x.subs(t=0)) # long time (5s) 2*B[(0, 0, 0)] + B[(-1, -1, 0)] + B[(-1, 1, 0)] + B[(-1, 0, -1)] + B[(-1, 0, 1)] + B[(1, -1, 0)] + B[(1, 1, 0)] + B[(1, 0, -1)] + B[(1, 0, 1)] + B[(0, -1, -1)] + B[(0, -1, 1)] + B[(0, 1, -1)] + B[(0, 1, 1)]
sage: R = RootSystem(['C',3,1]) sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]) sage: E[-2*omega[1]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (15s) True sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2]) sage: E[-omega[1]-omega[2]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (45s) True
sage: R = RootSystem(['C',2,1]) sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: for d in range(1,3): # long time (10s) ....: for x,y in IntegerVectors(d,2): ....: weight = x*La[1]+y*La[2] ....: weight0 = -x*omega[1]-y*omega[2] ....: LS = crystals.ProjectedLevelZeroLSPaths(weight) ....: assert E[weight0].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q)
sage: R = RootSystem(['B',3,1]) sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]) sage: E[-2*omega[1]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (23s) True sage: B = crystals.KirillovReshetikhin(['B',3,1],1,1) sage: T = crystals.TensorProduct(B,B) sage: T.one_dimensional_configuration_sum(q) == LS.one_dimensional_configuration_sum(q) # long time (2s) True
sage: R = RootSystem(['BC',3,2]) sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]) sage: E[-2*omega[1]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (21s) True
sage: R = RootSystem(CartanType(['BC',3,2]).dual()) sage: KL = R.weight_space(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]) sage: g = E[-2*omega[1]].map_coefficients(lambda x:x.subs(t=0)) # long time (30s) sage: f = LS.one_dimensional_configuration_sum(q) # long time (1.5s) sage: P = g.support()[0].parent() # long time sage: B = P.algebra(q.parent()) # long time sage: sum(p[1]*B(P(p[0])) for p in f) == g # long time True
sage: C = CartanType(['G',2,1]) sage: R = RootSystem(C.dual()) sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL, q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]) sage: E[-2*omega[1]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (149s) True sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2]) sage: E[-omega[1]-omega[2]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time (23s) True
The next test breaks if the energy is not scaled by the translation factor for dual type
:
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2]) sage: E[-2*omega[1]-omega[2]].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q) # long time True sage: R = RootSystem(['D',4,1]) sage: KL = R.weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL, q, t,-1) sage: omega = E.keys().fundamental_weights() sage: La = R.weight_space().basis() sage: for d in range(1,2): # long time (41s) ....: for a,b,c,d in IntegerVectors(d,4): ....: weight = a*La[1]+b*La[2]+c*La[3]+d*La[4] ....: weight0 = -a*omega[1]-b*omega[2]-c*omega[3]-d*omega[4] ....: LS = crystals.ProjectedLevelZeroLSPaths(weight) ....: assert E[weight0].map_coefficients(lambda x:x.subs(t=0)) == LS.one_dimensional_configuration_sum(q)
TESTS:
Calculations checked with Bogdan Ion 2013/04/18:
sage: K = QQ['q,t'].fraction_field() sage: q,t=K.gens() sage: E = NonSymmetricMacdonaldPolynomials(["B",2,1], q=q,q1=t,q2=-1/t) sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: E[omega[1]] ((-q*t^4+q*t^2)/(-q*t^6+1))*B[(0, 0)] + B[(1, 0)] sage: E[omega[2]] B[(1/2, 1/2)] sage: E[-omega[1]] ((-q^2*t^8+q^2*t^6-q*t^6+2*q*t^4-q*t^2+t^2-1)/(-q^3*t^8+q^2*t^6+q*t^2-1))*B[(0, 0)] + B[(-1, 0)] + ((-q*t^8+q*t^6+t^2-1)/(-q^3*t^8+q^2*t^6+q*t^2-1))*B[(1, 0)] + ((-t^2+1)/(-q*t^2+1))*B[(0, -1)] + ((t^2-1)/(q*t^2-1))*B[(0, 1)] sage: E[L0([0,1])] ((-q*t^4+q*t^2)/(-q*t^4+1))*B[(0, 0)] + ((-t^2+1)/(-q*t^4+1))*B[(1, 0)] + B[(0, 1)] sage: E[L0([1,1])] ((q*t^2-q)/(q*t^2-1))*B[(0, 0)] + ((-q*t^2+q)/(-q*t^2+1))*B[(1, 0)] + B[(1, 1)] + ((-q*t^2+q)/(-q*t^2+1))*B[(0, 1)] sage: E = NonSymmetricMacdonaldPolynomials(["A",2,1], q=q,q1=t,q2=-1/t) sage: L0 = E.keys() sage: factor(E[L0([-1,0,1])][L0.zero()]) (t - 1) * (t + 1) * (q*t^2 - 1)^-3 * (q*t^2 + 1)^-1 * (q^3*t^6 + 2*q^2*t^6 - 3*q^2*t^4 - 2*q*t^2 - t^2 + q + 2)
Checking step by step calculations in type
with Bogdan Ion 2013/04/18:
sage: K = QQ['q,t'].fraction_field() sage: q,t=K.gens() sage: E = NonSymmetricMacdonaldPolynomials(["BC",1,2], q=q,q1=t,q2=-1/t) sage: KL0 = E.domain() sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: e = L0.basis() sage: E._T_Y[1] ( KL0.monomial(e[0]) ) 1/t*B[(-1)] sage: E._T_Y[0] ( KL0.monomial(L0.zero()) ) t*B[(0)] sage: E._T_Y[0] ( KL0.monomial(-e[0])) ((-t^2+1)/(q*t))*B[(0)] + 1/(q^2*t)*B[(1)] sage: Y = E.Y() sage: alphacheck = Y.keys().simple_roots() sage: Y0 = Y[alphacheck[0]] sage: Y1 = Y[alphacheck[1]] sage: Y0 Generic endomorphism of Group algebra of the Ambient space of the Root system of type ['C', 1] over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field sage: Y0.word, Y0.signs, Y0.scalar ((0, 1), (-1, -1), 1/q) sage: Y1.word, Y1.signs, Y1.scalar ((1, 0), (1, 1), 1) sage: T0_check = E._T[0]
Comparing with Bogdan Ion’s hand calculations for type
, 2013/05/13:
Todo
add his notes in latex
