Coxeter Group Algebras¶

class sage.categories.coxeter_group_algebras.CoxeterGroupAlgebras(category, *args)[source]¶

Bases: AlgebrasCategory

class ParentMethods[source]¶

Bases: object

demazure_lusztig_eigenvectors(q1, q2)[source]¶

Return the family of eigenvectors for the Cherednik operators.

INPUT:

self – a finite Coxeter group \(W\)
q1, q2 – two elements of the ground ring \(K\)

The affine Hecke algebra \(H_{q_1,q_2}(\tilde W)\) acts on the group algebra of \(W\) through the Demazure-Lusztig operators \(T_i\). Its Cherednik operators \(Y^\lambda\) can be simultaneously diagonalized as long as \(q_1/q_2\) is not a small root of unity [HST2008].

This method returns the family of joint eigenvectors, indexed by \(W\).

See also

EXAMPLES:

Sage

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: E = KW.demazure_lusztig_eigenvectors(q1,q2)
sage: E.keys()                                                          # needs sage.rings.number_field
Weyl Group of type ['B', 2] (as a matrix group acting on the ambient space)
sage: w = W.an_element()
sage: E[w]                                                              # needs sage.rings.number_field
(q2/(-q1+q2))*2121 + ((-q2)/(-q1+q2))*121 - 212 + 12

Python

>>> from sage.all import *
>>> W = WeylGroup(["B",Integer(2)])
>>> W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
>>> K = QQ['q1,q2'].fraction_field()
>>> q1, q2 = K.gens()
>>> KW = W.algebra(K)
>>> E = KW.demazure_lusztig_eigenvectors(q1,q2)
>>> E.keys()                                                          # needs sage.rings.number_field
Weyl Group of type ['B', 2] (as a matrix group acting on the ambient space)
>>> w = W.an_element()
>>> E[w]                                                              # needs sage.rings.number_field
(q2/(-q1+q2))*2121 + ((-q2)/(-q1+q2))*121 - 212 + 12

Sage Live

W = WeylGroup(["B",2])
W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
K = QQ['q1,q2'].fraction_field()
q1, q2 = K.gens()
KW = W.algebra(K)
E = KW.demazure_lusztig_eigenvectors(q1,q2)
E.keys()                                                          # needs sage.rings.number_field
w = W.an_element()
E[w]                                                              # needs sage.rings.number_field

demazure_lusztig_operator_on_basis(w, i, q1, q2, side='right')[source]¶

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

INPUT:

w – an element of the Coxeter group
i – an element of the index set
q1, q2 – two elements of the ground ring
bar – boolean (default: False)

See demazure_lusztig_operators() for details.

EXAMPLES:

Sage

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: w = W.an_element()
sage: KW.demazure_lusztig_operator_on_basis(w, 0, q1, q2)
(-q2)*323123 + (q1+q2)*123
sage: KW.demazure_lusztig_operator_on_basis(w, 1, q1, q2)
q1*1231
sage: KW.demazure_lusztig_operator_on_basis(w, 2, q1, q2)
q1*1232
sage: KW.demazure_lusztig_operator_on_basis(w, 3, q1, q2)
(q1+q2)*123 + (-q2)*12

Python

>>> from sage.all import *
>>> W = WeylGroup(["B",Integer(3)])
>>> W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
>>> K = QQ['q1,q2']
>>> q1, q2 = K.gens()
>>> KW = W.algebra(K)
>>> w = W.an_element()
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(0), q1, q2)
(-q2)*323123 + (q1+q2)*123
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(1), q1, q2)
q1*1231
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(2), q1, q2)
q1*1232
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(3), q1, q2)
(q1+q2)*123 + (-q2)*12

Sage Live

W = WeylGroup(["B",3])
W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
K = QQ['q1,q2']
q1, q2 = K.gens()
KW = W.algebra(K)
w = W.an_element()
KW.demazure_lusztig_operator_on_basis(w, 0, q1, q2)
KW.demazure_lusztig_operator_on_basis(w, 1, q1, q2)
KW.demazure_lusztig_operator_on_basis(w, 2, q1, q2)
KW.demazure_lusztig_operator_on_basis(w, 3, q1, q2)

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

Sage

sage: KW.demazure_lusztig_operator_on_basis(w, 0, 1, 0)
123
sage: KW.demazure_lusztig_operator_on_basis(w, 1, 1, 0)
1231
sage: KW.demazure_lusztig_operator_on_basis(w, 2, 1, 0)
1232
sage: KW.demazure_lusztig_operator_on_basis(w, 3, 1, 0)
123

