def maximal_elements(self):
r"""
Return the maximal elements of ``self`` with respect to Bruhat order.
The current implementation is via a conjectural type-free
formula. Use maximal_elements_combinatorial() for proven
type-specific implementations. To compare type-free and
type-specific (combinatorial) implementations, use method
:meth:`_test_maximal_elements`.
EXAMPLES::
sage: W = WeylGroup(['A',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 4, 3, 2], [1, 0, 4, 3], [2, 1, 0, 4], [3, 2, 1, 0], [4, 3, 2, 1]]
sage: W = WeylGroup(RootSystem(["C",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [1, 2, 3, 2, 1, 0],
[2, 1, 0, 1, 2, 3], [2, 3, 2, 1, 0, 1], [3, 2, 1, 0, 1, 2]]
sage: W = WeylGroup(RootSystem(["B",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 3, 2, 0], [1, 0, 2, 3, 2], [1, 2, 3, 2, 1],
[2, 1, 0, 2, 3], [2, 3, 2, 1, 0], [3, 2, 1, 0, 2]]
sage: W = WeylGroup(['D',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 4, 3, 2, 0], [1, 0, 2, 4, 3, 2], [1, 2, 4, 3, 2, 1],
[2, 1, 0, 2, 4, 3], [2, 4, 3, 2, 1, 0], [3, 2, 1, 0, 2, 3],
[4, 2, 1, 0, 2, 4], [4, 3, 2, 1, 0, 2]]
"""
ct = self.W.cartan_type()
s = ct.translation_factors()[1]
R = RootSystem(ct).weight_space()
Lambda = R.fundamental_weights()
orbit = [
R.reduced_word_of_translation(x) for x in (
s * (Lambda[1] - Lambda[1].level() * Lambda[0]))._orbit_iter()
]
return [self.W.from_reduced_word(x) for x in orbit]
def __init__(self, cartan_type, starting_weight):
"""
EXAMPLES::
sage: C = CrystalOfLSPaths(['A',2,1],[-1,0,1]); C
The crystal of LS paths of type ['A', 2, 1] and weight (-1, 0, 1)
sage: C.R
Root system of type ['A', 2, 1]
sage: C.weight
-Lambda[0] + Lambda[2]
sage: C.weight.parent()
Extended weight space over the Rational Field of the Root system of type ['A', 2, 1]
sage: C.module_generators
[(-Lambda[0] + Lambda[2],)]
"""
self._cartan_type = CartanType(cartan_type)
self.R = RootSystem(cartan_type)
self._name = "The crystal of LS paths of type %s and weight %s"%(cartan_type,starting_weight)
if self._cartan_type.is_affine():
self.extended = True
if all(i>=0 for i in starting_weight):
Parent.__init__(self, category = HighestWeightCrystals())
else:
Parent.__init__(self, category = Crystals())
else:
self.extended = False
Parent.__init__(self, category = FiniteCrystals())
Lambda = self.R.weight_space(extended = self.extended).basis()
offset = self.R.index_set()[Integer(0)]
zero_weight = self.R.weight_space(extended = self.extended).zero()
self.weight = sum([zero_weight]+[starting_weight[j-offset]*Lambda[j] for j in self.R.index_set()])
if self.weight == zero_weight:
initial_element = self(tuple([]))
else:
initial_element = self(tuple([self.weight]))
self.module_generators = [initial_element]
def weight_in_root_lattice(self):
r"""
Return the weight of ``self`` as an element of the root lattice.
EXAMPLES::
sage: M = crystals.infinity.NakajimaMonomials(['F',4])
sage: m = M.module_generators[0].f_string([3,3,1,2,4])
sage: m.weight_in_root_lattice()
-alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]
sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
sage: mg = M.module_generators[0]
sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
sage: m.weight_in_root_lattice()
-3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
"""
Q = RootSystem(self.parent().cartan_type()).root_lattice()
alpha = Q.simple_roots()
path = self.to_highest_weight()
return Q(sum(-alpha[j] for j in path[1]))
def __classcall_private__(cls, starting_weight, cartan_type=None):
"""
Classcall to mend the input.
