Extended Affine Weyl Groups#
AUTHORS:
Daniel Bump (2012): initial version
Daniel Orr (2012): initial version
Anne Schilling (2012): initial version
Mark Shimozono (2012): initial version
Nicolas M. Thiery (2012): initial version
Mark Shimozono (2013): twisted affine root systems, multiple realizations, GL_n
- sage.combinat.root_system.extended_affine_weyl_group.ExtendedAffineWeylGroup(cartan_type, general_linear=None, **print_options)#
The extended affine Weyl group.
INPUT:
cartan_type
– An affine or finite Cartan type (a finite Cartan type is an abbreviation for its untwisted affinization)general_linear
– (default: None) If True andcartan_type
indicates untwisted type A, returns the universal central extensionprint_options
– Special instructions for printing elements (see below)
Mnemonics
“P” – subgroup of translations
“Pv” – subgroup of translations in a dual form
“W0” – classical Weyl group
“W” – affine Weyl group
“F” – fundamental group of length zero elements
There are currently six realizations: “PW0”, “W0P, “WF”, “FW”, “PvW0”, and “W0Pv”.
“PW0” means the semidirect product of “P” with “W0” acting from the right. “W0P” is similar but with “W0” acting from the left. “WF” is the semidirect product of “W” with “F” acting from the right, etc.
Recognized arguments for
print_options
are:print_tuple
– True or False (default: False) If True, elements are printed \((a,b)\), otherwise as \(a * b\)affine
– Prefix for simple reflections in the affine Weyl groupclassical
– Prefix for simple reflections in the classical Weyl grouptranslation
– Prefix for the translation elementsfundamental
– Prefix for the elements of the fundamental group
These options are not mutable.
The extended affine Weyl group was introduced in the following references.
REFERENCES:
[Iwahori]Iwahori, Generalized Tits system (Bruhat decomposition) on p-adic semisimple groups. 1966 Algebraic Groups and Discontinuous Subgroups (AMS Proc. Symp. Pure Math.., 1965) pp. 71-83 Amer. Math. Soc., Providence, R.I.
[Bour]Bourbaki, Lie Groups and Lie Algebras IV.2
Notation
\(R\) – An irreducible affine root system
\(I\) – Set of nodes of the Dynkin diagram of \(R\)
\(R_0\) – The classical subsystem of \(R\)
\(I_0\) – Set of nodes of the Dynkin diagram of \(R_0\)
\(E\) – Extended affine Weyl group of type \(R\)
\(W\) – Affine Weyl group of type \(R\)
\(W_0\) – finite (classical) Weyl group (of type \(R_0\))
\(M\) – translation lattice for \(W\)
\(L\) – translation lattice for \(E\)
\(F\) – Fundamental subgroup of \(E\) (the length zero elements)
\(P\) – Finite weight lattice
\(Q\) – Finite root lattice
\(P^\vee\) – Finite coweight lattice
\(Q^\vee\) – Finite coroot lattice
Translation lattices
The styles “PW0” and “W0P” use the following lattices:
Untwisted affine: \(L = P^\vee\), \(M = Q^\vee\)
Dual of untwisted affine: \(L = P\), \(M = Q\)
\(BC_n\) (\(A_{2n}^{(2)}\)): \(L = M = P\)
Dual of \(BC_n\) (\(A_{2n}^{(2)\dagger}\)): \(L = M = P^\vee\)
The styles “PvW0” and “W0Pv” use the following lattices:
Untwisted affine: The weight lattice of the dual finite Cartan type.
Dual untwisted affine: The same as for “PW0” and “W0P”.
For mixed affine type (\(A_{2n}^{(2)}\), aka \(\tilde{BC}_n\), and their affine duals) the styles “PvW0” and “W0Pv” are not implemented.
Finite and affine Weyl groups \(W_0\) and \(W\)
The finite Weyl group \(W_0\) is generated by the simple reflections \(s_i\) for \(i \in I_0\) where \(s_i\) is the reflection across a suitable hyperplane \(H_i\) through the origin in the real span \(V\) of the lattice \(M\).
\(R\) specifies another (affine) hyperplane \(H_0\). The affine Weyl group \(W\) is generated by \(W_0\) and the reflection \(S_0\) across \(H_0\).
Extended affine Weyl group \(E\)
The complement in \(V\) of the set \(H\) of hyperplanes obtained from the \(H_i\) by the action of \(W\), has connected components called alcoves. \(W\) acts freely and transitively on the set of alcoves. After the choice of a certain alcove (the fundamental alcove), there is an induced bijection from \(W\) to the set of alcoves under which the identity in \(W\) maps to the fundamental alcove.
Then \(L\) is the largest sublattice of \(V\), whose translations stabilize the set of alcoves.
There are isomorphisms
\[\begin{split}\begin{aligned} W &\cong M \rtimes W_0 \cong W_0 \ltimes M \\ E &\cong L \rtimes W_0 \cong W_0 \ltimes L \end{aligned}\end{split}\]Fundamental group of affine Dynkin automorphisms
Since \(L\) acts on the set of alcoves, the group \(F = L/M\) may be viewed as a subgroup of the symmetries of the fundamental alcove or equivalently the symmetries of the affine Dynkin diagram. \(F\) acts on the set of alcoves and hence on \(W\). Conjugation by an element of \(F\) acts on \(W\) by permuting the indices of simple reflections.
There are isomorphisms
\[E \cong F \ltimes W \cong W \rtimes F\]An affine Dynkin node is special if it is conjugate to the zero node under some affine Dynkin automorphism.
There is a bijection \(i\) \(\mapsto\) \(\pi_i\) from the set of special nodes to the group \(F\), where \(\pi_i\) is the unique element of \(F\) that sends \(0\) to \(i\). When \(L=P\) (resp. \(L=P^\vee\)) the element \(\pi_i\) is induced (under the isomorphism \(F \cong L/M\)) by addition of the coset of the \(i\)-th fundamental weight (resp. coweight).
The length function of the Coxeter group \(W\) may be extended to \(E\) by \(\ell(w \pi) = \ell(w)\) where \(w \in W\) and \(\pi\in F\). This is the number of hyperplanes in \(H\) separating the fundamental alcove from its image by \(w \pi\) (or equivalently \(w\)).
It is known that if \(G\) is the compact Lie group of adjoint type with root system \(R_0\) then \(F\) is isomorphic to the fundamental group of \(G\), or to the center of its simply-connected covering group. That is why we call \(F\) the fundamental group.
In the future we may want to build an element of the group from an appropriate linear map f on some of the root lattice realizations for this Cartan type: W.from_endomorphism(f).
