Hecke algebra representations¶

class sage.combinat.root_system.hecke_algebra_representation.CherednikOperatorsEigenvectors(T, T_Y=None, normalized=True)¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.sage_object.SageObject

A class for the family of eigenvectors of the $Y$ Cherednik operators for a module over a (Double) Affine Hecke algebra

INPUT:

T – a family $(T_i)_{i\in I}$ implementing the action of the generators of an affine Hecke algebra on self. The intertwiner operators are built from these.
T_Y – a family $(T^Y_i)_{i\in I}$ implementing the action of the generators of an affine Hecke algebra on self. By default, this is T. But this can be used to get the action of the full Double Affine Hecke Algebra. The $Y$ operators are built from the T_Y.

This returns a function $\mu\mapsto E_\mu$ which uses intertwining operators to calculate recursively eigenvectors $E_\mu$ for the action of the torus of the affine Hecke algebra with eigenvalue given by $f$ . Namely:

$E_\mu.Y^{\lambda^\vee} = f(\lambda^\vee, \mu) E_\mu$

Assumptions:

seed(mu) initializes the recurrence by returning an appropriate eigenvector $E_\mu$ for $\mu$ trivial enough. For example, for nonsymmetric Macdonald polynomials seed(mu) returns the monomial $X^\mu$ for a minuscule weight $\mu$ .
$f$ is almost equivariant. Namely, $f(\lambda^\vee,\mu) = f(\lambda^\vee s_i, twist(\mu,i))$ whenever $i$ is a descent of $\mu$ .
$twist(\mu, i)$ maps $\mu$ closer to the dominant chamber whenever $i$ is a descent of $\mu$ .

Todo

Add tests for the above assumptions, and also that the classical operators $T_1, \ldots, T_n$ from $T$ and $T_Y$ coincide.

Y()¶

Return the Cherednik operators.

EXAMPLES:

sage: W = WeylGroup(["B",2])
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: E.Y()
Lazy family (...)_{i in Coroot lattice of the Root system of type ['B', 2, 1]}

affine_lift(mu)¶

Lift the index \mu to a space admitting an action of the affine Weyl group.

INPUT:

mu – an element $\mu$ of the indexing set

In this space, one should have first_descent and apply_simple_reflection act properly.

EXAMPLES:

sage: W = WeylGroup(["A",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: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: w = W.an_element(); w
123
sage: E.affine_lift(w)
121

affine_retract(mu)¶

Retract $\mu$ from a space admitting an action of the affine Weyl group.

EXAMPLES:

sage: W = WeylGroup(["A",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: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: w = W.an_element(); w
123
sage: E.affine_retract(E.affine_lift(w)) == w
True

cartan_type()¶

Return Cartan type of self.

EXAMPLES:

sage: W = WeylGroup(["B",3])
sage: K = QQ['q1,q2']
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: E.cartan_type()
['B', 3, 1]

sage: NonSymmetricMacdonaldPolynomials(["B", 2, 1]).cartan_type()
['B', 2, 1]

domain()¶

The module on which the affine Hecke algebra acts.

EXAMPLES:

sage: W = WeylGroup(["B",3])
sage: K = QQ['q1,q2']
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: E.domain()
Group algebra of Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space) over Multivariate Polynomial Ring in q1, q2 over Rational Field

eigenvalue(mu, l)¶

Return the eigenvalue of $Y_{\lambda^\vee}$ on $E_\mu$ computed by applying $Y_{\lambda^\vee}$ on $E_\mu$ .

INPUT:

mu – the index $\mu$ of an eigenvector, or a tentative eigenvector
l – the index $\lambda^\vee$ of a Cherednik operator in self.Y_index_set()

This default implementation applies explicitly $Y_\mu$ to $E_\lambda$ .

EXAMPLES:

sage: W = WeylGroup(["B",2])
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: w0 = W.long_element()
sage: Y = E.Y()
sage: alphacheck = Y.keys().simple_roots()
sage: E.eigenvalue(w0, alphacheck[1])
q1/(-q2)
sage: E.eigenvalue(w0, alphacheck[2])
q1/(-q2)
sage: E.eigenvalue(w0, alphacheck[0])
q2^2/q1^2

The following checks that all $E_w$ are eigenvectors, with eigenvalue given by Lemma 7.5 of [HST2008] (checked for $A_2$ , $A_3$ ):

sage: Pcheck = Y.keys()
sage: Wcheck = Pcheck.weyl_group()
sage: P0check = Pcheck.classical()
sage: def height(root):
....:     return sum(P0check(root).coefficients())
sage: def eigenvalue(w, mu):
....:     return (-q2/q1)^height(Wcheck.from_reduced_word(w.reduced_word()).action(mu))
sage: all(E.eigenvalue(w, a) == eigenvalue(w, a) for w in E.keys() for a in Y.keys().simple_roots()) # long time (2.5s)
True

eigenvalues(mu)¶

Return the eigenvalues of $Y_{\alpha_0},\dots,Y_{\alpha_n}$ on $E_\mu$ .