Internally, the
:class:`~sage.combinat.crystals.littlemann_path.CrystalOfLSPaths` code
works with a ``starting_weight`` that is in the weight space associated
to the crystal. The user can, however, also input a ``cartan_type``
and the coefficients of the fundamental weights as
``starting_weight``. This code transforms the input into the right
format (also necessary for UniqueRepresentation).
TESTS::
sage: crystals.LSPaths(['A',2,1],[-1,0,1])
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]
sage: R = RootSystem(['B',2,1])
sage: La = R.weight_space().basis()
sage: C = crystals.LSPaths(['B',2,1],[0,0,1])
sage: B = crystals.LSPaths(La[2])
sage: B is C
True
"""
if cartan_type is not None:
cartan_type, starting_weight = CartanType(
starting_weight), cartan_type
if cartan_type.is_affine():
extended = True
else:
extended = False
R = RootSystem(cartan_type)
P = R.weight_space(extended=extended)
Lambda = P.basis()
offset = R.index_set()[Integer(0)]
starting_weight = P.sum(starting_weight[j - offset] * Lambda[j]
for j in R.index_set())
return super(CrystalOfLSPaths, cls).__classcall__(cls, starting_weight)
def __classcall_private__(cls, crystals, weight):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: L = RootSystem(['A',2,1]).weight_lattice()
sage: C = crystals.KyotoPathModel(B, L.fundamental_weight(0))
sage: C2 = crystals.KyotoPathModel((B,), L.fundamental_weight(0))
sage: C3 = crystals.KyotoPathModel([B], L.fundamental_weight(0))
sage: C is C2 and C2 is C3
True
sage: L = RootSystem(['A',2,1]).weight_space()
sage: C = KyotoPathModel(B, L.fundamental_weight(0))
Traceback (most recent call last):
...
ValueError: Lambda[0] is not in the weight lattice
"""
if isinstance(crystals, list):
crystals = tuple(crystals)
elif not isinstance(crystals, tuple):
crystals = (crystals, )
if any(not B.is_perfect() for B in crystals):
raise ValueError("all crystals must be perfect")
level = crystals[0].level()
if any(B.level() != level for B in crystals[1:]):
raise ValueError("all crystals must have the same level")
ct = crystals[0].cartan_type()
P = RootSystem(ct).weight_lattice()
if weight.parent() is not P:
raise ValueError("{} is not in the weight lattice".format(weight))
if sum(ct.dual().c()[i] * weight.scalar(h)
for i, h in enumerate(P.simple_coroots())) != level:
raise ValueError("{} is not a level {} weight".format(
weight, level))
return super(KyotoPathModel, cls).__classcall__(cls, crystals, weight)
def __init__(self, cartan_type, highest_weight):
"""
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
sage: C.list()
[[], [0], [0, 1], [0, 1, 2]]
sage: TestSuite(C).run()
"""
Parent.__init__(self, category=ClassicalCrystals())
self._cartan_type = CartanType(cartan_type)
self._name = "The crystal of alcove paths for type %s" % cartan_type
self.chain_cache = {}
self.endweight_cache = {}
self.R = RootSystem(cartan_type)
alpha = self.R.root_space().simple_roots()
Lambda = self.R.weight_space().basis()
self.positive_roots = sorted(self.R.root_space().positive_roots())
self.weight = Lambda[Integer(1)] - Lambda[Integer(1)]
offset = self.R.index_set()[Integer(0)]
for j in self.R.index_set():
self.weight = self.weight + highest_weight[j - offset] * Lambda[j]
self.initial_element = self([])
self.initial_element.chain = self.get_initial_chain(self.weight)
rho = (Integer(1) / Integer(2)) * sum(self.positive_roots)
self.initial_element.endweight = rho
self.chain_cache[str([])] = self.initial_element.chain
self.endweight_cache[str([])] = self.initial_element.endweight
self.module_generators = [self.initial_element]
self._list = super(ClassicalCrystalOfAlcovePaths, self).list()
self._digraph = super(ClassicalCrystalOfAlcovePaths, self).digraph()
self._digraph_closure = self.digraph().transitive_closure()
def _reflections(self):
"""
A dictionary of reflections to a pair of the associated root
and coroot.