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(["A",2,1]); E Extended affine Weyl group of type ['A', 2, 1] sage: type(E) <class 'sage.combinat.root_system.extended_affine_weyl_group.ExtendedAffineWeylGroup_Class_with_category'> sage: PW0=E.PW0(); PW0 Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Coweight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) sage: W0P = E.W0P(); W0P Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) acting on Multiplicative form of Coweight lattice of the Root system of type ['A', 2] sage: PvW0 = E.PvW0(); PvW0 Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Weight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) sage: W0Pv = E.W0Pv(); W0Pv Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) acting on Multiplicative form of Weight lattice of the Root system of type ['A', 2] sage: WF = E.WF(); WF Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) acted upon by Fundamental group of type ['A', 2, 1] sage: FW = E.FW(); FW Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Fundamental group of type ['A', 2, 1] acting on Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice)
When the realizations are constructed from each other as above, there are built-in coercions between them.
sage: F = E.fundamental_group() sage: x = WF.from_reduced_word([0,1,2]) * WF(F(2)); x S0*S1*S2 * pi[2] sage: FW(x) pi[2] * S1*S2*S0 sage: W0P(x) s1*s2*s1 * t[-2*Lambdacheck[1] - Lambdacheck[2]] sage: PW0(x) t[Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2*s1 sage: PvW0(x) t[Lambda[1] + 2*Lambda[2]] * s1*s2*s1
The translation lattice and its distinguished basis are obtained from
E
:sage: L = E.lattice(); L Coweight lattice of the Root system of type ['A', 2] sage: b = E.lattice_basis(); b Finite family {1: Lambdacheck[1], 2: Lambdacheck[2]}
Translation lattice elements can be coerced into any realization:
sage: PW0(b[1]-b[2]) t[Lambdacheck[1] - Lambdacheck[2]] sage: FW(b[1]-b[2]) pi[2] * S0*S1
The dual form of the translation lattice and its basis are similarly obtained:
sage: Lv = E.dual_lattice(); Lv Weight lattice of the Root system of type ['A', 2] sage: bv = E.dual_lattice_basis(); bv Finite family {1: Lambda[1], 2: Lambda[2]} sage: FW(bv[1]-bv[2]) pi[2] * S0*S1
The abstract fundamental group is accessed from
E
:sage: F = E.fundamental_group(); F Fundamental group of type ['A', 2, 1]
Its elements are indexed by the set of special nodes of the affine Dynkin diagram:
sage: E.cartan_type().special_nodes() (0, 1, 2) sage: F.special_nodes() (0, 1, 2) sage: [F(i) for i in F.special_nodes()] [pi[0], pi[1], pi[2]]
There is a coercion from the fundamental group into each realization:
sage: F(2) pi[2] sage: WF(F(2)) pi[2] sage: W0P(F(2)) s2*s1 * t[-Lambdacheck[1]] sage: W0Pv(F(2)) s2*s1 * t[-Lambda[1]]
Using
E
one may access the classical and affine Weyl groups and their morphisms into each realization:sage: W0 = E.classical_weyl(); W0 Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) sage: v = W0.from_reduced_word([1,2,1]); v s1*s2*s1 sage: PW0(v) s1*s2*s1 sage: WF(v) S1*S2*S1 sage: W = E.affine_weyl(); W Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) sage: w = W.from_reduced_word([2,1,0]); w S2*S1*S0 sage: WF(w) S2*S1*S0 sage: PW0(w) t[Lambdacheck[1] - 2*Lambdacheck[2]] * s1
Note that for untwisted affine type, the dual form of the classical Weyl group is isomorphic to the usual one, but acts on a different lattice and is therefore different to sage:
sage: W0v = E.dual_classical_weyl(); W0v Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) sage: v = W0v.from_reduced_word([1,2]) sage: x = PvW0(v); x s1*s2 sage: y = PW0(v); y s1*s2 sage: x.parent() == y.parent() False
However, because there is a coercion from
PvW0
toPW0
, the elementsx
andy
compare as equal:sage: x == y True
An element can be created directly from a reduced word:
sage: PW0.from_reduced_word([2,1,0]) t[Lambdacheck[1] - 2*Lambdacheck[2]] * s1
Here is a demonstration of the printing options:
sage: E = ExtendedAffineWeylGroup(["A",2,1], affine="sx", classical="Sx",translation="x",fundamental="pix") sage: PW0 = E.PW0() sage: y = PW0(E.lattice_basis()[1]) sage: y x[Lambdacheck[1]] sage: FW = E.FW() sage: FW(y) pix[1] * sx2*sx1 sage: PW0.an_element() x[2*Lambdacheck[1] + 2*Lambdacheck[2]] * Sx1*Sx2
Todo
Implement a “slow” action of \(E\) on any affine root or weight lattice realization.
Implement the level \(m\) actions of \(E\) and \(W\) on the lattices of finite type.
Implement the relevant methods from the usual affine Weyl group
Implementation by matrices: style “M”.
Use case: implement the Hecke algebra on top of this
The semidirect product construction in sage currently only admits multiplicative groups. Therefore for the styles involving “P” and “Pv”, one must convert the additive group of translations \(L\) into a multiplicative group by applying the
sage.groups.group_exp.GroupExp
functor.The general linear case
The general linear group is not semisimple. Sage can build its extended affine Weyl group:
sage: E = ExtendedAffineWeylGroup(['A',2,1], general_linear=True); E Extended affine Weyl group of GL(3)
If the Cartan type is
['A', n-1, 1]
and the parametergeneral_linear
is not True, the extended affine Weyl group that is built will be for \(SL_n\), not \(GL_n\). But ifgeneral_linear
is True, let \(W_a\) and \(W_e\) be the affine and extended affine Weyl groups. We make the following nonstandard definition: the extended affine Weyl group \(W_e(GL_n)\) is defined by\[W_e(GL_n) = P(GL_n) \rtimes W\]where \(W\) is the finite Weyl group (the symmetric group \(S_n\)) and \(P(GL_n)\) is the weight lattice of \(GL_n\), which is usually identified with the lattice \(\ZZ^n\) of \(n\)-tuples of integers:
sage: PW0 = E.PW0(); PW0 Extended affine Weyl group of GL(3) realized by Semidirect product of Multiplicative form of Ambient space of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space) sage: PW0.an_element() t[(2, 2, 3)] * s1*s2
There is an isomorphism
\[W_e(GL_n) = \ZZ \ltimes W_a\]where the group of integers \(\ZZ\) (with generator \(\pi\)) acts on \(W_a\) by
\[\pi\, s_i\, \pi^{-1} = s_{i+1}\]and the indices of the simple reflections are taken modulo \(n\):
sage: FW = E.FW(); FW Extended affine Weyl group of GL(3) realized by Semidirect product of Fundamental group of GL(3) acting on Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) sage: FW.an_element() pi[5] * S0*S1*S2
We regard \(\ZZ\) as the fundamental group of affine type \(GL_n\):
sage: F = E.fundamental_group(); F Fundamental group of GL(3) sage: F.special_nodes() Integer Ring sage: x = FW.from_fundamental(F(10)); x pi[10] sage: x*x pi[20] sage: E.PvW0()(x*x) t[(7, 7, 6)] * s2*s1
- class sage.combinat.root_system.extended_affine_weyl_group.ExtendedAffineWeylGroup_Class(cartan_type, general_linear, **print_options)#
Bases:
UniqueRepresentation
,Parent
The parent-with-realization class of an extended affine Weyl group.
- class ExtendedAffineWeylGroupFW(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the affine Weyl group by the fundamental group.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).FW() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Fundamental group of type ['A', 2, 1] acting on Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice)
- Element#
alias of
ExtendedAffineWeylGroupFWElement
- from_affine_weyl(w)#
Return the image of \(w\) under the map of the affine Weyl group into the right (affine Weyl group) factor in the “FW” style.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],print_tuple=True) sage: E.FW().from_affine_weyl(E.affine_weyl().from_reduced_word([0,2,1])) (pi[0], S0*S2*S1)
- from_fundamental(f)#
Return the image of the fundamental group element \(f\) into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],print_tuple=True) sage: E.FW().from_fundamental(E.fundamental_group()(2)) (pi[2], 1)
- simple_reflections()#
Return the family of simple reflections of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1],print_tuple=True).FW().simple_reflections() Finite family {0: (pi[0], S0), 1: (pi[0], S1), 2: (pi[0], S2)}
- class ExtendedAffineWeylGroupFWElement#
Bases:
GroupSemidirectProductElement
The element class for the “FW” realization.
