# Coxeter Group Algebras#

class sage.categories.coxeter_group_algebras.CoxeterGroupAlgebras(category, *args)#
class ParentMethods#

Bases: object

demazure_lusztig_eigenvectors(q1, q2)#

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 $$W$$.

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

demazure_lusztig_operator_on_basis(w, i, q1, q2, side='right')#

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 – a boolean (default False)

See demazure_lusztig_operators() for details.

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: 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


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

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


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

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

demazure_lusztig_operators(q1, q2, side='right', affine=True)#

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 – a 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: 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()


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.