sage: K = QQ['q,q1,q2'].fraction_field() sage: q,q1,q2=K.gens() sage: L = RootSystem(["A",4,2]).ambient_space() sage: L.cartan_type() ['BC', 2, 2] sage: L.null_root() 2*e['delta'] sage: L.simple_roots() Finite family {0: -e[0] + e['delta'], 1: e[0] - e[1], 2: 2*e[1]} sage: KL = L.algebra(K) sage: KL0 = KL.classical() sage: L0 = L.classical() sage: L0.cartan_type() ['C', 2] sage: E = NonSymmetricMacdonaldPolynomials(KL, q=q,q1=q1,q2=q2) sage: E.keys() Ambient space of the Root system of type ['C', 2] sage: E.keys().simple_roots() Finite family {1: (1, -1), 2: (0, 2)} sage: omega = E.keys().fundamental_weights() sage: E[0*omega[1]] B[(0, 0)] sage: E[omega[1]] ((-q*q1*q2^3-q*q2^4)/(q^2*q1^4-q2^4))*B[(0, 0)] + B[(1, 0)] sage: E[2*omega[2]] # long time # not checked against Bogdan's notes, but a good self-consistency test ((-q^12*q1^6-q^12*q1^5*q2+2*q^10*q1^5*q2+5*q^10*q1^4*q2^2+3*q^10*q1^3*q2^3+2*q^8*q1^5*q2+4*q^8*q1^4*q2^2+q^8*q1^3*q2^3-q^8*q1^2*q2^4+q^8*q1*q2^5+q^8*q2^6-q^6*q1^3*q2^3+q^6*q1^2*q2^4+4*q^6*q1*q2^5+2*q^6*q2^6+q^4*q1^3*q2^3+3*q^4*q1^2*q2^4+4*q^4*q1*q2^5+2*q^4*q2^6)/(-q^12*q1^6-q^10*q1^5*q2-q^8*q1^3*q2^3+q^6*q1^4*q2^2-q^6*q1^2*q2^4+q^4*q1^3*q2^3+q^2*q1*q2^5+q2^6))*B[(0, 0)] + ((q^7*q1^2*q2+2*q^7*q1*q2^2+q^7*q2^3+q^5*q1^2*q2+2*q^5*q1*q2^2+q^5*q2^3)/(-q^8*q1^3-q^6*q1^2*q2+q^2*q1*q2^2+q2^3))*B[(-1, 0)] + ((q^6*q1*q2+q^6*q2^2)/(-q^6*q1^2+q2^2))*B[(-1, -1)] + ((q^6*q1^2*q2+2*q^6*q1*q2^2+q^6*q2^3+q^4*q1^2*q2+2*q^4*q1*q2^2+q^4*q2^3)/(-q^8*q1^3-q^6*q1^2*q2+q^2*q1*q2^2+q2^3))*B[(-1, 1)] + ((q^3*q1*q2+q^3*q2^2)/(-q^6*q1^2+q2^2))*B[(-1, 2)] + ((-q^7*q1^3-q^7*q1^2*q2+q^7*q1*q2^2+q^7*q2^3+2*q^5*q1^2*q2+4*q^5*q1*q2^2+2*q^5*q2^3+2*q^3*q1^2*q2+4*q^3*q1*q2^2+2*q^3*q2^3)/(-q^8*q1^3-q^6*q1^2*q2+q^2*q1*q2^2+q2^3))*B[(1, 0)] + ((-q^6*q1^2*q2-2*q^6*q1*q2^2-q^6*q2^3-q^4*q1^2*q2-2*q^4*q1*q2^2-q^4*q2^3)/(q^8*q1^3+q^6*q1^2*q2-q^2*q1*q2^2-q2^3))*B[(1, -1)] + ((q^8*q1^3+q^8*q1^2*q2+q^6*q1^3+q^6*q1^2*q2-q^6*q1*q2^2-q^6*q2^3-2*q^4*q1^2*q2-4*q^4*q1*q2^2-2*q^4*q2^3-q^2*q1^2*q2-3*q^2*q1*q2^2-2*q^2*q2^3)/(q^8*q1^3+q^6*q1^2*q2-q^2*q1*q2^2-q2^3))*B[(1, 1)] + ((-q^5*q1^2-q^5*q1*q2+q^3*q1*q2+q^3*q2^2+q*q1*q2+q*q2^2)/(-q^6*q1^2+q2^2))*B[(1, 2)] + ((-q^6*q1^2-q^6*q1*q2+q^4*q1*q2+q^4*q2^2+q^2*q1*q2+q^2*q2^2)/(-q^6*q1^2+q2^2))*B[(2, 0)] + ((q^3*q1*q2+q^3*q2^2)/(-q^6*q1^2+q2^2))*B[(2, -1)] + ((-q^5*q1^2-q^5*q1*q2+q^3*q1*q2+q^3*q2^2+q*q1*q2+q*q2^2)/(-q^6*q1^2+q2^2))*B[(2, 1)] + B[(2, 2)] + ((-q^7*q1^2*q2-2*q^7*q1*q2^2-q^7*q2^3-q^5*q1^2*q2-2*q^5*q1*q2^2-q^5*q2^3)/(q^8*q1^3+q^6*q1^2*q2-q^2*q1*q2^2-q2^3))*B[(0, -1)] + ((q^7*q1^3+q^7*q1^2*q2-q^7*q1*q2^2-q^7*q2^3-2*q^5*q1^2*q2-4*q^5*q1*q2^2-2*q^5*q2^3-2*q^3*q1^2*q2-4*q^3*q1*q2^2-2*q^3*q2^3)/(q^8*q1^3+q^6*q1^2*q2-q^2*q1*q2^2-q2^3))*B[(0, 1)] + ((-q^6*q1^2-q^6*q1*q2+q^4*q1*q2+q^4*q2^2+q^2*q1*q2+q^2*q2^2)/(-q^6*q1^2+q2^2))*B[(0, 2)] sage: E.recursion(2*omega[2]) [0, 1, 0, 2, 1, 0, 2, 1, 0]
Some tests that the
s are implemented properly by hand defining the
s in terms of them:
sage: T = E._T_Y sage: Ye1 = T.Tw((1,2,1,0), scalar = (-1/(q1*q2))^2) sage: Ye2 = T.Tw((2,1,0,1), signs = (1,1,1,-1), scalar = (-1/(q1*q2))) sage: Yalpha0 = T.Tw((0,1,2,1), signs = (-1,-1,-1,-1), scalar = q^-1*(-q1*q2)^2) sage: Yalpha1 = T.Tw((1,2,0,1,2,0), signs=(1,1,-1,1,-1,1), scalar = -1/(q1*q2)) sage: Yalpha2 = T.Tw((2,1,0,1,2,1,0,1), signs = (1,1,1,-1,1,1,1,-1), scalar = (1/(q1*q2))^2) sage: Ye1(KL0.one()) q1^2/q2^2*B[(0, 0)] sage: Ye2(KL0.one()) ((-q1)/q2)*B[(0, 0)] sage: Yalpha0(KL0.one()) q2^2/(q*q1^2)*B[(0, 0)] sage: Yalpha1(KL0.one()) ((-q1)/q2)*B[(0, 0)] sage: Yalpha2(KL0.one()) q1^2/q2^2*B[(0, 0)]
Testing the
s directly:
sage: Y = E.Y() sage: Y.keys() Coroot lattice of the Root system of type ['BC', 2, 2] sage: alpha = Y.keys().simple_roots() sage: L(alpha[0]) -2*e[0] + e['deltacheck'] sage: L(alpha[1]) e[0] - e[1] sage: L(alpha[2]) e[1] sage: Y[alpha[0]].word (0, 1, 2, 1) sage: Y[alpha[0]].signs (-1, -1, -1, -1) sage: Y[alpha[0]].scalar # mind that Sage's q is the usual q^{1/2} q1^2*q2^2/q sage: Y[alpha[0]](KL0.one()) q2^2/(q*q1^2)*B[(0, 0)] sage: Y[alpha[1]].word (1, 2, 0, 1, 2, 0) sage: Y[alpha[1]].signs (1, 1, -1, 1, -1, 1) sage: Y[alpha[1]].scalar 1/(-q1*q2) sage: Y[alpha[2]].word # Bogdan says it should be the square of that; do we need to take translation factors into account or not? (2, 1, 0, 1) sage: Y[alpha[2]].signs (1, 1, 1, -1) sage: Y[alpha[2]].scalar 1/(-q1*q2)
Checking the provided nonsymmetric Macdonald polynomial:
sage: E10 = KL0.monomial(L0((1,0))) + KL0( q*(1-(-q1/q2)) / (1-q^2*(-q1/q2)^4) ) sage: E10 == E[omega[1]] True sage: E.eigenvalues(E10) # not checked [q*q1^2/q2^2, q2^3/(-q^2*q1^3), q1/(-q2)]
Checking T0check:
sage: T0check_on_basis = KL.T0_check_on_basis(q1,q2, convention="dominant") sage: T0check_on_basis.phi # note: this is in fact a0 phi (2, 0) sage: T0check_on_basis.v # what to match it with? (1,) sage: T0check_on_basis.j # what to match it with? 2 sage: T0check_on_basis(KL0.basis().keys().zero()) ((-q1^2)/q2)*B[(1, 0)] sage: T0check = E._T[0] sage: T0check(KL0.one()) ((-q1^2)/q2)*B[(1, 0)]
Systematic tests of nonsymmetric Macdonald polynomials in type
, in the weight lattice. Each time, we specify the eigenvalues for the action of
, and
:
sage: K = QQ['q','t'].fraction_field() sage: q,t = K.gens() sage: KL = RootSystem(["A",1,1]).weight_lattice(extended=True).algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t, -1) sage: omega = E.keys().fundamental_weights() sage: x = E[0*omega[1]]; x B[0] sage: E.eigenvalues(x) [1/(q*t), t] sage: x.is_one() True sage: x.parent() Group algebra of the Weight lattice of the Root system of type ['A', 1] over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field sage: E[omega[1]] B[Lambda[1]] sage: E.eigenvalues(_) [t, 1/(q*t)] sage: E[2*omega[1]] ((-q*t+q)/(-q*t+1))*B[0] + B[2*Lambda[1]] sage: E.eigenvalues(_) [q*t, 1/(q^2*t)] sage: E[3*omega[1]] ((-q^2*t+q^2)/(-q^2*t+1))*B[-Lambda[1]] + ((-q^2*t+q^2-q*t+q)/(-q^2*t+1))*B[Lambda[1]] + B[3*Lambda[1]] sage: E.eigenvalues(_) [q^2*t, 1/(q^3*t)] sage: E[4*omega[1]] ((q^5*t^2-q^5*t+q^4*t^2-2*q^4*t+q^3*t^2+q^4-2*q^3*t+q^3-q^2*t+q^2)/(q^5*t^2-q^3*t-q^2*t+1))*B[0] + ((-q^3*t+q^3)/(-q^3*t+1))*B[-2*Lambda[1]] + ((-q^3*t+q^3-q^2*t+q^2-q*t+q)/(-q^3*t+1))*B[2*Lambda[1]] + B[4*Lambda[1]] sage: E.eigenvalues(_) [q^3*t, 1/(q^4*t)] sage: E[6*omega[1]] ((-q^12*t^3+q^12*t^2-q^11*t^3+2*q^11*t^2-2*q^10*t^3-q^11*t+4*q^10*t^2-2*q^9*t^3-2*q^10*t+5*q^9*t^2-2*q^8*t^3-4*q^9*t+6*q^8*t^2-q^7*t^3+q^9-5*q^8*t+5*q^7*t^2-q^6*t^3+q^8-6*q^7*t+4*q^6*t^2+2*q^7-5*q^6*t+2*q^5*t^2+2*q^6-4*q^5*t+q^4*t^2+2*q^5-2*q^4*t+q^4-q^3*t+q^3)/(-q^12*t^3+q^9*t^2+q^8*t^2+q^7*t^2-q^5*t-q^4*t-q^3*t+1))*B[0] + ((-q^5*t+q^5)/(-q^5*t+1))*B[-4*Lambda[1]] + ((q^9*t^2-q^9*t+q^8*t^2-2*q^8*t+q^7*t^2+q^8-2*q^7*t+q^6*t^2+q^7-2*q^6*t+q^5*t^2+q^6-2*q^5*t+q^5-q^4*t+q^4)/(q^9*t^2-q^5*t-q^4*t+1))*B[-2*Lambda[1]] + ((q^9*t^2-q^9*t+q^8*t^2-2*q^8*t+2*q^7*t^2+q^8-3*q^7*t+2*q^6*t^2+q^7-4*q^6*t+2*q^5*t^2+2*q^6-4*q^5*t+q^4*t^2+2*q^5-3*q^4*t+q^3*t^2+2*q^4-2*q^3*t+q^3-q^2*t+q^2)/(q^9*t^2-q^5*t-q^4*t+1))*B[2*Lambda[1]] + ((q^5*t-q^5+q^4*t-q^4+q^3*t-q^3+q^2*t-q^2+q*t-q)/(q^5*t-1))*B[4*Lambda[1]] + B[6*Lambda[1]] sage: E.eigenvalues(_) [q^5*t, 1/(q^6*t)] sage: E[-omega[1]] B[-Lambda[1]] + ((-t+1)/(-q*t+1))*B[Lambda[1]] sage: E.eigenvalues(_) [(-1)/(-q^2*t), q*t]
As expected,
is not an eigenvector:
sage: E.eigenvalues(KL.classical().monomial(-omega[1])) Traceback (most recent call last): ... AssertionError
We proceed by comparing against the examples from the appendix of [HHL06] in type
:
sage: K = QQ['q','t'].fraction_field() sage: q,t = K.gens() sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t, -1) sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: P = K['x1,x2,x3'] sage: def EE(weight): return E[L0(weight)].expand(P.gens()) sage: EE([0,0,0]) 1 sage: EE([1,0,0]) x1 sage: EE([0,1,0]) ((-t + 1)/(-q*t^2 + 1))*x1 + x2 sage: EE([0,0,1]) ((-t + 1)/(-q*t + 1))*x1 + ((-t + 1)/(-q*t + 1))*x2 + x3 sage: EE([1,1,0]) x1*x2 sage: EE([1,0,1]) ((-t + 1)/(-q*t^2 + 1))*x1*x2 + x1*x3 sage: EE([0,1,1]) ((-t + 1)/(-q*t + 1))*x1*x2 + ((-t + 1)/(-q*t + 1))*x1*x3 + x2*x3 sage: EE([2,0,0]) x1^2 + ((-q*t + q)/(-q*t + 1))*x1*x2 + ((-q*t + q)/(-q*t + 1))*x1*x3 sage: EE([0,2,0]) ((-t + 1)/(-q^2*t^2 + 1))*x1^2 + ((-q^2*t^3 + q^2*t^2 - q*t^2 + 2*q*t - q + t - 1)/(-q^3*t^3 + q^2*t^2 + q*t - 1))*x1*x2 + x2^2 + ((q*t^2 - 2*q*t + q)/(q^3*t^3 - q^2*t^2 - q*t + 1))*x1*x3 + ((-q*t + q)/(-q*t + 1))*x2*x3
Systematic checks with Sage’s implementation of [HHL06]:
sage: import sage.combinat.sf.ns_macdonald as NS sage: assert all(EE([x,y,z]) == NS.E([x,y,z]) for d in range(5) for x,y,z in IntegerVectors(d,3)) # long time (9s)
We check that we get eigenvectors for generic
,
:
sage: K = QQ['q,q1,q2'].fraction_field() sage: q,q1,q2 = K.gens() sage: KL = RootSystem(["A",2,1]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, q1, q2) sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: E[2*omega[2]] ((q*q1+q*q2)/(q*q1+q2))*B[(1, 2, 1)] + ((q*q1+q*q2)/(q*q1+q2))*B[(2, 1, 1)] + B[(2, 2, 0)] sage: for d in range(4): # long time (9s) ... for weight in IntegerVectors(d,3).map(list).map(L0): ... eigenvalues = E.eigenvalues(E[L0(weight)])
Some type
calculations:
sage: K = QQ['q','t'].fraction_field() sage: q, t = K.gens() sage: KL = RootSystem(["C",2,1]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t, -1) sage: L0 = E.keys() sage: omega = L0.fundamental_weights() sage: E[0*omega[1]] B[(0, 0)] sage: E.eigenvalues(_) # checked for i=0 with previous calculation [1/(q*t^3), t, t] sage: E[omega[1]] B[(1, 0)] sage: E.eigenvalues(_) # not checked [t, 1/(q*t^3), t] sage: E[-omega[1]] # consistent with before refactoring B[(-1, 0)] + ((-t+1)/(-q*t+1))*B[(1, 0)] + ((-t+1)/(-q*t+1))*B[(0, -1)] + ((t-1)/(q*t-1))*B[(0, 1)] sage: E.eigenvalues(_) # not checked [(-1)/(-q^2*t^3), q*t, t] sage: E[-omega[1]+omega[2]] # consistent with before refactoring ((-t+1)/(-q*t^3+1))*B[(1, 0)] + B[(0, 1)] sage: E.eigenvalues(_) # not checked [t, q*t^3, (-1)/(-q*t^2)] sage: E[omega[1]-omega[2]] # consistent with before refactoring ((-t+1)/(-q*t^2+1))*B[(1, 0)] + B[(0, -1)] + ((-t+1)/(-q*t^2+1))*B[(0, 1)] sage: E.eigenvalues(_) # not checked [1/(q^2*t^3), 1/(q*t), q*t^2] sage: E[-omega[2]] ((-q^2*t^4+q^2*t^3-q*t^3+2*q*t^2-q*t+t-1)/(-q^3*t^4+q^2*t^3+q*t-1))*B[(0, 0)] + B[(-1, -1)] + ((-t+1)/(-q*t+1))*B[(-1, 1)] + ((t-1)/(q*t-1))*B[(1, -1)] + ((-q*t^4+q*t^3+t-1)/(-q^3*t^4+q^2*t^3+q*t-1))*B[(1, 1)] sage: E.eigenvalues(_) # not checked # long time (1s) [1/(q^3*t^3), t, q*t] sage: E[-omega[2]].map_coefficients(lambda c: c.subs(t=0)) # checking againsts crystals B[(0, 0)] + B[(-1, -1)] + B[(-1, 1)] + B[(1, -1)] + B[(1, 1)] sage: E[2*omega[2]] ((-q^6*t^7+q^6*t^6-q^5*t^6+2*q^5*t^5-q^4*t^5-q^5*t^3+3*q^4*t^4-3*q^4*t^3+q^3*t^4+q^4*t^2-2*q^3*t^2+q^3*t-q^2*t+q^2)/(-q^6*t^7+q^5*t^6+q^4*t^4+q^3*t^4-q^3*t^3-q^2*t^3-q*t+1))*B[(0, 0)] + ((-q^3*t^2+q^3*t)/(-q^3*t^3+1))*B[(-1, -1)] + ((-q^3*t^3+2*q^3*t^2-q^3*t)/(-q^4*t^4+q^3*t^3+q*t-1))*B[(-1, 1)] + ((-q^3*t^3+2*q^3*t^2-q^3*t)/(-q^4*t^4+q^3*t^3+q*t-1))*B[(1, -1)] + ((-q^4*t^4+q^4*t^3-q^3*t^3+2*q^3*t^2-q^2*t^3-q^3*t+2*q^2*t^2-q^2*t+q*t-q)/(-q^4*t^4+q^3*t^3+q*t-1))*B[(1, 1)] + ((q*t-q)/(q*t-1))*B[(2, 0)] + B[(2, 2)] + ((-q*t+q)/(-q*t+1))*B[(0, 2)] sage: E.eigenvalues(_) # not checked [q^3*t^3, t, (-1)/(-q^2*t^2)]
The following computations were calculated by hand:
sage: KL0 = KL.classical() sage: E11 = KL0.sum_of_terms([[L0([1,1]), 1], [L0([0,0]), (-q*t^2 + q*t)/(1-q*t^3)]]) sage: E11 == E[omega[2]] True sage: E.eigenvalues(E11) [q*t^3, t, (-1)/(-q*t^2)] sage: E1m1 = KL0.sum_of_terms([[L0([1,-1]), 1], [L0([1,1]), (1-t)/(1-q*t^2)], [L0([0,0]), q*t*(1-t)/(1-q*t^2)] ]) sage: E1m1 == E[2*omega[1]-omega[2]] True sage: E.eigenvalues(E1m1) [1/(q*t), 1/(q^2*t^3), q*t^2]
Now we present an example for a twisted affine root system. The results are eigenvectors:
sage: K = QQ['q','t'].fraction_field() sage: q, t = K.gens() sage: KL = RootSystem("C2~*").ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, t, -1) sage: omega = E.keys().fundamental_weights() sage: E[0*omega[1]] B[(0, 0)] sage: E.eigenvalues(_) [1/(q*t^2), t, t] sage: E[omega[1]] ((-q*t+q)/(-q*t^2+1))*B[(0, 0)] + B[(1, 0)] sage: E.eigenvalues(_) [q*t^2, 1/(q^2*t^3), t] sage: E[-omega[1]] ((-q*t+q-t+1)/(-q^2*t+1))*B[(0, 0)] + B[(-1, 0)] + ((-t+1)/(-q^2*t+1))*B[(1, 0)] + ((-t+1)/(-q^2*t+1))*B[(0, -1)] + ((t-1)/(q^2*t-1))*B[(0, 1)] sage: E.eigenvalues(_) [(-1)/(-q^3*t^2), q^2*t, t] sage: E[-omega[1]+omega[2]] B[(-1/2, 1/2)] + ((-t+1)/(-q^2*t^3+1))*B[(1/2, -1/2)] + ((-q*t^3+q*t^2-t+1)/(-q^2*t^3+1))*B[(1/2, 1/2)] sage: E.eigenvalues(_) [(-1)/(-q^2*t^2), q^2*t^3, (-1)/(-q*t)] sage: E[omega[1]-omega[2]] B[(1/2, -1/2)] + ((-t+1)/(-q*t^2+1))*B[(1/2, 1/2)] sage: E.eigenvalues(_) [t, 1/(q^2*t^3), q*t^2]
Type BC, comparison with calculations with Maple by Bogdan Ion:
sage: K = QQ['q','t'].fraction_field() sage: q,t = K.gens() sage: def to_SR(x): return x.expand([SR.var('x%s'%i) for i in range(1,x.parent().basis().keys().dimension()+1)]).subs(q=SR.var('q'), t=SR.var('t')) sage: var('x1,x2,x3') (x1, x2, x3) sage: E = NonSymmetricMacdonaldPolynomials(["BC",2,2], q=q, q1=t^2,q2=-1) sage: omega=E.keys().fundamental_weights() sage: expected = (t-1)*(t+1)*(2+q^4+2*q^2-2*t^2-2*q^2*t^2-t^4*q^2-q^4*t^4+t^4-3*q^6*t^6-2*q^4*t^6+2*q^6*t^8+2*q^4*t^8+t^10*q^8)*q^4/((q^2*t^3-1)*(q^2*t^3+1)*(t*q-1)*(t*q+1)*(t^2*q^3+1)*(t^2*q^3-1))+(t-1)^2*(t+1)^2*(2*q^2+q^4+2+q^4*t^2)*q^3*x1/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1))+(t-1)^2*(t+1)^2*(q^2+1)*q^5/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1)*x1)+(t-1)^2*(t+1)^2*(q^2+1)*q^4*x2/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1)*x1)+(t-1)^2*(t+1)^2*(2*q^2+q^4+2+q^4*t^2)*q^3*x2/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1))+(t-1)^2*(t+1)^2*(q^2+1)*q^5/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1)*x2)+x1^2*x2^2+(t-1)*(t+1)*(-2*q^2-q^4-2+2*q^2*t^2+t^2+q^6*t^4+q^4*t^4)*q^2*x2*x1/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1))+(t-1)*(t+1)*(q^2+1+q^4*t^2)*q*x2^2*x1/((t^2*q^3-1)*(t^2*q^3+1))+(t-1)*(t+1)*q^3*x1^2/((t^2*q^3-1)*(t^2*q^3+1)*x2)+(t-1)*(t+1)*(q^2+1+q^4*t^2)*q*x2*x1^2/((t^2*q^3-1)*(t^2*q^3+1))+(t-1)*(t+1)*q^6/((t^2*q^3+1)*(t^2*q^3-1)*x1*x2)+(t-1)*(t+1)*(q^2+1+q^4*t^2)*q^2*x1^2/((t^2*q^3-1)*(t^2*q^3+1))+(t-1)*(t+1)*(q^2+1+q^4*t^2)*q^2*x2^2/((t^2*q^3-1)*(t^2*q^3+1))+(t-1)*(t+1)*q^3*x2^2/((t^2*q^3-1)*(t^2*q^3+1)*x1)+(t-1)^2*(t+1)^2*(q^2+1)*q^4*x1/((t^2*q^3+1)*(t^2*q^3-1)*(t*q-1)*(t*q+1)*x2) sage: to_SR(E[2*omega[2]]) - expected # long time (3.5s) 0 sage: E = NonSymmetricMacdonaldPolynomials(["BC",3,2], q=q, q1=t^2,q2=-1) sage: omega=E.keys().fundamental_weights() sage: mu = -3*omega[1] + 3*omega[2] - omega[3]; mu (-1, 2, -1) sage: expected = (t-1)^2*(t+1)^2*(3*q^2+q^4+1+t^2*q^4+q^2*t^2-3*t^4*q^2-5*t^6*q^4+2*t^8*q^4-4*t^8*q^6-q^8*t^10+2*t^10*q^6-2*q^8*t^12+t^14*q^8-t^14*q^10+q^10*t^16+q^8*t^16+q^10*t^18+t^18*q^12)*x2*x1/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(q^2*t^6+2*t^6*q^4-q^4*t^4+t^4*q^2-q^2*t^2+t^2-2-q^2)*q^2*x1/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)^2*(t+1)^2*(-q^2-1+t^4*q^2-q^4*t^4+2*t^6*q^4)*x1^2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t+1)*(t-1)*x2^2*x3/((t*q-1)*(t*q+1)*x1)+(t-1)^2*(t+1)^2*(3*q^2+q^4+2+t^2*q^4+2*q^2*t^2-4*t^4*q^2+q^4*t^4-6*t^6*q^4+t^8*q^4-4*t^8*q^6-q^8*t^10+t^10*q^6-3*q^8*t^12-2*t^14*q^10+2*t^14*q^8+2*q^10*t^16+q^8*t^16+t^18*q^12+2*q^10*t^18)*q*x2/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(1+q^4+2*q^2+t^2*q^4-3*t^4*q^2+q^2*t^6-5*t^6*q^4+3*t^8*q^4-4*t^8*q^6+2*t^10*q^6-q^8*t^12-t^14*q^10+t^14*q^8+q^10*t^16+t^18*q^12)*x3*x1/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(2*q^2+1+q^4+t^2*q^4-t^2+q^2*t^2-4*t^4*q^2+q^4*t^4+q^2*t^6-5*t^6*q^4+3*t^8*q^4-4*t^8*q^6+2*t^10*q^6+q^6*t^12-2*q^8*t^12-2*t^14*q^10+2*t^14*q^8+q^10*t^16+t^18*q^12)*q*x3/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(1+t^2+t^4*q^2)*q*x3*x2^2/((t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1))+(t-1)^2*(t+1)^2*(-q^2-2-q^2*t^2+t^4-q^4*t^4-t^4*q^2+3*q^2*t^6-t^6*q^4-t^8*q^6+t^8*q^4+t^10*q^4+2*q^6*t^12-q^8*t^12+t^14*q^8)*q*x3*x2*x1/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t-1)*(t+1)*x1^2/((q^3*t^5-1)*(q^3*t^5+1)*x3*x2)+(t-1)*(t+1)*(-q^2-1+t^4*q^2-q^4*t^4+2*t^6*q^4)*x2^2/((t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1))+(t-1)*(t+1)*(t^3*q-1)*(t^3*q+1)*x3*x2^2*x1/((t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1))+(t-1)^2*(t+1)^2*(q^2+1)*q*x1/((t*q+1)*(t*q-1)*(q^3*t^5+1)*(q^3*t^5-1)*x3*x2)+(t-1)^2*(t+1)^2*(t^3*q-1)*(t^3*q+1)*x3*x2*x1^2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t-1)^2*(t+1)^2*q^3*x3/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x1*x2)+(t-1)*(t+1)*(-1-q^2+q^2*t^2+t^10*q^6)*q*x2/((t*q+1)*(t*q-1)*(q^3*t^5+1)*(q^3*t^5-1)*x3*x1)+x2^2/(x1*x3)+(t-1)*(t+1)*q*x2^2/((t*q-1)*(t*q+1)*x3)+(t-1)^3*(t+1)^3*(1+t^2+t^4*q^2)*q*x2*x1^2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t-1)^2*(t+1)^2*q*x1^2/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t-1)^2*(t+1)^2*(q^2*t^6+2*t^