- action_on_affine_roots(beta)#
Act by
self
on the affine root lattice elementbeta
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],affine="s") sage: x = E.FW().an_element(); x pi[2] * s0*s1*s2 sage: v = RootSystem(['A',2,1]).root_lattice().an_element(); v 2*alpha[0] + 2*alpha[1] + 3*alpha[2] sage: x.action_on_affine_roots(v) alpha[0] + alpha[1]
- has_descent(i, side='right', positive=False)#
Return whether
self
has descent at \(i\).INPUT:
\(i\) – an affine Dynkin index.
OPTIONAL:
side
– ‘left’ or ‘right’ (default: ‘right’)positive
– True or False (default: False)
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.FW().an_element(); x pi[2] * S0*S1*S2 sage: [(i, x.has_descent(i)) for i in E.cartan_type().index_set()] [(0, False), (1, False), (2, True)]
- to_affine_weyl_right()#
Project
self
to the right (affine Weyl group) factor in the “FW” style.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.FW().from_translation(E.lattice_basis()[1]); x pi[1] * S2*S1 sage: x.to_affine_weyl_right() S2*S1
- to_fundamental_group()#
Return the projection of
self
to the fundamental group in the “FW” style.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.FW().from_translation(E.lattice_basis()[2]); x pi[2] * S1*S2 sage: x.to_fundamental_group() pi[2]
- class ExtendedAffineWeylGroupPW0(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the translation lattice by the finite Weyl group.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).PW0() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Coweight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice)
- Element#
alias of
ExtendedAffineWeylGroupPW0Element
- S0()#
Return the affine simple reflection.
EXAMPLES:
sage: ExtendedAffineWeylGroup(['B',2]).PW0().S0() t[Lambdacheck[2]] * s2*s1*s2
- from_classical_weyl(w)#
Return the image of \(w\) under the homomorphism of the classical Weyl group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup("A3",print_tuple=True) sage: E.PW0().from_classical_weyl(E.classical_weyl().from_reduced_word([1,2])) (t[0], s1*s2)
- from_translation(la)#
Map the translation lattice element
la
intoself
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1], translation="tau", print_tuple = True) sage: la = E.lattice().an_element(); la 2*Lambdacheck[1] + 2*Lambdacheck[2] sage: E.PW0().from_translation(la) (tau[2*Lambdacheck[1] + 2*Lambdacheck[2]], 1)
- simple_reflection(i)#
Return the \(i\)-th simple reflection in
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup("G2") sage: [(i, E.PW0().simple_reflection(i)) for i in E.cartan_type().index_set()] [(0, t[Lambdacheck[2]] * s2*s1*s2*s1*s2), (1, s1), (2, s2)]
- simple_reflections()#
Return a family for the simple reflections of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup("A3").PW0().simple_reflections() Finite family {0: t[Lambdacheck[1] + Lambdacheck[3]] * s1*s2*s3*s2*s1, 1: s1, 2: s2, 3: s3}
- class ExtendedAffineWeylGroupPW0Element#
Bases:
GroupSemidirectProductElement
The element class for the “PW0” realization.
- action(la)#
Return the action of
self
on an elementla
of the translation lattice.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]); PW0=E.PW0() sage: x = PW0.an_element(); x t[2*Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2 sage: la = E.lattice().an_element(); la 2*Lambdacheck[1] + 2*Lambdacheck[2] sage: x.action(la) -2*Lambdacheck[1] + 4*Lambdacheck[2]
- has_descent(i, side='right', positive=False)#
Return whether
self
has \(i\) as a descent.INPUT:
\(i\) – an affine Dynkin node
OPTIONAL:
side
– ‘left’ or ‘right’ (default: ‘right’)positive
– True or False (default: False)
EXAMPLES:
sage: w = ExtendedAffineWeylGroup(['A',4,2]).PW0().from_reduced_word([0,1]); w t[Lambda[1]] * s1*s2 sage: w.has_descent(0, side='left') True
- to_classical_weyl()#
Return the image of
self
under the homomorphism that projects to the classical Weyl group factor after rewriting it in either style “PW0” or “W0P”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).PW0().S0(); s t[Lambdacheck[1] + Lambdacheck[2]] * s1*s2*s1 sage: s.to_classical_weyl() s1*s2*s1
- to_translation_left()#
The image of
self
under the map that projects to the translation lattice factor after factoring it to the left as in style “PW0”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).PW0().S0(); s t[Lambdacheck[1] + Lambdacheck[2]] * s1*s2*s1 sage: s.to_translation_left() Lambdacheck[1] + Lambdacheck[2]
- class ExtendedAffineWeylGroupPvW0(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the dual form of the translation lattice by the finite Weyl group.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).PvW0() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Weight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice)
- Element#
alias of
ExtendedAffineWeylGroupPvW0Element
- from_dual_classical_weyl(w)#
Return the image of \(w\) under the homomorphism of the dual form of the classical Weyl group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1],print_tuple=True) sage: E.PvW0().from_dual_classical_weyl(E.dual_classical_weyl().from_reduced_word([1,2])) (t[0], s1*s2)
- from_dual_translation(la)#
Map the dual translation lattice element
la
intoself
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1], translation="tau", print_tuple = True) sage: la = E.dual_lattice().an_element(); la 2*Lambda[1] + 2*Lambda[2] sage: E.PvW0().from_dual_translation(la) (tau[2*Lambda[1] + 2*Lambda[2]], 1)
- simple_reflections()#
Return a family for the simple reflections of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PvW0().simple_reflections() Finite family {0: t[Lambda[1] + Lambda[3]] * s1*s2*s3*s2*s1, 1: s1, 2: s2, 3: s3}
- class ExtendedAffineWeylGroupPvW0Element#
Bases:
GroupSemidirectProductElement
The element class for the “PvW0” realization.
- dual_action(la)#
Return the action of
self
on an elementla
of the dual version of the translation lattice.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.PvW0().an_element(); x t[2*Lambda[1] + 2*Lambda[2]] * s1*s2 sage: la = E.dual_lattice().an_element(); la 2*Lambda[1] + 2*Lambda[2] sage: x.dual_action(la) -2*Lambda[1] + 4*Lambda[2]
- has_descent(i, side='right', positive=False)#
Return whether
self
has \(i\) as a descent.INPUT:
\(i\) - an affine Dynkin index
OPTIONAL:
side
– ‘left’ or ‘right’ (default: ‘right’)positive
– True or False (default: False)
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',4,2]) sage: w = E.PvW0().from_reduced_word([0,1]); w t[Lambda[1]] * s1*s2 sage: [(i, w.has_descent(i, side='left')) for i in E.cartan_type().index_set()] [(0, True), (1, False), (2, False)]
- to_dual_classical_weyl()#
Return the image of
self
under the homomorphism that projects to the dual classical Weyl group factor after rewriting it in either style “PvW0” or “W0Pv”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).PvW0().simple_reflection(0); s t[Lambda[1] + Lambda[2]] * s1*s2*s1 sage: s.to_dual_classical_weyl() s1*s2*s1
- to_dual_translation_left()#
The image of
self
under the map that projects to the dual translation lattice factor after factoring it to the left as in style “PvW0”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).PvW0().simple_reflection(0); s t[Lambda[1] + Lambda[2]] * s1*s2*s1 sage: s.to_dual_translation_left() Lambda[1] + Lambda[2]
- class ExtendedAffineWeylGroupW0P(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the finite Weyl group by the translation lattice.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).W0P() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) acting on Multiplicative form of Coweight lattice of the Root system of type ['A', 2]
- Element#
alias of
ExtendedAffineWeylGroupW0PElement
- S0()#
Return the zero-th simple reflection in style “W0P”.