EXAMPLES::
sage: R = algebras.RationalCherednik(['B',2], [1,2], 1, QQ)
sage: [R._reflections[k] for k in sorted(R._reflections, key=str)]
[(alpha[1], alphacheck[1], 1),
(alpha[1] + alpha[2], 2*alphacheck[1] + alphacheck[2], 2),
(alpha[2], alphacheck[2], 2),
(alpha[1] + 2*alpha[2], alphacheck[1] + alphacheck[2], 1)]
"""
d = {}
for r in RootSystem(self._cartan_type).root_lattice().positive_roots():
s = self._weyl.from_reduced_word(r.associated_reflection())
if r.is_short_root():
c = self._c[1]
else:
c = self._c[0]
d[s] = (r, r.associated_coroot(), c)
return d
def __classcall_private__(cls, cartan_type, La):
r"""
Normalize input to ensure a unique representation.
INPUT:
- ``ct`` -- Cartan type
- ``La`` -- an element of ``weight_lattice``
EXAMPLES::
sage: La = RootSystem(['E',8,1]).weight_lattice().fundamental_weights()
sage: M = CrystalOfNakajimaMonomials(['E',8,1],La[0]+La[8])
sage: M1 = CrystalOfNakajimaMonomials(CartanType(['E',8,1]),La[0]+La[8])
sage: M2 = CrystalOfNakajimaMonomials(['E',8,1],M.Lambda()[0] + M.Lambda()[8])
sage: M is M1 is M2
True
"""
cartan_type = CartanType(cartan_type)
La = RootSystem(cartan_type).weight_lattice()(La)
return super(CrystalOfNakajimaMonomials,
cls).__classcall__(cls, cartan_type, La)
def __init__(self,
abstract_polynomial_ring,
group_code,
group_type,
basis_name,
basis_repr=None,
extra_category=None):
r"""
TESTS::
sage: # Fix a nice test
"""
if (basis_repr is None):
basis_repr = abstract_polynomial_ring._main_repr_var
self._root_system = RootSystem(group_code)
self._group_type = group_type
self._basis_keys = MonomialKeyWrapper(self._root_system)
FiniteRankPolynomialRingWithBasis.__init__(
self,
abstract_polynomial_ring,
abstract_polynomial_ring.polynomial_ring_tower(
).monomial_basis_with_type(group_type),
self._basis_keys,
basis_name,
basis_repr,
extra_category=extra_category)
monomial_basis = self.abstract_algebra().monomial_basis()
self._to_monomial_morphism = self._module_morphism(
self._to_monomial_on_basis, codomain=monomial_basis)
self._from_monomial_morphism = monomial_basis._module_morphism(
self._from_monomial_on_basis, codomain=self)
#temp#
self._to_monomial_morphism.register_as_coercion()
self._from_monomial_morphism.register_as_coercion()
def __init__(self, cartan_type, prefix, finite=True):
r"""
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1])
sage: F in Groups().Commutative().Finite()
True
sage: TestSuite(F).run()
"""
def leading_support(beta):
r"""
Given a dictionary with one key, return this key
"""
supp = beta.support()
assert len(supp) == 1
return supp[0]
self._cartan_type = cartan_type
self._prefix = prefix
special_node = cartan_type.special_node()
self._special_nodes = cartan_type.special_nodes()
# initialize dictionaries with the entries for the
# distinguished special node
# dictionary of inverse elements
inverse_dict = {}
inverse_dict[special_node] = special_node
# dictionary for the action of special automorphisms by
# permutations of the affine Dynkin nodes
auto_dict = {}
for i in cartan_type.index_set():
auto_dict[special_node, i] = i
# dictionary for the finite Weyl component of the special automorphisms
reduced_words_dict = {}
reduced_words_dict[0] = tuple([])
if cartan_type.dual().is_untwisted_affine():
# this combines the computations for an untwisted affine
# type and its affine dual
cartan_type = cartan_type.dual()
if cartan_type.is_untwisted_affine():
cartan_type_classical = cartan_type.classical()
I = [i for i in cartan_type_classical.index_set()]
Q = RootSystem(cartan_type_classical).root_lattice()
alpha = Q.simple_roots()
omega = RootSystem(
cartan_type_classical).weight_lattice().fundamental_weights()
W = Q.weyl_group(prefix="s")
for i in self._special_nodes:
if i == special_node:
continue
antidominant_weight, reduced_word = omega[
i].to_dominant_chamber(reduced_word=True, positive=False)
reduced_words_dict[i] = tuple(reduced_word)
w0i = W.from_reduced_word(reduced_word)
idual = leading_support(-antidominant_weight)
inverse_dict[i] = idual
auto_dict[i, special_node] = i
for j in I:
if j == idual:
auto_dict[i, j] = special_node
else:
auto_dict[i, j] = leading_support(w0i.action(alpha[j]))
self._action = Family(
self._special_nodes, lambda i: Family(cartan_type.index_set(),
lambda j: auto_dict[i, j]))
self._dual_node = Family(self._special_nodes, inverse_dict.__getitem__)
self._reduced_words = Family(self._special_nodes,
reduced_words_dict.__getitem__)
if finite:
cat = Category.join(
(Groups().Commutative().Finite(), EnumeratedSets()))
else:
cat = Groups().Commutative().Infinite()
Parent.__init__(self, category=cat)
def energy_function(self):
r"""
Return the energy function of ``self``.