INPUT:

mu – the index $\mu$ of an eigenvector or a tentative eigenvector

EXAMPLES:

sage: W = WeylGroup(["B",2])
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: w0 = W.long_element()
sage: E.eigenvalues(w0)
[q2^2/q1^2, q1/(-q2), q1/(-q2)]
sage: w = W.an_element()
sage: E.eigenvalues(w)
[(-q2)/q1, (-q2^2)/(-q1^2), q1^3/(-q2^3)]

hecke_parameters(i)¶

Return the Hecke parameters for index i.

EXAMPLES:

sage: W = WeylGroup(["B",3])
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: E.hecke_parameters(1)
(q1, q2)
sage: E.hecke_parameters(2)
(q1, q2)
sage: E.hecke_parameters(0)
(q1, q2)

keys()¶

The index set for the eigenvectors.

By default, this assumes that the eigenvectors span the full affine Hecke algebra module and that the eigenvectors have the same indexing as the basis of this module.

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: K = QQ['q1,q2']
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: E.keys()
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)

recursion(mu)¶

Return the indices used in the recursion.

INPUT:

mu – the index $\mu$ of an eigenvector

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: w0 = W.long_element()
sage: E.recursion(w0)
[]
sage: w = W.an_element(); w
123
sage: E.recursion(w)
[1, 2, 1]

seed(mu)¶

Return the eigenvector for $\mu$ minuscule.

INPUT:

mu – an element $\mu$ of the indexing set

OUTPUT: an element of T.domain()

This default implementation returns the monomial indexed by $\mu$ .

EXAMPLES:

sage: W = WeylGroup(["A",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: E = KW.demazure_lusztig_eigenvectors(q1, q2)
sage: E.seed(W.long_element())
B[123121]

twist(mu, i)¶

Act by $s_i$ on $\mu$ .

By default, this calls the method apply_simple_reflection.

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

class sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation(domain, on_basis, cartan_type, q1, q2, q=1, side='right')¶

Bases: sage.structure.sage_object.SageObject

A representation of an (affine) Hecke algebra given by the action of the $T$ generators

Let $F_i$ be a family of operators implementing an action of the operators $(T_i)_{i\in I}$ of the Hecke algebra on some vector space domain, given by their action on the basis of domain. This constructs the family of operators $(F_w)_{w\in W}$ describing the action of all elements of the basis $(T_w)_{w\in W}$ of the Hecke algebra. This is achieved by linearity on the first argument, and applying recursively the $F_i$ along a reduced word for $w=s_{i_1}\cdots s_{i_k}$ :

$F_w (x) = F_{i_k}\circ\cdots\circ F_{i_1}(x) .$

INPUT:

domain – a vector space
f – a function f(l,i) taking a basis element $l$ of domain and an index $i$ , and returning $F_i$
cartan_type – The Cartan type of the Hecke algebra
q1,q2 – The eigenvalues of the generators $T$ of the Hecke algebra
side – “left” or “right” (default: “right”) whether this is a left or right representation

EXAMPLES:

sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = WeylGroup(["A",3]).algebra(QQ)
sage: H = KW.demazure_lusztig_operators(q1,q2); H
A representation of the (q1, q2)-Hecke algebra of type ['A', 3, 1]
on Group algebra of Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space) over Rational Field

Among other things, it implements the $T_w$ operators, their inverses and compositions thereof:

sage: H.Tw((1,2))
Generic endomorphism of Group algebra of Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space) over Rational Field

and the Cherednik operators $Y^{\lambda^\vee}$ :

sage: H.Y()
Lazy family (...)_{i in Coroot lattice of the Root system of type ['A', 3, 1]}

REFERENCES:

[HST2008]

(1, 2, 3, 4, 5) F. Hivert, A. Schilling, N. Thiery, Hecke group algebras as quotients of affine Hecke algebras at level 0, Journal of Combinatorial Theory, Series A 116 (2009) 844-863 ( arXiv:0804.3781 [math.RT] )

Ti_inverse_on_basis(x, i)¶

The $T_i^{-1}$ operators, on basis elements

INPUT:

x – the index of a basis element
i – the index of a generator

EXAMPLES:

sage: W = WeylGroup("A3")
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: rho = KW.demazure_lusztig_operators(q1,q2)
sage: w = W.an_element()
sage: rho.Ti_inverse_on_basis(w, 1)
-1/q2*B[1231] + ((q1+q2)/(q1*q2))*B[123]

Ti_on_basis(x, i)¶

The $T_i$ operators, on basis elements.