EXAMPLES:
sage: ExtendedAffineWeylGroup(["A",3,1]).W0P().S0() s1*s2*s3*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[3]]
- from_classical_weyl(w)#
Return the image of the classical Weyl group element \(w\) in
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],print_tuple=True) sage: E.W0P().from_classical_weyl(E.classical_weyl().from_reduced_word([2,1])) (s2*s1, t[0])
- from_translation(la)#
Return the image of the lattice element
la
inself
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],print_tuple=True) sage: E.W0P().from_translation(E.lattice().an_element()) (1, t[2*Lambdacheck[1] + 2*Lambdacheck[2]])
- simple_reflection(i)#
Return the \(i\)-th simple reflection in
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); W0P = E.W0P() sage: [(i, W0P.simple_reflection(i)) for i in E.cartan_type().index_set()] [(0, s1*s2*s3*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[3]]), (1, s1), (2, s2), (3, s3)]
- simple_reflections()#
Return the family of simple reflections.
EXAMPLES:
sage: ExtendedAffineWeylGroup(["A",3,1]).W0P().simple_reflections() Finite family {0: s1*s2*s3*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[3]], 1: s1, 2: s2, 3: s3}
- class ExtendedAffineWeylGroupW0PElement#
Bases:
GroupSemidirectProductElement
The element class for the W0P realization.
- has_descent(i, side='right', positive=False)#
Return whether
self
has \(i\) as a descent.INPUT:
\(i\) - an index.
OPTIONAL:
side
- ‘left’ or ‘right’ (default: ‘right’)positive
- True or False (default: False)
EXAMPLES:
sage: W0P = ExtendedAffineWeylGroup(['A',4,2]).W0P() sage: w = W0P.from_reduced_word([0,1]); w s1*s2 * t[Lambda[1] - Lambda[2]] sage: w.has_descent(0, side='left') True
- to_classical_weyl()#
Project
self
into the classical Weyl group.EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',2,1]).W0P().simple_reflection(0); x s1*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[2]] sage: x.to_classical_weyl() s1*s2*s1
- to_translation_right()#
Project onto the right (translation) factor in the “W0P” style.
EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',2,1]).W0P().simple_reflection(0); x s1*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[2]] sage: x.to_translation_right() -Lambdacheck[1] - Lambdacheck[2]
- class ExtendedAffineWeylGroupW0Pv(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the finite Weyl group, acting on the dual form of the translation lattice.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).W0Pv() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) acting on Multiplicative form of Weight lattice of the Root system of type ['A', 2]
- Element#
alias of
ExtendedAffineWeylGroupW0PvElement
- from_dual_classical_weyl(w)#
Return the image of \(w\) under the homomorphism of the dual form of the classical Weyl group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1],print_tuple=True) sage: E.W0Pv().from_dual_classical_weyl(E.dual_classical_weyl().from_reduced_word([1,2])) (s1*s2, t[0])
- from_dual_translation(la)#
Map the dual translation lattice element
la
intoself
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1], translation="tau", print_tuple = True) sage: la = E.dual_lattice().an_element(); la 2*Lambda[1] + 2*Lambda[2] sage: E.W0Pv().from_dual_translation(la) (1, tau[2*Lambda[1] + 2*Lambda[2]])
- simple_reflections()#
Return a family for the simple reflections of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).W0Pv().simple_reflections() Finite family {0: s1*s2*s3*s2*s1 * t[-Lambda[1] - Lambda[3]], 1: s1, 2: s2, 3: s3}
- class ExtendedAffineWeylGroupW0PvElement#
Bases:
GroupSemidirectProductElement
The element class for the “W0Pv” realization.
- dual_action(la)#
Return the action of
self
on an elementla
of the dual version of the translation lattice.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.W0Pv().an_element(); x s1*s2 * t[2*Lambda[1] + 2*Lambda[2]] sage: la = E.dual_lattice().an_element(); la 2*Lambda[1] + 2*Lambda[2] sage: x.dual_action(la) -8*Lambda[1] + 4*Lambda[2]
- has_descent(i, side='right', positive=False)#
Return whether
self
has \(i\) as a descent.INPUT:
\(i\) - an affine Dynkin index
OPTIONAL:
side
- ‘left’ or ‘right’ (default: ‘right’)positive
- True or False (default: False)
EXAMPLES:
sage: w = ExtendedAffineWeylGroup(['A',4,2]).W0Pv().from_reduced_word([0,1]); w s1*s2 * t[Lambda[1] - Lambda[2]] sage: w.has_descent(0, side='left') True
- to_dual_classical_weyl()#
Return the image of
self
under the homomorphism that projects to the dual classical Weyl group factor after rewriting it in either style “PvW0” or “W0Pv”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).W0Pv().simple_reflection(0); s s1*s2*s1 * t[-Lambda[1] - Lambda[2]] sage: s.to_dual_classical_weyl() s1*s2*s1
- to_dual_translation_right()#
The image of
self
under the map that projects to the dual translation lattice factor after factoring it to the right as in style “W0Pv”.EXAMPLES:
sage: s = ExtendedAffineWeylGroup(['A',2,1]).W0Pv().simple_reflection(0); s s1*s2*s1 * t[-Lambda[1] - Lambda[2]] sage: s.to_dual_translation_right() -Lambda[1] - Lambda[2]
- class ExtendedAffineWeylGroupWF(E)#
Bases:
GroupSemidirectProduct
,BindableClass
Extended affine Weyl group, realized as the semidirect product of the affine Weyl group by the fundamental group.
INPUT:
\(E\) – A parent with realization in
ExtendedAffineWeylGroup_Class
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).WF() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) acted upon by Fundamental group of type ['A', 2, 1]
- Element#
alias of
ExtendedAffineWeylGroupWFElement
- from_affine_weyl(w)#
Return the image of the affine Weyl group element \(w\) in
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['C',2,1],print_tuple=True) sage: E.WF().from_affine_weyl(E.affine_weyl().from_reduced_word([1,2,1,0])) (S1*S2*S1*S0, pi[0])
- from_fundamental(f)#
Return the image of \(f\) under the homomorphism from the fundamental group into the right (fundamental group) factor in “WF” style.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['E',6,1],print_tuple=True); WF = E.WF(); F = E.fundamental_group() sage: [(x,WF.from_fundamental(x)) for x in F] [(pi[0], (1, pi[0])), (pi[1], (1, pi[1])), (pi[6], (1, pi[6]))]
- simple_reflections()#
Return the family of simple reflections.
EXAMPLES:
sage: ExtendedAffineWeylGroup(["A",3,1],affine="r").WF().simple_reflections() Finite family {0: r0, 1: r1, 2: r2, 3: r3}
- class ExtendedAffineWeylGroupWFElement#
Bases:
GroupSemidirectProductElement
Element class for the “WF” realization.