The energy function `D(\pi)` of the level zero LS path `\pi \in \mathbb{B}_\mathrm{cl}(\lambda)`
requires a series of definitions; for simplicity the root system is assumed to be untwisted affine.
The LS path `\pi` is a piecewise linear map from the unit interval `[0,1]` to the weight lattice.
It is specified by "times" `0=\sigma_0<\sigma_1<\dotsm<\sigma_s=1` and "direction vectors"
`x_u \lambda` where `x_u \in W/W_J` for `1\le u\le s`, and `W_J` is the
stabilizer of `\lambda` in the finite Weyl group `W`. Precisely,
.. MATH::
\pi(t)=\sum_{u'=1}^{u-1} (\sigma_{u'}-\sigma_{u'-1})x_{u'}\lambda+(t-\sigma_{u-1})x_{u}\lambda
for `1\le u\le s` and `\sigma_{u-1} \le t \le \sigma_{u}`.
For any `x,y\in W/W_J` let
.. MATH::
d: x= w_{0} \stackrel{\beta_{1}}{\leftarrow}
w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=y
be a shortest directed path in the parabolic quantum Bruhat graph. Define
.. MATH::
\mathrm{wt}(d):=\sum_{\substack{1\le k\le n \\ \ell(w_{k-1})<\ell(w_k)}}
\beta_{k}^{\vee}
It can be shown that `\mathrm{wt}(d)` depends only on `x,y`;
call its value `\mathrm{wt}(x,y)`. The energy function `D(\pi)` is defined by
.. MATH::
D(\pi)=-\sum_{u=1}^{s-1} (1-\sigma_{u}) \langle \lambda,\mathrm{wt}(x_u,x_{u+1}) \rangle
For more information, see [LNSSS2013]_.
REFERENCES:
.. [LNSSS2013] C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono,
A uniform model for Kirillov-Reshetikhin crystals. Extended abstract.
DMTCS proc, to appear ( {{{:arXiv:`1211.6019`}}} )
.. NOTE::
In the dual-of-untwisted case the parabolic quantum Bruhat graph that is used is obtained by
exchanging the roles of roots and coroots. Moreover, in the computation of the
pairing the short roots must be doubled (or tripled for type `G`). This factor
is determined by the translation factor of the corresponding root.
Type `BC` is viewed as untwisted type, whereas the dual of `BC` is viewed as twisted.
Except for the untwisted cases, these formulas are currently still conjectural.