6*q^4-q^4*t^4+t^4*q^2-q^2*t^2+t^2-2-q^2)*q^3/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)*(t+1)*(q^2+2-t^2+q^4*t^4-t^4*q^2-3*t^6*q^4+t^8*q^4-2*t^10*q^6-q^8*t^12+q^6*t^12+q^8*t^16+q^10*t^16)*q^2*x2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x1)+(t-1)^2*(t+1)^2*(q^2+1)*q^2/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3*x2)+(t-1)*(t+1)*(1+q^4+2*q^2-2*q^2*t^2+t^4*q^6-q^4*t^4-3*q^6*t^6-t^6*q^4+2*t^8*q^6-t^10*q^6-q^8*t^10-t^14*q^10+t^14*q^8+2*q^10*t^16)*x2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t-1)^2*(t+1)^2*(-q^2-2-q^2*t^2-q^4*t^4+2*t^6*q^4+t^10*q^6+q^8*t^12+t^14*q^8)*q^3/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x1)+(t-1)^2*(t+1)^2*(-1-q^2-q^2*t^2+t^2+t^4*q^2-q^4*t^4+2*t^6*q^4)*q^2*x3/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)*(t+1)*q*x2^2/((t*q-1)*(t*q+1)*x1)+(t-1)^2*(t+1)^2*(1+t^2+t^4*q^2)*q*x2^2*x1/((t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1))+(t-1)^2*(t+1)^2*q*x1^2/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)^2*(t+1)^2*(-1-q^4-2*q^2-t^2*q^4-q^2*t^2+t^4*q^2-t^4*q^6-2*q^4*t^4+3*t^6*q^4-q^6*t^6-t^8*q^8+t^8*q^6+2*t^10*q^6-q^10*t^12+3*q^8*t^12+2*t^14*q^10)*x3*x2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t-1)*(t+1)*(q^2+1-t^2+q^4*t^4-t^4*q^2+q^2*t^6-3*t^6*q^4+t^8*q^4-t^10*q^6+q^6*t^12-q^8*t^12+q^10*t^16)*q^2*x3/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x1)+(t-1)*(t+1)*(-1-q^2+q^2*t^2+t^10*q^6)*q^2/((t*q-1)*(t*q+1)*(q^3*t^5+1)*(q^3*t^5-1)*x1*x3)+(t-1)*(t+1)*(1+q^4+2*q^2-3*q^2*t^2+t^4*q^6-q^4*t^4-3*q^6*t^6-t^6*q^4+t^8*q^4+2*t^8*q^6-t^10*q^6+t^14*q^8-t^14*q^10+q^10*t^16)*x1/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t-1)^2*(t+1)^2*(3*q^2+q^4+2+q^2*t^2-t^2+t^2*q^4-6*t^4*q^2+q^4*t^4-7*t^6*q^4+q^2*t^6+3*t^8*q^4-4*t^8*q^6+t^10*q^4+3*t^10*q^6-q^8*t^12-t^14*q^10+t^14*q^8+q^8*t^16+q^10*t^18)*q*x1/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(-q^2-2-q^2*t^2-q^4*t^4+2*t^6*q^4+t^10*q^6+q^6*t^12+t^14*q^8)*q*x2*x1/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t+1)*(t-1)*x2^2*x1/((t*q-1)*(t*q+1)*x3)+(t-1)^3*(t+1)^3*(1+t^2+t^4*q^2)*q*x3*x1^2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1))+(t-1)*(t+1)*q^3/((q^3*t^5+1)*(q^3*t^5-1)*x1*x2*x3)+(t-1)^2*(t+1)^2*(3+3*q^2+q^4+2*q^2*t^2-t^2+t^2*q^4-6*t^4*q^2+q^4*t^4-8*t^6*q^4+q^2*t^6+2*t^8*q^4-4*t^8*q^6+t^10*q^4+2*t^10*q^6-2*q^8*t^12-t^14*q^10+t^14*q^8+q^8*t^16+q^10*t^16+2*q^10*t^18)*q^2/((q^3*t^5+1)*(q^3*t^5-1)*(t*q-1)*(t*q+1)*(t^3*q^2+1)*(t^3*q^2-1)*(t^2*q-1)*(t^2*q+1))+(t-1)^2*(t+1)^2*(-q^4-2*q^2-1-t^2*q^4-t^4*q^6+2*q^6*t^6+t^6*q^4+t^10*q^6+q^8*t^12+t^14*q^10)*q/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t-1)^2*(t+1)^2*(-1-q^2-q^2*t^2+t^2+t^4*q^2-q^4*t^4+2*t^6*q^4)*q*x3*x1/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)^2*(t+1)^2*x2*x1^2/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x3)+(t-1)^2*(t+1)^2*x3*x1^2/((t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x2)+(t-1)^2*(t+1)^2*q^4/((t*q+1)*(t*q-1)*(q^3*t^5+1)*(q^3*t^5-1)*x1*x2)+(t-1)^2*(t+1)^2*(-q^2-1-q^2*t^2-q^4*t^4+t^6*q^4+t^10*q^6+q^8*t^12+t^14*q^10)*q*x3*x2/((t^3*q^2-1)*(t^3*q^2+1)*(t*q+1)*(t*q-1)*(q^3*t^5-1)*(q^3*t^5+1)*x1) sage: to_SR(E[mu]) - expected # long time (20s) 0 sage: E = NonSymmetricMacdonaldPolynomials(["BC",1,2], q=q, q1=t^2,q2=-1) sage: omega=E.keys().fundamental_weights() sage: mu = -4*omega[1]; mu (-4) sage: expected = (t-1)*(t+1)*(-1+q^2*t^2-q^2-3*q^10-7*q^26*t^8+5*t^2*q^6-q^16-3*q^4+4*t^10*q^30-4*t^6*q^22-10*q^20*t^6+2*q^32*t^10-3*q^6-4*q^8+q^34*t^10-4*t^8*q^24-2*q^12-q^14+2*q^22*t^10+4*q^26*t^10+4*q^28*t^10+t^6*q^30-2*q^32*t^8-2*t^8*q^22+2*q^24*t^10-q^20*t^2-2*t^6*q^12+t^8*q^14+2*t^4*q^24-4*t^8*q^30+2*t^8*q^20-9*t^6*q^16+3*q^26*t^6+q^28*t^6+3*t^2*q^4+2*q^18*t^8-6*t^6*q^14+4*t^4*q^22-2*q^24*t^6+3*t^2*q^12+7*t^4*q^20-t^2*q^16+11*q^18*t^4-2*t^2*q^18+9*q^16*t^4-t^4*q^6+6*q^8*t^2+5*q^10*t^2-6*q^28*t^8+q^12*t^4+8*t^4*q^14-10*t^6*q^18-q^4*t^4+q^16*t^8-2*t^4*q^8)/((t*q^4-1)*(t*q^4+1)*(q^7*t^2-1)*(q^7*t^2+1)*(t*q^3-1)*(t*q^3+1)*(q^5*t^2+1)*(q^5*t^2-1))+(q^2+1)*(q^4+1)*(t-1)*(t+1)*(-1+q^2*t^2-q^2+t^2*q^6-q^4+t^6*q^22+3*q^10*t^4+t^2-q^8-2*t^8*q^24+q^22*t^10+q^26*t^10-2*t^8*q^22+q^24*t^10-4*t^6*q^12-2*t^8*q^20-3*t^6*q^16+2*t^2*q^4-t^6*q^10-2*t^6*q^14+t^8*q^12-t^2*q^12+2*q^16*t^4+q^8*t^2-q^10*t^2+3*q^12*t^4+2*t^4*q^14+t^6*q^18-2*q^4*t^4+q^16*t^8+q^20*t^10)*q*x1/((t*q^4-1)*(t*q^4+1)*(q^7*t^2-1)*(q^7*t^2+1)*(t*q^3-1)*(t*q^3+1)*(q^5*t^2+1)*(q^5*t^2-1))+(q^2+1)*(q^4+1)*(t-1)*(t+1)*(1+q^8+q^4+q^2-q^8*t^2-2*t^2*q^4-t^2*q^6+t^2*q^12-t^2+t^4*q^6-2*q^16*t^4-t^4*q^14-2*q^12*t^4+t^6*q^12+t^6*q^16+t^6*q^18+t^6*q^14)*q/((t*q^4-1)*(t*q^4+1)*(q^7*t^2-1)*(q^7*t^2+1)*(t*q^3-1)*(t*q^3+1)*x1)+(t-1)*(t+1)*(-1-q^2-q^6-q^4-q^8+t^2*q^4-t^2*q^14+t^2*q^6-q^10*t^2+q^8*t^2-t^2*q^12+q^12*t^4+q^10*t^4+q^16*t^4+2*t^4*q^14)*(q^4+1)/((q^7*t^2+1)*(q^7*t^2-1)*(t*q^4-1)*(t*q^4+1)*x1^2)+(t-1)*(t+1)*(q^4+1)*(q^2+1)*q/((t*q^4-1)*(t*q^4+1)*x1^3)+(q^4+1)*(t-1)*(t+1)*(1+q^6+q^8+q^2+q^4-q^2*t^2-3*t^2*q^4+q^10*t^2+t^2*q^12-2*t^2*q^6-q^8*t^2-2*q^16*t^4+q^4*t^4+t^4*q^6-q^10*t^4-2*q^12*t^4-2*t^4*q^14+t^6*q^12+t^6*q^18+2*t^6*q^16+t^6*q^14)*x1^2/((t*q^4-1)*(t*q^4+1)*(q^7*t^2-1)*(q^7*t^2+1)*(t*q^3-1)*(t*q^3+1))+(t-1)*(t+1)*(-1-t^2*q^6+t^2+t^4*q^8)*(q^4+1)*(q^2+1)*q*x1^3/((q^7*t^2+1)*(q^7*t^2-1)*(t*q^4-1)*(t*q^4+1))+1/x1^4+(t-1)*(t+1)*x1^4/((t*q^4-1)*(t*q^4+1)) sage: to_SR(E[mu]) - expected 0
Type
dual, comparison with hand calculations by Bogdan Ion:
sage: K = QQ['q,q1,q2'].fraction_field() sage: q,q1,q2 = K.gens() sage: ct = CartanType(["BC",2,2]).dual() sage: E = NonSymmetricMacdonaldPolynomials(ct, q=q, q1=q1, q2=q2) sage: KL = E.domain(); KL Group algebra of the Ambient space of the Root system of type ['B', 2] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field sage: alpha = E.keys().simple_roots(); alpha Finite family {1: (1, -1), 2: (0, 1)} sage: omega=E.keys().fundamental_weights(); omega Finite family {1: (1, 0), 2: (1/2, 1/2)} sage: epsilon = E.keys().basis(); epsilon Finite family {0: (1, 0), 1: (0, 1)}
Note: Sage’s
is the usual
:
sage: E.L().null_root() e['delta'] sage: E.L().null_coroot() 2*e['deltacheck']
Some eigenvectors:
sage: E[0*omega[1]] B[(0, 0)] sage: E[omega[1]] ((-q^2*q1^3*q2-q^2*q1^2*q2^2)/(q^2*q1^4-q2^4))*B[(0, 0)] + B[(1, 0)] sage: Eomega1 = KL.one() * (q^2*(-q1/q2)^2*(1-(-q1/q2))) / (1-q^2*(-q1/q2)^4) + KL.monomial(omega[1]) sage: E[omega[1]] == Eomega1 True
Checking the
s:
sage: Y = E.Y() sage: alphacheck = Y.keys().simple_roots() sage: Y0 = Y[alphacheck[0]] sage: Y1 = Y[alphacheck[1]] sage: Y2 = Y[alphacheck[2]] sage: Y0.word, Y0.signs, Y0.scalar ((0, 1, 2, 1, 0, 1, 2, 1), (-1, -1, -1, -1, -1, -1, -1, -1), q1^4*q2^4/q^2) sage: Y1.word, Y1.signs, Y1.scalar ((1, 2, 0, 1, 2, 0), (1, 1, -1, 1, -1, 1), 1/(-q1*q2)) sage: Y2.word, Y2.signs, Y2.scalar ((2, 1, 0, 1), (1, 1, 1, -1), 1/(-q1*q2)) sage: E.eigenvalues(0*omega[1]) [q2^4/(q^2*q1^4), q1/(-q2), q1/(-q2)]
Checking the
and
s:
sage: T = E._T_Y sage: Tinv0 = T.Tw_inverse([0]) sage: Tinv1 = T.Tw_inverse([1]) sage: Tinv2 = T.Tw_inverse([2]) sage: for x in [0*epsilon[0], -epsilon[0], -epsilon[1], epsilon[0], epsilon[1]]: ....: x = KL.monomial(x) ....: assert Tinv0(T[0](x)) == x and T[0](Tinv0(x)) == x ....: assert Tinv1(T[1](x)) == x and T[1](Tinv1(x)) == x ....: assert Tinv2(T[2](x)) == x and T[2](Tinv2(x)) == x sage: start = E[omega[1]]; start ((-q^2*q1^3*q2-q^2*q1^2*q2^2)/(q^2*q1^4-q2^4))*B[(0, 0)] + B[(1, 0)] sage: Tinv1(Tinv2(Tinv1(Tinv0(Tinv1(Tinv2(Tinv1(Tinv0(start)))))))) * (q1*q2)^4/q^2 == Y0(start) True sage: Y0(start) == q^2*q1^4/q2^4 * start True
Checking the relation between the
s:
sage: q^2 * Y0(Y1(Y1(Y2(Y2(start))))) == start True sage: for x in [0*epsilon[0], -epsilon[0], -epsilon[1], epsilon[0], epsilon[1]]: ....: x = KL.monomial(x) ....: assert q^2 * Y0(Y1(Y1(Y2(Y2(start))))) == start
-
KL0
()¶ Return the group algebra where the nonsymmetric Macdonald polynomials live.
EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials("B2~").KL0() Group algebra of the Ambient space of the Root system of type ['B', 2] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field sage: NonSymmetricMacdonaldPolynomials("B2~*").KL0() Group algebra of the Ambient space of the Root system of type ['C', 2] over Fraction Field of Multivariate Polynomial Ring in q, q1, q2 over Rational Field
-
L
()¶ Return the affinization of the classical weight space.
EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials(["B", 2, 1]).L() Ambient space of the Root system of type ['B', 2, 1]
-
L0
()¶ Return the space indexing the monomials of the nonsymmetric Macdonald polynomials.
EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials("B2~").L0() Ambient space of the Root system of type ['B', 2] sage: NonSymmetricMacdonaldPolynomials("B2~*").L0() Ambient space of the Root system of type ['C', 2]
-
L_check
()¶ Return the other affinization of the classical weight space.
Todo
should this just return
in the simply laced case?
EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials(["B", 2, 1]).L_check() Coambient space of the Root system of type ['C', 2, 1] sage: NonSymmetricMacdonaldPolynomials(["B", 2, 1]).L_check().classical() Ambient space of the Root system of type ['B', 2]
-
L_prime
()¶ The affine space where classical weights are lifted for the recursion.
Also the parent of
.
EXAMPLES:
In the twisted case, this is the affinization of the classical ambient space:
sage: NonSymmetricMacdonaldPolynomials("B2~*").L() Ambient space of the Root system of type ['B', 2, 1]^* sage: NonSymmetricMacdonaldPolynomials("B2~*").L().classical() Ambient space of the Root system of type ['C', 2] sage: NonSymmetricMacdonaldPolynomials("B2~*").L_prime() Ambient space of the Root system of type ['B', 2, 1]^* sage: NonSymmetricMacdonaldPolynomials("B2~*").L_prime().classical() Ambient space of the Root system of type ['C', 2]
In the untwisted case, this is the other affinization of the classical ambient space:
sage: NonSymmetricMacdonaldPolynomials("B2~").L() Ambient space of the Root system of type ['B', 2, 1] sage: NonSymmetricMacdonaldPolynomials("B2~").L().classical() Ambient space of the Root system of type ['B', 2] sage: NonSymmetricMacdonaldPolynomials("B2~").L_prime() Coambient space of the Root system of type ['C', 2, 1] sage: NonSymmetricMacdonaldPolynomials("B2~").L_prime().classical() Ambient space of the Root system of type ['B', 2]
For simply laced, the two affinizations coincide:
sage: NonSymmetricMacdonaldPolynomials("A2~").L() Ambient space of the Root system of type ['A', 2, 1] sage: NonSymmetricMacdonaldPolynomials("A2~").L().classical() Ambient space of the Root system of type ['A', 2] sage: NonSymmetricMacdonaldPolynomials("A2~").L_prime() Coambient space of the Root system of type ['A', 2, 1] sage: NonSymmetricMacdonaldPolynomials("A2~").L_prime().classical() Ambient space of the Root system of type ['A', 2]
Note
do we want the coambient space of type
instead?
For type BC:
sage: NonSymmetricMacdonaldPolynomials(["BC",3,2]).L_prime() Ambient space of the Root system of type ['BC', 3, 2]
-
Q_to_Qcheck
()¶ The reindexing of the index set of the Y’s by the coroot lattice.