- bruhat_le(x)#
Return whether
self
is less than or equal to \(x\) in the Bruhat order.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],affine="s", print_tuple=True); WF=E.WF() sage: r = E.affine_weyl().from_reduced_word sage: v = r([1,0]) sage: w = r([1,2,0]) sage: v.bruhat_le(w) True sage: vv = WF.from_affine_weyl(v); vv (s1*s0, pi[0]) sage: ww = WF.from_affine_weyl(w); ww (s1*s2*s0, pi[0]) sage: vv.bruhat_le(ww) True sage: f = E.fundamental_group()(2); f pi[2] sage: ff = WF.from_fundamental(f); ff (1, pi[2]) sage: vv.bruhat_le(ww*ff) False sage: (vv*ff).bruhat_le(ww*ff) True
- has_descent(i, side='right', positive=False)#
Return whether
self
has \(i\) as a descent.INPUT:
\(i\) – an affine Dynkin index
OPTIONAL:
side
– ‘left’ or ‘right’ (default: ‘right’)positive
– True or False (default: False)
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.WF().an_element(); x S0*S1*S2 * pi[2] sage: [(i, x.has_descent(i)) for i in E.cartan_type().index_set()] [(0, True), (1, False), (2, False)]
- to_affine_weyl_left()#
Project
self
to the left (affine Weyl group) factor in the “WF” style.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.WF().from_translation(E.lattice_basis()[1]); x S0*S2 * pi[1] sage: x.to_affine_weyl_left() S0*S2
- to_fundamental_group()#
Project
self
to the right (fundamental group) factor in the “WF” style.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: x = E.WF().from_translation(E.lattice_basis()[1]); x S0*S2 * pi[1] sage: x.to_fundamental_group() pi[1]
- FW()#
Realizes
self
in “FW”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).FW() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Fundamental group of type ['A', 2, 1] acting on Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice)
- PW0()#
Realizes
self
in “PW0”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).PW0() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Coweight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice)
- PW0_to_WF_func(x)#
Implements coercion from style “PW0” to “WF”.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(["A", 2, 1]) sage: x = E.PW0().an_element(); x t[2*Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2 sage: E.PW0_to_WF_func(x) S0*S1*S2*S0*S1*S0
Warning
This function cannot use coercion, because it is used to define the coercion maps.
- PvW0()#
Realizes
self
in “PvW0”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).PvW0() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Weight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice)
- class Realizations(parent_with_realization)#
Bases:
Category_realization_of_parent
The category of the realizations of an extended affine Weyl group
- class ElementMethods#
Bases:
object
- action(la)#
Action of
self
on a lattice elementla
.INPUT:
self
– an element of the extended affine Weyl groupla
– an element of the translation lattice of the extended affine Weyl group, the lattice denoted by the mnemonic “P” in the documentation forExtendedAffineWeylGroup()
.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],affine="s") sage: x = E.FW().an_element(); x pi[2] * s0*s1*s2 sage: la = E.lattice().an_element(); la 2*Lambdacheck[1] + 2*Lambdacheck[2] sage: x.action(la) 5*Lambdacheck[1] - 3*Lambdacheck[2] sage: E = ExtendedAffineWeylGroup(['C',2,1],affine="s") sage: x = E.PW0().from_translation(E.lattice_basis()[1]) sage: x.action(E.lattice_basis()[2]) Lambdacheck[1] + Lambdacheck[2]
Warning
Must be implemented by style “PW0”.
- action_on_affine_roots(beta)#
Act by
self
on the affine root lattice elementbeta
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: beta = E.cartan_type().root_system().root_lattice().an_element(); beta 2*alpha[0] + 2*alpha[1] + 3*alpha[2] sage: x = E.FW().an_element(); x pi[2] * S0*S1*S2 sage: x.action_on_affine_roots(beta) alpha[0] + alpha[1]
Warning
Must be implemented by style “FW”.
- alcove_walk_signs()#
Return a signed alcove walk for
self
.INPUT:
An element
self
of the extended affine Weyl group.
OUTPUT:
A 3-tuple (\(g\),
rw
,signs
).
ALGORITHM:
The element
self
can be uniquely writtenself
= \(g\) * \(w\) where \(g\) has length zero and \(w\) is an element of the nonextended affine Weyl group. Let \(w\) have reduced wordrw
. Starting with \(g\) and applying simple reflections fromrw
, one obtains a sequence of extended affine Weyl group elements (that is, alcoves) and simple roots. The signs give the sequence of sides on which the alcoves lie, relative to the face indicated by the simple roots.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); FW=E.FW() sage: w = FW.from_reduced_word([0,2,1,3,0])*FW.from_fundamental(1); w pi[1] * S3*S1*S2*S0*S3 sage: w.alcove_walk_signs() (pi[1], [3, 1, 2, 0, 3], [-1, 1, -1, -1, 1])
- apply_simple_projection(i, side='right', length_increasing=True)#
Return the product of
self
by the simple reflection \(s_i\) if that product is of greater length thanself
and otherwise returnself
.INPUT:
self
– an element of the extended affine Weyl group\(i\) – a Dynkin node (index of a simple reflection \(s_i\))
side
– ‘right’ or ‘left’ (default: ‘right’) according to which side ofself
the reflection \(s_i\) should be multipliedlength_increasing
– True or False (default True). If False do the above with the word “greater” replaced by “less”.
EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',3,1]).WF().an_element(); x S0*S1*S2*S3 * pi[3] sage: x.apply_simple_projection(1) S0*S1*S2*S3*S0 * pi[3] sage: x.apply_simple_projection(1, length_increasing=False) S0*S1*S2*S3 * pi[3]
- apply_simple_reflection(i, side='right')#
Apply the \(i\)-th simple reflection to
self
.EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',3,1]).WF().an_element(); x S0*S1*S2*S3 * pi[3] sage: x.apply_simple_reflection(1) S0*S1*S2*S3*S0 * pi[3] sage: x.apply_simple_reflection(0, side='left') S1*S2*S3 * pi[3]
- bruhat_le(x)#
Return whether
self
<= \(x\) in Bruhat order.INPUT:
self
– an element of the extended affine Weyl group\(x\) – another element with the same parent as
self
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],print_tuple=True); WF=E.WF() sage: W = E.affine_weyl() sage: v = W.from_reduced_word([2,1,0]) sage: w = W.from_reduced_word([2,0,1,0]) sage: v.bruhat_le(w) True sage: vx = WF.from_affine_weyl(v); vx (S2*S1*S0, pi[0]) sage: wx = WF.from_affine_weyl(w); wx (S2*S0*S1*S0, pi[0]) sage: vx.bruhat_le(wx) True sage: F = E.fundamental_group() sage: f = WF.from_fundamental(F(2)) sage: vx.bruhat_le(wx*f) False sage: (vx*f).bruhat_le(wx*f) True
Warning
Must be implemented by “WF”.