EXAMPLES::
sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[1]+La[3])
sage: b = LS.module_generators[0]
sage: c = b.f(1).f(3).f(2)
sage: c.energy_function()
0
sage: c=b.e(0)
sage: c.energy_function()
1
sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1])
sage: b = LS.module_generators[0]
sage: c = b.e(0)
sage: c.energy_function()
1
sage: [c.energy_function() for c in sorted(LS.list())]
[0, 1, 0, 0, 0, 1, 0, 1, 0]
The next test checks that the energy function is constant on classically connected components::
sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=[1,2])
sage: C = G.connected_components()
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True]
sage: R = RootSystem(['D',4,2])
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[2])
sage: J = R.cartan_type().classical().index_set()
sage: hw = [x for x in LS if x.is_highest_weight(J)]
sage: [(x.weight(), x.energy_function()) for x in hw]
[(-2*Lambda[0] + Lambda[2], 0), (-2*Lambda[0] + Lambda[1], 1), (0, 2)]
sage: G = LS.digraph(index_set=J)
sage: C = G.connected_components()
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True]
sage: R = RootSystem(CartanType(['G',2,1]).dual())
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[1]+La[2])
sage: G = LS.digraph(index_set=[1,2])
sage: C = G.connected_components()
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: ct = CartanType(['BC',2,2]).dual()
sage: R = RootSystem(ct)
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
sage: C = G.connected_components()
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True, True, True, True, True, True, True, True]
sage: R = RootSystem(['BC',2,2])
sage: La = R.weight_space().basis()
sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1]+La[2])
sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
sage: C = G.connected_components()
sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
"""
weight = self.parent().weight
P = weight.parent()
c_weight = P.classical()(weight)
ct = P.cartan_type()
cartan = ct.classical()
Qv = RootSystem(cartan).coroot_lattice()
W = WeylGroup(cartan,prefix='s')
J = tuple(weight.weyl_stabilizer())
L = self.weyl_group_representation()
if ct.is_untwisted_affine() or ct.type() == 'BC':
untwisted = True
G = W.quantum_bruhat_graph(J)
else:
untwisted = False
cartan_dual = cartan.dual()
Wd = WeylGroup(cartan_dual, prefix='s')
G = Wd.quantum_bruhat_graph(J)
Qd = RootSystem(cartan_dual).root_lattice()
dualize = lambda x: Qv.from_vector(x.to_vector())
L = [Wd.from_reduced_word(x.reduced_word()) for x in L]
def stretch_short_root(a):
# stretches roots by translation factor
if ct.dual().type() == 'BC':
return ct.c()[a.to_simple_root()]*a
return ct.dual().c()[a.to_simple_root()]*a
#if a.is_short_root():
# if cartan_dual.type() == 'G':
# return 3*a
# else:
# return 2*a
#return a
paths = [G.shortest_path(L[i+1],L[i]) for i in range(len(L)-1)]
paths_labels = [[G.edge_label(p[i],p[i+1]) for i in range(len(p)-1) if p[i].length()+1 != p[i+1].length()] for p in paths]
scalars = self.scalar_factors()
if untwisted:
s = sum((1-scalars[i])*c_weight.scalar( Qv.sum(root.associated_coroot()
for root in paths_labels[i]) ) for i in range(len(paths_labels)))
if ct.type() == 'BC':
return 2*s
else:
return s
else:
s = sum((1-scalars[i])*c_weight.scalar( dualize (Qd.sum(stretch_short_root(root) for root in paths_labels[i])) ) for i in range(len(paths_labels)))
if ct.dual().type() == 'BC':
return s/2
else:
return s
def Shi(self, data, K=QQ, names=None, m=1):
r"""
Return the Shi arrangement.
INPUT:
- ``data`` -- either an integer or a Cartan type (or coercible
into; see "CartanType")
- ``K`` -- field (default:``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
- ``m`` -- integer (default: 1)
OUTPUT:
- If ``data`` is an integer `n`, return the Shi arrangement in
dimension `n`, i.e. the set of `n(n-1)` hyperplanes:
`\{ x_i - x_j = 0,1 : 1 \leq i \leq j \leq n \}`. This corresponds
to the Shi arrangement of Cartan type `A_{n-1}`.
- If ``data`` is a Cartan type, return the Shi arrangement of given
type.
- If `m > 1`, return the `m`-extended Shi arrangement of given type.
The `m`-extended Shi arrangement of a given crystallographic
Cartan type is defined by the inner product
`\langle a,x \rangle = k` for `-m < k \leq m` and
`a \in \Phi^+` is a positive root of the root system `\Phi`.
EXAMPLES::
sage: hyperplane_arrangements.Shi(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3")
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3",m=2)
Arrangement of 24 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("B4")
Arrangement of 32 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("B4",m=3)
Arrangement of 96 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("C3")
Arrangement of 18 hyperplanes of dimension 3 and rank 3
sage: hyperplane_arrangements.Shi("D4",m=3)
Arrangement of 72 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("E6")
Arrangement of 72 hyperplanes of dimension 8 and rank 6
sage: hyperplane_arrangements.Shi("E6",m=2)
Arrangement of 144 hyperplanes of dimension 8 and rank 6
If the Cartan type is not crystallographic, the Shi arrangement
is not defined::
sage: hyperplane_arrangements.Shi("H4")
Traceback (most recent call last):
...