INPUT:

x – the index of a basis element
i – the index of a generator

EXAMPLES:

sage: W = WeylGroup("A3")
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: rho = KW.demazure_lusztig_operators(q1,q2)
sage: w = W.an_element()
sage: rho.Ti_on_basis(w,1)
q1*B[1231]

Tw(word, signs=None, scalar=None)¶

Return $T_w$ .

INPUT:

word – a word $i_1,\dots,i_k$ for some element $w$ of the Weyl group. See straighten_word() for how this word can be specified.
signs – a list $\epsilon_1,\dots,\epsilon_k$ of the same length as word with $\epsilon_i =\pm 1$ or None for $1,\dots,1$ (default: None)
scalar – an element $c$ of the base ring or None for $1$ (default: None)

OUTPUT:

a module morphism implementing

$T_w = T_{i_k} \circ \cdots \circ T_{i_1}$

in left action notation; that is $T_{i_1}$ is applied first, then $T_{i_2}$ , etc.

More generally, if scalar or signs is specified, the morphism implements

$c T_{i_k}^{\epsilon_k} \circ \cdots \circ T_{i_1}^{\epsilon_k}$

EXAMPLES:

sage: W = WeylGroup("A3")
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: x = KW.an_element(); x
2*B[12321] + 3*B[1231] + B[123] + B[]

sage: T = KW.demazure_lusztig_operators(q1,q2)
sage: T12 = T.Tw( (1,2) )
sage: T12(KW.one())
q1^2*B[12]

This is $T_2 \circ T_1$ :

sage: T[2](T[1](KW.one()))
q1^2*B[12]
sage: T[1](T[2](KW.one()))
q1^2*B[21]
sage: T12(x) == T[2](T[1](x))
True

Now with signs and scalar coefficient we construct $3 T_2 \circ T_1^{-1}$ :

sage: phi = T.Tw((1,2), (-1,1), 3)
sage: phi(KW.one())
((-3*q1)/q2)*B[12] + ((3*q1+3*q2)/q2)*B[2]
sage: phi(T[1](x)) == 3*T[2](x)
True

For debugging purposes, one can recover the input data:

sage: phi.word
(1, 2)
sage: phi.signs
(-1, 1)
sage: phi.scalar
3

Tw_inverse(word)¶

Return $T_w^{-1}$ .

This is essentially a shorthand for Tw() with all minus signs.

Todo

Add an example where $T_i\ne T_i^{-1}$

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: W.element_class._repr_ = lambda x: "".join(str(i) for i in x.reduced_word())
sage: KW = W.algebra(QQ)
sage: rho = KW.demazure_lusztig_operators(1, -1)
sage: x = KW.monomial(W.an_element()); x
B[123]
sage: word = [1,2]
sage: rho.Tw(word)(x)
B[12312]
sage: rho.Tw_inverse(word)(x)
B[12321]

sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: rho = KW.demazure_lusztig_operators(q1, q2)
sage: x = KW.monomial(W.an_element()); x
B[123]
sage: rho.Tw_inverse(word)(x)
1/q2^2*B[12321] + ((-q1-q2)/(q1*q2^2))*B[1231] + ((-q1-q2)/(q1*q2^2))*B[1232] + ((q1^2+2*q1*q2+q2^2)/(q1^2*q2^2))*B[123]
sage: rho.Tw(word)(_)
B[123]

Y(base_ring=Integer Ring)¶

Return the Cherednik operators $Y$ for this representation of an affine Hecke algebra.

INPUT:

self – a representation of an affine Hecke algebra
base_ring – the base ring of the coroot lattice

This is a family of operators indexed by the coroot lattice for this Cartan type. In practice this is currently indexed instead by the affine coroot lattice, even if this indexing is not one to one, in order to allow for $Y[\alpha^\vee_0]$ .

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: rho = KW.demazure_lusztig_operators(q2, q1)
sage: Y = rho.Y(); Y
Lazy family (...(i))_{i in Coroot lattice of the Root system of type ['A', 3, 1]}

Y_eigenvectors()¶

Return the family of eigenvectors for the Cherednik operators $Y$ of this representation of an affine Hecke algebra.

INPUT:

self – a representation of an affine Hecke algebra
base_ring – the base ring of the coroot lattice

EXAMPLES:

sage: W = WeylGroup(["B",2])
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)
sage: rho = KW.demazure_lusztig_operators(q1, q2, affine=True)
sage: E = rho.Y_eigenvectors()
sage: E.keys()
Weyl Group of type ['B', 2] (as a matrix group acting on the ambient space)
sage: w0 = W.long_element()

To set the recurrence up properly, one often needs to customize the CherednikOperatorsEigenvectors.affine_lift() and CherednikOperatorsEigenvectors.affine_retract() methods. This would usually be done by subclassing CherednikOperatorsEigenvectors; here we just override the methods directly.