- coset_representative(index_set, side='right')#
Return the minimum length representative in the coset of
self
with respect to the subgroup generated by the reflections given byindex_set
.INPUT:
self
– an element of the extended affine Weyl groupindex_set
– a subset of the set of Dynkin nodesside
– ‘right’ or ‘left’ (default: ‘right’) the side on which the subgroup acts
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); WF = E.WF() sage: b = E.lattice_basis() sage: I0 = E.cartan_type().classical().index_set() sage: [WF.from_translation(x).coset_representative(index_set=I0) for x in b] [pi[1], pi[2], pi[3]]
- dual_action(la)#
Action of
self
on a dual lattice elementla
.INPUT:
self
– an element of the extended affine Weyl groupla
– an element of the dual translation lattice of the extended affine Weyl group, the lattice denoted by the mnemonic “Pv” in the documentation forExtendedAffineWeylGroup()
.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1],affine="s") sage: x = E.FW().an_element(); x pi[2] * s0*s1*s2 sage: la = E.dual_lattice().an_element(); la 2*Lambda[1] + 2*Lambda[2] sage: x.dual_action(la) 5*Lambda[1] - 3*Lambda[2] sage: E = ExtendedAffineWeylGroup(['C',2,1],affine="s") sage: x = E.PvW0().from_dual_translation(E.dual_lattice_basis()[1]) sage: x.dual_action(E.dual_lattice_basis()[2]) Lambda[1] + Lambda[2]
Warning
Must be implemented by style “PvW0”.
- face_data(i)#
Return a description of one of the bounding hyperplanes of the alcove of an extended affine Weyl group element.
INPUT:
self
– An element of the extended affine Weyl group\(i\) – an affine Dynkin node
OUTPUT:
A 2-tuple \((m,\beta)\) defined as follows.
ALGORITHM:
Each element of the extended affine Weyl group corresponds to an alcove, and each alcove has a face for each affine Dynkin node. Given the data of
self
and \(i\), let the extended affine Weyl group elementself
act on the affine simple root \(\alpha_i\), yielding a real affine root, which can be expressed uniquely as\[``self`` \cdot \alpha_i = m \delta + \beta\]where \(m\) is an integer (the height of the \(i\)-th bounding hyperplane of the alcove of
self
) and \(\beta\) is a classical root (the normal vector for the hyperplane which points towards the alcove).EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',2,1]).PW0().an_element(); x t[2*Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2 sage: x.face_data(0) (-1, alpha[1])
- first_descent(side='right', positive=False, index_set=None)#
Return the first descent of
self
.INPUT:
side
– ‘left’ or ‘right’ (default: ‘right’)positive
– True or False (default: False)index_set
– an optional subset of Dynkin nodes
If
index_set
is not None, then the descent must be in theindex_set
.EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',3,1]).WF().an_element(); x S0*S1*S2*S3 * pi[3] sage: x.first_descent() 0 sage: x.first_descent(side='left') 0 sage: x.first_descent(positive=True) 1 sage: x.first_descent(side='left',positive=True) 1
- has_descent(i, side='right', positive=False)#
Return whether
self
* \(s_i\) <self
where \(s_i\) is the \(i\)-th simple reflection in the realized group.INPUT:
i
– an affine Dynkin index
OPTIONAL:
side
– ‘right’ or ‘left’ (default: ‘right’)positive
– True or False (default: False)
If
side``='left' then the reflection acts on the left. If ``positive
= True then the inequality is reversed.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); WF=E.WF() sage: F = E.fundamental_group() sage: x = WF.an_element(); x S0*S1*S2*S3 * pi[3] sage: I = E.cartan_type().index_set() sage: [(i, x.has_descent(i)) for i in I] [(0, True), (1, False), (2, False), (3, False)] sage: [(i, x.has_descent(i,side='left')) for i in I] [(0, True), (1, False), (2, False), (3, False)] sage: [(i, x.has_descent(i,positive=True)) for i in I] [(0, False), (1, True), (2, True), (3, True)]
Warning
This method is abstract because it is used in the recursive coercions between “PW0” and “WF” and other methods use this coercion.
- is_affine_grassmannian()#
Return whether
self
is affine Grassmannian.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]); PW0=E.PW0() sage: F = E.fundamental_group() sage: [(x,PW0.from_fundamental(x).is_affine_grassmannian()) for x in F] [(pi[0], True), (pi[1], True), (pi[2], True)] sage: b = E.lattice_basis() sage: [(-x,PW0.from_translation(-x).is_affine_grassmannian()) for x in b] [(-Lambdacheck[1], True), (-Lambdacheck[2], True)]
- is_grassmannian(index_set, side='right')#
Return whether
self
is of minimum length in its coset with respect to the subgroup generated by the reflections ofindex_set
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0=E.PW0() sage: x = PW0.from_translation(E.lattice_basis()[1]); x t[Lambdacheck[1]] sage: I = E.cartan_type().index_set() sage: [(i, x.is_grassmannian(index_set=[i])) for i in I] [(0, True), (1, False), (2, True), (3, True)] sage: [(i, x.is_grassmannian(index_set=[i], side='left')) for i in I] [(0, False), (1, True), (2, True), (3, True)]
- is_translation()#
Return whether
self
is a translation element or not.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]); FW=E.FW() sage: F = E.fundamental_group() sage: FW.from_affine_weyl(E.affine_weyl().from_reduced_word([1,2,1,0])).is_translation() True sage: FW.from_translation(E.lattice_basis()[1]).is_translation() True sage: FW.simple_reflection(0).is_translation() False
- length()#
Return the length of
self
in the Coxeter group sense.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0=E.PW0() sage: I0 = E.cartan_type().classical().index_set() sage: [PW0.from_translation(E.lattice_basis()[i]).length() for i in I0] [3, 4, 3]
- to_affine_grassmannian()#
Return the unique affine Grassmannian element in the same coset of
self
with respect to the finite Weyl group acting on the right.EXAMPLES:
sage: elts = ExtendedAffineWeylGroup(['A',2,1]).PW0().some_elements() sage: [(x, x.to_affine_grassmannian()) for x in elts] [(t[2*Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2, t[2*Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2*s1)]
- to_affine_weyl_left()#
Return the projection of
self
to the affine Weyl group on the left, after factorizing using the style “WF”.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0 = E.PW0() sage: b = E.lattice_basis() sage: [(x,PW0.from_translation(x).to_affine_weyl_left()) for x in b] [(Lambdacheck[1], S0*S3*S2), (Lambdacheck[2], S0*S3*S1*S0), (Lambdacheck[3], S0*S1*S2)]
Warning
Must be implemented in style “WF”.
- to_affine_weyl_right()#
Return the projection of
self
to the affine Weyl group on the right, after factorizing using the style “FW”.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0=E.PW0() sage: b = E.lattice_basis() sage: [(x,PW0.from_translation(x).to_affine_weyl_right()) for x in b] [(Lambdacheck[1], S3*S2*S1), (Lambdacheck[2], S2*S3*S1*S2), (Lambdacheck[3], S1*S2*S3)]
Warning
Must be implemented in style “FW”.
- to_classical_weyl()#
Return the image of
self
under the homomorphism to the classical Weyl group.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).WF().simple_reflection(0).to_classical_weyl() s1*s2*s3*s2*s1
Warning
Must be implemented in style “PW0”.
- to_dual_classical_weyl()#
Return the image of
self
under the homomorphism to the dual form of the classical Weyl group.EXAMPLES:
sage: x = ExtendedAffineWeylGroup(['A',3,1]).WF().simple_reflection(0).to_dual_classical_weyl(); x s1*s2*s3*s2*s1 sage: x.parent() Weyl Group of type ['A', 3] (as a matrix group acting on the weight lattice)
Warning
Must be implemented in style “PvW0”.