NotImplementedError: Shi arrangements are not defined for non crystallographic Cartan types
The characteristic polynomial is pre-computed using the results
of [Ath1996]_::
sage: hyperplane_arrangements.Shi("A3").characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: hyperplane_arrangements.Shi("A3",m=2).characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: hyperplane_arrangements.Shi("C3").characteristic_polynomial()
x^3 - 18*x^2 + 108*x - 216
sage: hyperplane_arrangements.Shi("E6").characteristic_polynomial()
x^8 - 72*x^7 + 2160*x^6 - 34560*x^5 + 311040*x^4 - 1492992*x^3 + 2985984*x^2
sage: hyperplane_arrangements.Shi("B4",m=3).characteristic_polynomial()
x^4 - 96*x^3 + 3456*x^2 - 55296*x + 331776
TESTS::
sage: h = hyperplane_arrangements.Shi(4)
sage: h.characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h = hyperplane_arrangements.Shi("A3",m=2)
sage: h.characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: h.characteristic_polynomial.clear_cache()
sage: h.characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: h = hyperplane_arrangements.Shi("B3",m=3)
sage: h.characteristic_polynomial()
x^3 - 54*x^2 + 972*x - 5832
sage: h.characteristic_polynomial.clear_cache()
sage: h.characteristic_polynomial()
x^3 - 54*x^2 + 972*x - 5832
"""
if data in NN:
cartan_type = CartanType(["A", data - 1])
else:
cartan_type = CartanType(data)
if not cartan_type.is_crystallographic():
raise NotImplementedError(
"Shi arrangements are not defined for non crystallographic Cartan types"
)
n = cartan_type.rank()
h = cartan_type.coxeter_number()
Ra = RootSystem(cartan_type).ambient_space()
PR = Ra.positive_roots()
d = Ra.dimension()
H = make_parent(K, d, names)
x = H.gens()
hyperplanes = []
for a in PR:
for const in range(-m + 1, m + 1):
hyperplanes.append(sum(a[j] * x[j] for j in range(d)) - const)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x**(d - n) * (x - m * h)**n
A.characteristic_polynomial.set_cache(charpoly)
return A
def product_on_basis(self, left, right):
r"""
Return ``left`` multiplied by ``right`` in ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: a2 = R.algebra_generators()['a2']
sage: ac1 = R.algebra_generators()['ac1']
sage: a2 * ac1 # indirect doctest
a2*ac1
sage: ac1 * a2
-I + a2*ac1 - s1 - s2 + 1/2*s1*s2*s1
sage: x = R.an_element()
sage: [y * x for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
3*a1*ac1 + 2*a1*s1 + a1^2,
3*a2*ac1 + 2*a2*s1 + a1*a2,
3*s1*ac1 + 2*I - a1*s1,
3*s2*ac1 + 2*s2*s1 + a1*s2 + a2*s2,
3*ac1^2 - 2*s1*ac1 + 2*I + a1*ac1 + 2*s1 + 1/2*s2 + 1/2*s1*s2*s1,
3*ac1*ac2 + 2*s1*ac1 + 2*s1*ac2 - I + a1*ac2 - s1 - s2 + 1/2*s1*s2*s1]
sage: [x * y for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
6*I + 3*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 - 2*a1*s1 + a1^2,
-3*I + 3*a2*ac1 - 3*s1 - 3*s2 + 3/2*s1*s2*s1 + 2*a1*s1 + 2*a2*s1 + a1*a2,
-3*s1*ac1 + 2*I + a1*s1,
3*s2*ac1 + 3*s2*ac2 + 2*s1*s2 + a1*s2,
3*ac1^2 + 2*s1*ac1 + a1*ac1,
3*ac1*ac2 + 2*s1*ac2 + a1*ac2]
"""
# Make copies of the internal dictionaries
dl = dict(left[2]._monomial)
dr = dict(right[0]._monomial)
# If there is nothing to commute
if not dl and not dr:
return self.monomial((left[0], left[1] * right[1], right[2]))
R = self.base_ring()
I = self._cartan_type.index_set()
P = PolynomialRing(R, 'x', len(I))
G = P.gens()
gens_dict = {a: G[i] for i, a in enumerate(I)}