EXAMPLES:
sage: E = NonSymmetricMacdonaldPolynomials("C2~") sage: alphacheck = E.Y().keys().simple_roots() sage: E.Q_to_Qcheck(alphacheck[0]) alphacheck[0] - alphacheck[2] sage: E.Q_to_Qcheck(alphacheck[1]) alphacheck[1] sage: E.Q_to_Qcheck(alphacheck[2]) alphacheck[2] sage: x = alphacheck[1] + 2*alphacheck[2] sage: x.parent() Root lattice of the Root system of type ['B', 2, 1] sage: E.Q_to_Qcheck(x) alphacheck[1] + 2*alphacheck[2] sage: _.parent() Coroot lattice of the Root system of type ['C', 2, 1]
-
Y
()¶ Return the family of
operators whose eigenvectors are the nonsymmetric Macdonald polynomials.
EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials("C2~").Y() Lazy family (<lambda>(i))_{i in Root lattice of the Root system of type ['B', 2, 1]} sage: _.keys().classical() Root lattice of the Root system of type ['B', 2] sage: NonSymmetricMacdonaldPolynomials("C2~*").Y() Lazy family (<...Y_lambdacheck...>(i))_{i in Coroot lattice of the Root system of type ['C', 2, 1]^*} sage: _.keys().classical() Root lattice of the Root system of type ['C', 2] sage: NonSymmetricMacdonaldPolynomials(["BC", 3, 2]).Y() Lazy family (<...Y_lambdacheck...>(i))_{i in Coroot lattice of the Root system of type ['BC', 3, 2]} sage: _.keys().classical() Root lattice of the Root system of type ['B', 3]
-
affine_lift
(mu)¶ Return the affinization of
in
.
INPUT:
mu
– a classical weight
See also
EXAMPLES:
In the untwisted case, this is the other affinization at level 1:
sage: E = NonSymmetricMacdonaldPolynomials("B2~") sage: L0 = E.keys(); L0 Ambient space of the Root system of type ['B', 2] sage: omega = L0.fundamental_weights() sage: E.affine_lift(omega[1]) e[0] + e['deltacheck'] sage: E.affine_lift(omega[1]).parent() Coambient space of the Root system of type ['C', 2, 1]
In the twisted case, this is the usual affinization at level 1:
sage: E = NonSymmetricMacdonaldPolynomials("B2~*") sage: L0 = E.keys(); L0 Ambient space of the Root system of type ['C', 2] sage: omega = L0.fundamental_weights() sage: E.affine_lift(omega[1]) e[0] + e['deltacheck'] sage: E.affine_lift(omega[1]).parent() Ambient space of the Root system of type ['B', 2, 1]^*
-
affine_retract
(mu)¶ Retract the affine weight
into a classical weight.
INPUT:
mu
– an affine weightin
See also
hecke_algebra_representation.HeckeAlgebraRepresentation.affine_retract()
affine_lift()
L_prime()
EXAMPLES:
sage: E = NonSymmetricMacdonaldPolynomials("B2~") sage: L0 = E.keys(); L0 Ambient space of the Root system of type ['B', 2] sage: omega = L0.fundamental_weights() sage: E.affine_lift(omega[1]) e[0] + e['deltacheck'] sage: E.affine_retract(E.affine_lift(omega[1])) (1, 0)
-
cartan_type
()¶ Return Cartan type of
self
.EXAMPLES:
sage: NonSymmetricMacdonaldPolynomials(["B", 2, 1]).cartan_type() ['B', 2, 1]
-
eigenvalue_experimental
(mu, l)¶ Return the eigenvalue of
acting on the macdonald polynomial
.
INPUT:
mu
– the indexof an eigenvector
– an index
of some
Note
- This method is currently not used; most tests below even test the naive method. They are left here as a basis for a future implementation.
- This is actually equivariant, as long as
does not fix
.
- This method is only really needed for
with
.
See Corollary 6.11 of [Haiman06].
EXAMPLES:
sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: q1 = t sage: q2 = -1 sage: KL = RootSystem(["A",1,1]).ambient_space().algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, q1, q2) sage: L0 = E.keys() sage: E.eigenvalues(L0([0,0])) # Checked by hand by Mark and Arun [1/(q*t), t] sage: alpha = E.Y().keys().simple_roots() sage: E.eigenvalue_experimental(L0([0,0]), alpha[0]) # todo: not implemented 1/(q*t) sage: E.eigenvalue_experimental(L0([0,0]), alpha[1]) t
Some examples of eigenvalues (not mathematically checked!!!):
sage: E.eigenvalues(L0([1,0])) [t, 1/(q*t)] sage: E.eigenvalues(L0([0,1])) [1/(q^2*t), q*t] sage: E.eigenvalues(L0([1,1])) [1/(q*t), t] sage: E.eigenvalues(L0([2,1])) [t, 1/(q*t)] sage: E.eigenvalues(L0([-1,1])) [(-1)/(-q^3*t), q^2*t] sage: E.eigenvalues(L0([-2,1])) [(-1)/(-q^4*t), q^3*t] sage: E.eigenvalues(L0([-2,0])) [(-1)/(-q^3*t), q^2*t]
Some type
examples:
sage: K = QQ['q,t'].fraction_field() sage: q,t = K.gens() sage: q1 = t sage: q2 = -1 sage: L = RootSystem(["B",2,1]).ambient_space() sage: KL = L.algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, q1, q2) sage: L0 = E.keys() sage: alpha = L.simple_coroots() sage: E.eigenvalue(L0((0,0)), alpha[0]) # not checked # not tested q/t sage: E.eigenvalue(L0((1,0)), alpha[1]) # What Mark got by hand # not tested q sage: E.eigenvalue(L0((1,0)), alpha[2]) # not checked # not tested t sage: E.eigenvalue(L0((1,0)), alpha[0]) # not checked # not tested 1 sage: L = RootSystem("B2~*").ambient_space() sage: KL = L.algebra(K) sage: E = NonSymmetricMacdonaldPolynomials(KL,q, q1, q2) sage: L0 = E.keys() sage: alpha = L.simple_coroots() sage: E.eigenvalue(L0((0,0)), alpha[0]) # assuming Mark's calculation is correct, one should get # not tested 1/(q*t^2)
The expected value can more or less be read off from equation (37), Corollary 6.15 of [Haiman06]
Todo
- Use proposition 6.9 of [Haiman06] to check the action
of the
s on monomials.
- Generalize to any
,
.
- Check claim by Mark: all scalar products should occur in the finite weight lattice, with alpha 0 being the appropriate projection of the affine alpha 0. Question: can this be emulated by being at level 0?
-
rho_prime
()¶ Return the level 0 sum of the classical fundamental weights in
.
See also
EXAMPLES:
Untwisted case:
sage: NonSymmetricMacdonaldPolynomials("B2~").rho_prime() # CHECKME 3/2*e[0] + 1/2*e[1] sage: NonSymmetricMacdonaldPolynomials("B2~").rho_prime().parent() Coambient space of the Root system of type ['C', 2, 1]
Twisted case:
sage: NonSymmetricMacdonaldPolynomials("B2~*").rho_prime() # CHECKME 2*e[0] + e[1] sage: NonSymmetricMacdonaldPolynomials("B2~*").rho_prime().parent() Ambient space of the Root system of type ['B', 2, 1]^*
-
seed
(mu)¶ Return
for
minuscule, i.e. in the fundamental alcove.
INPUT:
mu
– the indexof an eigenvector
EXAMPLES:
sage: E = NonSymmetricMacdonaldPolynomials(["A",2,1]) sage: omega = E.keys().fundamental_weights() sage: E.seed(omega[1]) B[(1, 0, 0)]
-
symmetric_macdonald_polynomial
(mu)¶ Return the symmetric Macdonald polynomial indexed by
.
INPUT:
mu
– a dominant weight
Warning
The result is Weyl-symmetric only for Hecke parameters of the form
and
. In general the value of
below, should be the square root of
, but the use of
and
results in nonintegral powers of
.