In this particular case, we multiply by $w_0$ to take into account that $w_0$ is the seed for the recursion:

sage: E.affine_lift = w0._mul_
sage: E.affine_retract = w0._mul_

sage: E[w0]
B[2121]
sage: E.eigenvalues(E[w0])
[q2^2/q1^2, q1/(-q2), q1/(-q2)]

This step is taken care of automatically if one instead calls the specialization sage.coxeter_groups.CoxeterGroups.Algebras.demazure_lusztig_eigenvectors().

Now we can compute all eigenvectors:

sage: [E[w] for w in W]
[B[2121] - B[121] - B[212] + B[12] + B[21] - B[1] - B[2] + B[],
-B[2121] + B[212],
(q2/(q1-q2))*B[2121] + (q2/(-q1+q2))*B[121] + (q2/(-q1+q2))*B[212] - B[12] + ((-q2)/(-q1+q2))*B[21] + B[2],
((-q2^2)/(-q1^2+q1*q2-q2^2))*B[2121] - B[121] + (q2^2/(-q1^2+q1*q2-q2^2))*B[212] + B[21],
((q1^2+q2^2)/(-q1^2+q1*q2-q2^2))*B[2121] + ((-q1^2-q2^2)/(-q1^2+q1*q2-q2^2))*B[121] + ((-q2^2)/(-q1^2+q1*q2-q2^2))*B[212] + (q2^2/(-q1^2+q1*q2-q2^2))*B[12] - B[21] + B[1],
B[2121],
(q2/(-q1+q2))*B[2121] + ((-q2)/(-q1+q2))*B[121] - B[212] + B[12],
-B[2121] + B[121]]

Y_lambdacheck(lambdacheck)¶

Return the Cherednik operators $Y^{\lambda^\vee}$ for this representation of an affine Hecke algebra.

INPUT:

lambdacheck – an element of the coroot lattice for this cartan type

EXAMPLES:

sage: W = WeylGroup(["B",2])
sage: W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
sage: K = QQ['q1,q2'].fraction_field()
sage: q1, q2 = K.gens()
sage: KW = W.algebra(K)

We take $q_2$ and $q_1$ as eigenvalues to match with the notations of [HST2008]

sage: rho = KW.demazure_lusztig_operators(q2, q1)
sage: L = rho.Y().keys()
sage: alpha = L.simple_roots()
sage: Y0 = rho.Y_lambdacheck(alpha[0])
sage: Y1 = rho.Y_lambdacheck(alpha[1])
sage: Y2 = rho.Y_lambdacheck(alpha[2])

sage: x = KW.monomial(W.an_element()); x
B[12]
sage: Y1(x)
((-q1^2-2*q1*q2-q2^2)/(-q2^2))*B[2121] + ((q1^3+q1^2*q2+q1*q2^2+q2^3)/(-q1*q2^2))*B[121] + ((q1^2+q1*q2)/(-q2^2))*B[212] + ((-q1^2)/(-q2^2))*B[12]
sage: Y2(x)
((-q1^4-q1^3*q2-q1*q2^3-q2^4)/(-q1^3*q2))*B[2121] + ((q1^3+q1^2*q2+q1*q2^2+q2^3)/(-q1^2*q2))*B[121] + (q2^3/(-q1^3))*B[12]
sage: Y1(Y2(x))
((q1*q2+q2^2)/q1^2)*B[212] + ((-q2)/q1)*B[12]
sage: Y2(Y1(x))
((q1*q2+q2^2)/q1^2)*B[212] + ((-q2)/q1)*B[12]

The $Y$ operators commute:

sage: Y0(Y1(x)) - Y1(Y0(x))
0
sage: Y2(Y1(x)) - Y1(Y2(x))
0

In the classical root lattice, $\alpha_0+\alpha_1+\alpha_2=0$ :

sage: Y0(Y1(Y2(x)))
B[12]

Lemma 7.2 of [HST2008]:

sage: w0 = KW.monomial(W.long_element())
sage: rho.Tw(0)(w0)
q2*B[1]
sage: rho.Tw_inverse(1)(w0)
1/q2*B[212]
sage: rho.Tw_inverse(2)(w0)
1/q2*B[121]

Lemma 7.5 of [HST2008]:

sage: Y0(w0)
q1^2/q2^2*B[2121]
sage: Y1(w0)
(q2/(-q1))*B[2121]
sage: Y2(w0)
(q2/(-q1))*B[2121]

Todo

Add more tests

Add tests in type BC affine where the null coroot $\delta^\vee$ can have non trivial coefficient in term of $\alpha_0$

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