- to_dual_translation_left()#
Return the projection of
self
to the dual translation lattice after factorizing it to the left using the style “PvW0”.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PvW0().simple_reflection(0).to_dual_translation_left() Lambda[1] + Lambda[3]
Warning
Must be implemented in style “PvW0”.
- to_dual_translation_right()#
Return the projection of
self
to the dual translation lattice after factorizing it to the right using the style “W0Pv”.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PW0().simple_reflection(0).to_dual_translation_right() -Lambda[1] - Lambda[3]
Warning
Must be implemented in style “W0Pv”.
- to_fundamental_group()#
Return the image of
self
under the homomorphism to the fundamental group.EXAMPLES:
sage: PW0 = ExtendedAffineWeylGroup(['A',3,1]).PW0() sage: b = PW0.realization_of().lattice_basis() sage: [(x, PW0.from_translation(x).to_fundamental_group()) for x in b] [(Lambdacheck[1], pi[1]), (Lambdacheck[2], pi[2]), (Lambdacheck[3], pi[3])]
Warning
Must be implemented in style “WF”.
- to_translation_left()#
Return the projection of
self
to the translation lattice after factorizing it to the left using the style “PW0”.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PW0().simple_reflection(0).to_translation_left() Lambdacheck[1] + Lambdacheck[3]
Warning
Must be implemented in style “PW0”.
- to_translation_right()#
Return the projection of
self
to the translation lattice after factorizing it to the right using the style “W0P”.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PW0().simple_reflection(0).to_translation_right() -Lambdacheck[1] - Lambdacheck[3]
Warning
Must be implemented in style “W0P”.
- class ParentMethods#
Bases:
object
- from_affine_weyl(w)#
Return the image of \(w\) under the homomorphism from the affine Weyl group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0=E.PW0() sage: W = E.affine_weyl() sage: w = W.from_reduced_word([2,1,3,0]) sage: x = PW0.from_affine_weyl(w); x t[Lambdacheck[1] - 2*Lambdacheck[2] + Lambdacheck[3]] * s3*s1 sage: FW = E.FW() sage: y = FW.from_affine_weyl(w); y S2*S3*S1*S0 sage: FW(x) == y True
Warning
Must be implemented in style “WF” and “FW”.
- from_classical_weyl(w)#
Return the image of \(w\) from the finite Weyl group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PW0=E.PW0() sage: W0 = E.classical_weyl() sage: w = W0.from_reduced_word([2,1,3]) sage: y = PW0.from_classical_weyl(w); y s2*s3*s1 sage: y.parent() == PW0 True sage: y.to_classical_weyl() == w True sage: W0P = E.W0P() sage: z = W0P.from_classical_weyl(w); z s2*s3*s1 sage: z.parent() == W0P True sage: W0P(y) == z True sage: FW = E.FW() sage: x = FW.from_classical_weyl(w); x S2*S3*S1 sage: x.parent() == FW True sage: FW(y) == x True sage: FW(z) == x True
Warning
Must be implemented in style “PW0” and “W0P”.
- from_dual_classical_weyl(w)#
Return the image of \(w\) from the finite Weyl group of dual form into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]); PvW0 = E.PvW0() sage: W0v = E.dual_classical_weyl() sage: w = W0v.from_reduced_word([2,1,3]) sage: y = PvW0.from_dual_classical_weyl(w); y s2*s3*s1 sage: y.parent() == PvW0 True sage: y.to_dual_classical_weyl() == w True sage: x = E.FW().from_dual_classical_weyl(w); x S2*S3*S1 sage: PvW0(x) == y True
Warning
Must be implemented in style “PvW0” and “W0Pv”.
- from_dual_translation(la)#
Return the image of
la
under the homomorphism of the dual version of the translation lattice intoself
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]); PvW0 = E.PvW0() sage: bv = E.dual_lattice_basis(); bv Finite family {1: Lambda[1], 2: Lambda[2]} sage: x = PvW0.from_dual_translation(2*bv[1]-bv[2]); x t[2*Lambda[1] - Lambda[2]] sage: FW = E.FW() sage: y = FW.from_dual_translation(2*bv[1]-bv[2]); y S0*S2*S0*S1 sage: FW(x) == y True
- from_fundamental(x)#
Return the image of \(x\) under the homomorphism from the fundamental group into
self
.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',3,1]) sage: PW0=E.PW0() sage: F = E.fundamental_group() sage: Is = F.special_nodes() sage: [(i, PW0.from_fundamental(F(i))) for i in Is] [(0, 1), (1, t[Lambdacheck[1]] * s1*s2*s3), (2, t[Lambdacheck[2]] * s2*s3*s1*s2), (3, t[Lambdacheck[3]] * s3*s2*s1)] sage: [(i, E.W0P().from_fundamental((F(i)))) for i in Is] [(0, 1), (1, s1*s2*s3 * t[-Lambdacheck[3]]), (2, s2*s3*s1*s2 * t[-Lambdacheck[2]]), (3, s3*s2*s1 * t[-Lambdacheck[1]])] sage: [(i, E.WF().from_fundamental(F(i))) for i in Is] [(0, 1), (1, pi[1]), (2, pi[2]), (3, pi[3])]
Warning
This method must be implemented by the “WF” and “FW” realizations.
- from_reduced_word(word)#
Converts an affine or finite reduced word into a group element.
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).PW0().from_reduced_word([1,0,1,2]) t[-Lambdacheck[1] + 2*Lambdacheck[2]]
- from_translation(la)#
Return the element of translation by
la
inself
.INPUT:
self
– a realization of the extended affine Weyl groupla
– an element of the translation lattice
In the notation of the documentation for
ExtendedAffineWeylGroup()
,la
must be an element of “P”.EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]); PW0=E.PW0() sage: b = E.lattice_basis(); b Finite family {1: Lambdacheck[1], 2: Lambdacheck[2]} sage: x = PW0.from_translation(2*b[1]-b[2]); x t[2*Lambdacheck[1] - Lambdacheck[2]] sage: FW = E.FW() sage: y = FW.from_translation(2*b[1]-b[2]); y S0*S2*S0*S1 sage: FW(x) == y True
Since the implementation as a semidirect product requires wrapping the lattice group to make it multiplicative, we cannot declare that this map is a morphism for sage
Groups()
.Warning
This method must be implemented by the “PW0” and “W0P” realizations.
- simple_reflection(i)#
Return the \(i\)-th simple reflection in
self
.INPUT:
self
– a realization of the extended affine Weyl group\(i\) – An affine Dynkin node
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).PW0().simple_reflection(0) t[Lambdacheck[1] + Lambdacheck[3]] * s1*s2*s3*s2*s1 sage: ExtendedAffineWeylGroup(['C',2,1]).WF().simple_reflection(0) S0 sage: ExtendedAffineWeylGroup(['D',3,2]).PvW0().simple_reflection(1) s1
- simple_reflections()#
Return a family from the set of affine Dynkin nodes to the simple reflections in the realization of the extended affine Weyl group.