EXAMPLES:
sage: K = QQ['q,v,t'].fraction_field() sage: q,v,t = K.gens() sage: E = NonSymmetricMacdonaldPolynomials(['A',2,1], q, v, -1/v) sage: om = E.L0().fundamental_weights() sage: E.symmetric_macdonald_polynomial(om[2]) B[(1, 1, 0)] + B[(1, 0, 1)] + B[(0, 1, 1)] sage: E.symmetric_macdonald_polynomial(2*om[1]) ((q*v^2+v^2-q-1)/(q*v^2-1))*B[(1, 1, 0)] + ((q*v^2+v^2-q-1)/(q*v^2-1))*B[(1, 0, 1)] + B[(2, 0, 0)] + ((q*v^2+v^2-q-1)/(q*v^2-1))*B[(0, 1, 1)] + B[(0, 2, 0)] + B[(0, 0, 2)] sage: f = E.symmetric_macdonald_polynomial(E.L0()((2,1,0))); f ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*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)]
We compare with the type
Macdonald polynomials coming from symmetric functions:
sage: P = SymmetricFunctions(K).macdonald().P() sage: g = P[2,1].expand(3); g x0^2*x1 + x0*x1^2 + x0^2*x2 + ((-2*q*t^2 + q*t - t^2 + q - t + 2)/(-q*t^2 + 1))*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 sage: fe =f.expand(g.parent().gens()); fe x0^2*x1 + x0*x1^2 + x0^2*x2 + ((2*q*v^4 + v^4 - q*v^2 + v^2 - q - 2)/(q*v^4 - 1))*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 sage: g.map_coefficients(lambda x: x.subs(t=v*v)) == fe True sage: E = NonSymmetricMacdonaldPolynomials(['C',3,1], q, v, -1/v) sage: om = E.L0().fundamental_weights() sage: E.symmetric_macdonald_polynomial(om[1]+om[2]) B[(-2, -1, 0)] + B[(-2, 1, 0)] + B[(-2, 0, -1)] + B[(-2, 0, 1)] + ((4*q^3*v^14+2*q^2*v^14-2*q^3*v^12+2*q^2*v^12-2*q^3*v^10+q*v^12-5*q^2*v^10-5*q*v^4+q^2*v^2-2*v^4+2*q*v^2-2*v^2+2*q+4)/(q^3*v^14-q^2*v^10-q*v^4+1))*B[(-1, 0, 0)] + B[(-1, -2, 0)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(-1, -1, -1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(-1, -1, 1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(-1, 1, -1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(-1, 1, 1)] + B[(-1, 2, 0)] + B[(-1, 0, -2)] + B[(-1, 0, 2)] + ((4*q^3*v^14+2*q^2*v^14-2*q^3*v^12+2*q^2*v^12-2*q^3*v^10+q*v^12-5*q^2*v^10-5*q*v^4+q^2*v^2-2*v^4+2*q*v^2-2*v^2+2*q+4)/(q^3*v^14-q^2*v^10-q*v^4+1))*B[(1, 0, 0)] + B[(1, -2, 0)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(1, -1, -1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(1, -1, 1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(1, 1, -1)] + ((2*q*v^4+v^4-q*v^2+v^2-q-2)/(q*v^4-1))*B[(1, 1, 1)] + B[(1, 2, 0)] + B[(1, 0, -2)] + B[(1, 0, 2)] + B[(2, -1, 0)] + B[(2, 1, 0)] + B[(2, 0, -1)] + B[(2, 0, 1)] + B[(0, -2, -1)] + B[(0, -2, 1)] + ((-4*q^3*v^14-2*q^2*v^14+2*q^3*v^12-2*q^2*v^12+2*q^3*v^10-q*v^12+5*q^2*v^10+5*q*v^4-q^2*v^2+2*v^4-2*q*v^2+2*v^2-2*q-4)/(-q^3*v^14+q^2*v^10+q*v^4-1))*B[(0, -1, 0)] + B[(0, -1, -2)] + B[(0, -1, 2)] + ((-4*q^3*v^14-2*q^2*v^14+2*q^3*v^12-2*q^2*v^12+2*q^3*v^10-q*v^12+5*q^2*v^10+5*q*v^4-q^2*v^2+2*v^4-2*q*v^2+2*v^2-2*q-4)/(-q^3*v^14+q^2*v^10+q*v^4-1))*B[(0, 1, 0)] + B[(0, 1, -2)] + B[(0, 1, 2)] + B[(0, 2, -1)] + B[(0, 2, 1)] + ((4*q^3*v^14+2*q^2*v^14-2*q^3*v^12+2*q^2*v^12-2*q^3*v^10+q*v^12-5*q^2*v^10-5*q*v^4+q^2*v^2-2*v^4+2*q*v^2-2*v^2+2*q+4)/(q^3*v^14-q^2*v^10-q*v^4+1))*B[(0, 0, -1)] + ((4*q^3*v^14+2*q^2*v^14-2*q^3*v^12+2*q^2*v^12-2*q^3*v^10+q*v^12-5*q^2*v^10-5*q*v^4+q^2*v^2-2*v^4+2*q*v^2-2*v^2+2*q+4)/(q^3*v^14-q^2*v^10-q*v^4+1))*B[(0, 0, 1)]
An example for type
:
sage: E = NonSymmetricMacdonaldPolynomials(['G',2,1], q, v, -1/v) sage: om = E.L0().fundamental_weights() sage: E.symmetric_macdonald_polynomial(2*om[1]) ((3*q^6*v^22+3*q^5*v^22-3*q^6*v^20+q^4*v^22-4*q^5*v^20+q^4*v^18-q^5*v^16+q^3*v^18-2*q^4*v^16+q^5*v^14-q^3*v^16+q^4*v^14-4*q^4*v^12+q^2*v^14+q^5*v^10-8*q^3*v^12+4*q^4*v^10-4*q^2*v^12+8*q^3*v^10-q*v^12-q^4*v^8+4*q^2*v^10-q^2*v^8+q^3*v^6-q*v^8+2*q^2*v^6-q^3*v^4+q*v^6-q^2*v^4+4*q*v^2-q^2+3*v^2-3*q-3)/(q^6*v^22-q^5*v^20-q^4*v^12-q^3*v^12+q^3*v^10+q^2*v^10+q*v^2-1))*B[(0, 0, 0)] + ((q*v^2+v^2-q-1)/(q*v^2-1))*B[(-2, 1, 1)] + B[(-2, 2, 0)] + B[(-2, 0, 2)] + ((-q*v^2-v^2+q+1)/(-q*v^2+1))*B[(-1, -1, 2)] + ((2*q^4*v^12+2*q^3*v^12-2*q^4*v^10-2*q^3*v^10+q^2*v^8-q^3*v^6+q*v^8-2*q^2*v^6+q^3*v^4-q*v^6+q^2*v^4-2*q*v^2-2*v^2+2*q+2)/(q^4*v^12-q^3*v^10-q*v^2+1))*B[(-1, 1, 0)] + ((-q*v^2-v^2+q+1)/(-q*v^2+1))*B[(-1, 2, -1)] + ((2*q^4*v^12+2*q^3*v^12-2*q^4*v^10-2*q^3*v^10+q^2*v^8-q^3*v^6+q*v^8-2*q^2*v^6+q^3*v^4-q*v^6+q^2*v^4-2*q*v^2-2*v^2+2*q+2)/(q^4*v^12-q^3*v^10-q*v^2+1))*B[(-1, 0, 1)] + ((-q*v^2-v^2+q+1)/(-q*v^2+1))*B[(1, -2, 1)] + ((-2*q^4*v^12-2*q^3*v^12+2*q^4*v^10+2*q^3*v^10-q^2*v^8+q^3*v^6-q*v^8+2*q^2*v^6-q^3*v^4+q*v^6-q^2*v^4+2*q*v^2+2*v^2-2*q-2)/(-q^4*v^12+q^3*v^10+q*v^2-1))*B[(1, -1, 0)] + ((-q*v^2-v^2+q+1)/(-q*v^2+1))*B[(1, 1, -2)] + ((-2*q^4*v^12-2*q^3*v^12+2*q^4*v^10+2*q^3*v^10-q^2*v^8+q^3*v^6-q*v^8+2*q^2*v^6-q^3*v^4+q*v^6-q^2*v^4+2*q*v^2+2*v^2-2*q-2)/(-q^4*v^12+q^3*v^10+q*v^2-1))*B[(1, 0, -1)] + B[(2, -2, 0)] + ((q*v^2+v^2-q-1)/(q*v^2-1))*B[(2, -1, -1)] + B[(2, 0, -2)] + B[(0, -2, 2)] + ((-2*q^4*v^12-2*q^3*v^12+2*q^4*v^10+2*q^3*v^10-q^2*v^8+q^3*v^6-q*v^8+2*q^2*v^6-q^3*v^4+q*v^6-q^2*v^4+2*q*v^2+2*v^2-2*q-2)/(-q^4*v^12+q^3*v^10+q*v^2-1))*B[(0, -1, 1)] + ((2*q^4*v^12+2*q^3*v^12-2*q^4*v^10-2*q^3*v^10+q^2*v^8-q^3*v^6+q*v^8-2*q^2*v^6+q^3*v^4-q*v^6+q^2*v^4-2*q*v^2-2*v^2+2*q+2)/(q^4*v^12-q^3*v^10-q*v^2+1))*B[(0, 1, -1)] + B[(0, 2, -2)]
-
twist
(mu, i)¶ Act by
on the affine weight
.
This calls
simple_reflection
; which is semantically the same as the default implementation.EXAMPLES:
sage: W = WeylGroup(["B",3]) sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word()) sage: K = QQ['q1,q2'] sage: q1, q2 = K.gens() sage: KW = W.algebra(K) sage: T = KW.demazure_lusztig_operators(q1, q2, affine=True) sage: E = T.Y_eigenvectors() sage: w = W.an_element(); w 123 sage: E.twist(w,1) 1231