Python

>>> from sage.all import *
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(0), Integer(1), Integer(0))
123
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(1), Integer(1), Integer(0))
1231
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(2), Integer(1), Integer(0))
1232
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(3), Integer(1), Integer(0))
123

Sage Live

KW.demazure_lusztig_operator_on_basis(w, 0, 1, 0)
KW.demazure_lusztig_operator_on_basis(w, 1, 1, 0)
KW.demazure_lusztig_operator_on_basis(w, 2, 1, 0)
KW.demazure_lusztig_operator_on_basis(w, 3, 1, 0)

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

Sage

sage: KW.demazure_lusztig_operator_on_basis(w, 0, 1, -1)
323123
sage: KW.demazure_lusztig_operator_on_basis(w, 1, 1, -1)
1231
sage: KW.demazure_lusztig_operator_on_basis(w, 2, 1, -1)
1232
sage: KW.demazure_lusztig_operator_on_basis(w, 3, 1, -1)
12

Python

>>> from sage.all import *
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(0), Integer(1), -Integer(1))
323123
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(1), Integer(1), -Integer(1))
1231
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(2), Integer(1), -Integer(1))
1232
>>> KW.demazure_lusztig_operator_on_basis(w, Integer(3), Integer(1), -Integer(1))
12

Sage Live

KW.demazure_lusztig_operator_on_basis(w, 0, 1, -1)
KW.demazure_lusztig_operator_on_basis(w, 1, 1, -1)
KW.demazure_lusztig_operator_on_basis(w, 2, 1, -1)
KW.demazure_lusztig_operator_on_basis(w, 3, 1, -1)

demazure_lusztig_operators(q1, q2, side='right', affine=True)[source]¶

Return the Demazure Lusztig operators acting on self.

INPUT:

q1, q2 – two elements of the ground ring \(K\)
side – 'left' or 'right' (default: 'right'); which side to act upon
affine – boolean (default: True)

The Demazure-Lusztig operator \(T_i\) is the linear map \(R \to R\) obtained by interpolating between the simple projection \(\pi_i\) (see CoxeterGroups.ElementMethods.simple_projection()) and the simple reflection \(s_i\) so that \(T_i\) has eigenvalues \(q_1\) and \(q_2\):

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

The Demazure-Lusztig operators give the usual representation of the operators \(T_i\) of the \(q_1,q_2\) Hecke algebra associated to the Coxeter group.

For a finite Coxeter group, and if affine=True, the Demazure-Lusztig operators \(T_1,\dots,T_n\) are completed by \(T_0\) to implement the level \(0\) action of the affine Hecke algebra.

EXAMPLES:

Sage

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: x = KW.monomial(W.an_element()); x
123
sage: T[0](x)
(-q2)*323123 + (q1+q2)*123
sage: T[1](x)
q1*1231
sage: T[2](x)
q1*1232
sage: T[3](x)
(q1+q2)*123 + (-q2)*12

sage: T._test_relations()

Python

>>> from sage.all import *
>>> W = WeylGroup(["B",Integer(3)])
>>> W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
>>> K = QQ['q1,q2']
>>> q1, q2 = K.gens()
>>> KW = W.algebra(K)
>>> T = KW.demazure_lusztig_operators(q1, q2, affine=True)
>>> x = KW.monomial(W.an_element()); x
123
>>> T[Integer(0)](x)
(-q2)*323123 + (q1+q2)*123
>>> T[Integer(1)](x)
q1*1231
>>> T[Integer(2)](x)
q1*1232
>>> T[Integer(3)](x)
(q1+q2)*123 + (-q2)*12

>>> T._test_relations()

Sage Live

W = WeylGroup(["B",3])
W.element_class._repr_=lambda x: "".join(str(i) for i in x.reduced_word())
K = QQ['q1,q2']
q1, q2 = K.gens()
KW = W.algebra(K)
T = KW.demazure_lusztig_operators(q1, q2, affine=True)
x = KW.monomial(W.an_element()); x
T[0](x)
T[1](x)
T[2](x)
T[3](x)
T._test_relations()

Note

For a finite Weyl group \(W\), the level 0 action of the affine Weyl group \(\tilde W\) only depends on the Coxeter diagram of the affinization, not its Dynkin diagram. Hence it is possible to explore all cases using only untwisted affinizations.