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',3,1]).W0P().simple_reflections() Finite family {0: s1*s2*s3*s2*s1 * t[-Lambdacheck[1] - Lambdacheck[3]], 1: s1, 2: s2, 3: s3} sage: ExtendedAffineWeylGroup(['A',3,1]).WF().simple_reflections() Finite family {0: S0, 1: S1, 2: S2, 3: S3} sage: ExtendedAffineWeylGroup(['A',3,1], print_tuple=True).FW().simple_reflections() Finite family {0: (pi[0], S0), 1: (pi[0], S1), 2: (pi[0], S2), 3: (pi[0], S3)} sage: ExtendedAffineWeylGroup(['A',3,1],fundamental="f",print_tuple=True).FW().simple_reflections() Finite family {0: (f[0], S0), 1: (f[0], S1), 2: (f[0], S2), 3: (f[0], S3)} sage: ExtendedAffineWeylGroup(['A',3,1]).PvW0().simple_reflections() Finite family {0: t[Lambda[1] + Lambda[3]] * s1*s2*s3*s2*s1, 1: s1, 2: s2, 3: s3}
- super_categories()#
EXAMPLES:
sage: R = ExtendedAffineWeylGroup(['A',2,1]).Realizations(); R Category of realizations of Extended affine Weyl group of type ['A', 2, 1] sage: R.super_categories() [Category of associative inverse realizations of unital magmas]
- W0P()#
Realizes
self
in “W0P”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).W0P() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) acting on Multiplicative form of Coweight lattice of the Root system of type ['A', 2]
- W0Pv()#
Realizes
self
in “W0Pv”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).W0Pv() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) acting on Multiplicative form of Weight lattice of the Root system of type ['A', 2]
- WF()#
Realizes
self
in “WF”-style.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).WF() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) acted upon by Fundamental group of type ['A', 2, 1]
- WF_to_PW0_func(x)#
Coercion from style “WF” to “PW0”.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(["A", 2, 1]) sage: x = E.WF().an_element(); x S0*S1*S2 * pi[2] sage: E.WF_to_PW0_func(x) t[Lambdacheck[1] + 2*Lambdacheck[2]] * s1*s2*s1
Warning
Since this is used to define some coercion maps it cannot itself use coercion.
- a_realization()#
Return the default realization of an extended affine Weyl group.
EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).a_realization() Extended affine Weyl group of type ['A', 2, 1] realized by Semidirect product of Multiplicative form of Coweight lattice of the Root system of type ['A', 2] acted upon by Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice)
- affine_weyl()#
Return the affine Weyl group of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).affine_weyl() Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root lattice) sage: ExtendedAffineWeylGroup(['A',5,2]).affine_weyl() Weyl Group of type ['B', 3, 1]^* (as a matrix group acting on the root lattice) sage: ExtendedAffineWeylGroup(['A',4,2]).affine_weyl() Weyl Group of type ['BC', 2, 2] (as a matrix group acting on the root lattice) sage: ExtendedAffineWeylGroup(CartanType(['A',4,2]).dual()).affine_weyl() Weyl Group of type ['BC', 2, 2]^* (as a matrix group acting on the root lattice)
- cartan_type()#
The Cartan type of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(["D",3,2]).cartan_type() ['C', 2, 1]^*
- classical_weyl()#
Return the classical Weyl group of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).classical_weyl() Weyl Group of type ['A', 2] (as a matrix group acting on the coweight lattice) sage: ExtendedAffineWeylGroup(['A',5,2]).classical_weyl() Weyl Group of type ['C', 3] (as a matrix group acting on the weight lattice) sage: ExtendedAffineWeylGroup(['A',4,2]).classical_weyl() Weyl Group of type ['C', 2] (as a matrix group acting on the weight lattice) sage: ExtendedAffineWeylGroup(CartanType(['A',4,2]).dual()).classical_weyl() Weyl Group of type ['C', 2] (as a matrix group acting on the coweight lattice)
- classical_weyl_to_affine(w)#
The image of \(w\) under the homomorphism from the classical Weyl group into the affine Weyl group.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: W0 = E.classical_weyl() sage: w = W0.from_reduced_word([1,2]); w s1*s2 sage: v = E.classical_weyl_to_affine(w); v S1*S2
- dual_classical_weyl()#
Return the dual version of the classical Weyl group of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).dual_classical_weyl() Weyl Group of type ['A', 2] (as a matrix group acting on the weight lattice) sage: ExtendedAffineWeylGroup(['A',5,2]).dual_classical_weyl() Weyl Group of type ['C', 3] (as a matrix group acting on the weight lattice)
- dual_classical_weyl_to_affine(w)#
The image of \(w\) under the homomorphism from the dual version of the classical Weyl group into the affine Weyl group.
EXAMPLES:
sage: E = ExtendedAffineWeylGroup(['A',2,1]) sage: W0v = E.dual_classical_weyl() sage: w = W0v.from_reduced_word([1,2]); w s1*s2 sage: v = E.dual_classical_weyl_to_affine(w); v S1*S2
- dual_lattice()#
Return the dual version of the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).dual_lattice() Weight lattice of the Root system of type ['A', 2] sage: ExtendedAffineWeylGroup(['A',5,2]).dual_lattice() Weight lattice of the Root system of type ['C', 3]
- dual_lattice_basis()#
Return the distinguished basis of the dual version of the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).dual_lattice_basis() Finite family {1: Lambda[1], 2: Lambda[2]} sage: ExtendedAffineWeylGroup(['A',5,2]).dual_lattice_basis() Finite family {1: Lambda[1], 2: Lambda[2], 3: Lambda[3]}
- exp_dual_lattice()#
Return the multiplicative version of the dual version of the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).exp_dual_lattice() Multiplicative form of Weight lattice of the Root system of type ['A', 2]
- exp_lattice()#
Return the multiplicative version of the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).exp_lattice() Multiplicative form of Coweight lattice of the Root system of type ['A', 2]
- fundamental_group()#
Return the abstract fundamental group.
EXAMPLES:
sage: F = ExtendedAffineWeylGroup(['D',5,1]).fundamental_group(); F Fundamental group of type ['D', 5, 1] sage: [a for a in F] [pi[0], pi[1], pi[4], pi[5]]
- group_generators()#
Return a set of generators for the default realization of
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).group_generators() (t[Lambdacheck[1]], t[Lambdacheck[2]], s1, s2)
- lattice()#
Return the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).lattice() Coweight lattice of the Root system of type ['A', 2] sage: ExtendedAffineWeylGroup(['A',5,2]).lattice() Weight lattice of the Root system of type ['C', 3] sage: ExtendedAffineWeylGroup(['A',4,2]).lattice() Weight lattice of the Root system of type ['C', 2] sage: ExtendedAffineWeylGroup(CartanType(['A',4,2]).dual()).lattice() Coweight lattice of the Root system of type ['B', 2] sage: ExtendedAffineWeylGroup(CartanType(['A',2,1]), general_linear=True).lattice() Ambient space of the Root system of type ['A', 2]
- lattice_basis()#
Return the distinguished basis of the translation lattice for
self
.EXAMPLES:
sage: ExtendedAffineWeylGroup(['A',2,1]).lattice_basis() Finite family {1: Lambdacheck[1], 2: Lambdacheck[2]} sage: ExtendedAffineWeylGroup(['A',5,2]).lattice_basis() Finite family {1: Lambda[1], 2: Lambda[2], 3: Lambda[3]} sage: ExtendedAffineWeylGroup(['A',4,2]).lattice_basis() Finite family {1: Lambda[1], 2: Lambda[2]} sage: ExtendedAffineWeylGroup(CartanType(['A',4,2]).dual()).lattice_basis() Finite family {1: Lambdacheck[1], 2: Lambdacheck[2]}