Q = RootSystem(self._cartan_type).root_lattice()
alpha = Q.simple_roots()
alphacheck = Q.simple_coroots()
def commute_w_hd(w, al): # al is given as a dictionary
ret = P.one()
for k in al:
x = sum(c * gens_dict[i] for i, c in alpha[k].weyl_action(w))
ret *= x**al[k]
ret = ret.dict()
for k in ret:
yield (self._hd({I[i]: e
for i, e in enumerate(k) if e != 0}), ret[k])
# Do Lac Ra if they are both non-trivial
if dl and dr:
il = dl.keys()[0]
ir = dr.keys()[0]
# Compute the commutator
terms = self._product_coroot_root(il, ir)
# remove the generator from the elements
dl[il] -= 1
if dl[il] == 0:
del dl[il]
dr[ir] -= 1
if dr[ir] == 0:
del dr[ir]
# We now commute right roots past the left reflections: s Ra = Ra' s
cur = self._from_dict({(hd, s * right[1], right[2]): c * cc
for s, c in terms
for hd, cc in commute_w_hd(s, dr)})
cur = self.monomial((left[0], left[1], self._h(dl))) * cur
# Add back in the commuted h and hd elements
rem = self.monomial((left[0], left[1], self._h(dl)))
rem = rem * self.monomial(
(self._hd({ir: 1}), self._weyl.one(), self._h({il: 1})))
rem = rem * self.monomial((self._hd(dr), right[1], right[2]))
return cur + rem
if dl:
# We have La Ls Lac Rs Rac,
# so we must commute Lac Rs = Rs Lac'
# and obtain La (Ls Rs) (Lac' Rac)
ret = P.one()
for k in dl:
x = sum(c * gens_dict[i] for i, c in alphacheck[k].weyl_action(
right[1].reduced_word(), inverse=True))
ret *= x**dl[k]
ret = ret.dict()
w = left[1] * right[1]
return self._from_dict({
(left[0], w,
self._h({I[i]: e
for i, e in enumerate(k) if e != 0}) * right[2]):
ret[k]
for k in ret
})
# Otherwise dr is non-trivial and we have La Ls Ra Rs Rac,
# so we must commute Ls Ra = Ra' Ls
w = left[1] * right[1]
return self._from_dict({(left[0] * hd, w, right[2]): c
for hd, c in commute_w_hd(left[1], dr)})
def one_dimensional_configuration_sum(self, q=None, group_components=True):
r"""
Compute the one-dimensional configuration sum.
INPUT:
- ``q`` -- (default: ``None``) a variable or ``None``; if ``None``,
a variable ``q`` is set in the code
- ``group_components`` -- (default: ``True``) boolean; if ``True``,
then the terms are grouped by classical component
The one-dimensional configuration sum is the sum of the weights of all elements in the crystal
weighted by the energy function. For untwisted types it uses the parabolic quantum Bruhat graph, see [LNSSS2013]_.
In the dual-of-untwisted case, the parabolic quantum Bruhat graph is defined by
exchanging the roles of roots and coroots (which is still conjectural at this point).
EXAMPLES::
sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
sage: LS.one_dimensional_configuration_sum() # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]]
+ (q+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]] + B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]
sage: R.<t> = ZZ[]
sage: LS.one_dimensional_configuration_sum(t, False) # long time
B[-2*Lambda[1] + 2*Lambda[2]] + (t+1)*B[-Lambda[1]] + (t+1)*B[Lambda[1] - Lambda[2]]
+ B[2*Lambda[1]] + B[-2*Lambda[2]] + (t+1)*B[Lambda[2]]
TESTS::
sage: R = RootSystem(['B',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
sage: LS.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum(group_components=False) # long time
True
sage: K1 = crystals.KirillovReshetikhin(['B',3,1],1,1)
sage: K2 = crystals.KirillovReshetikhin(['B',3,1],2,1)
sage: T = crystals.TensorProduct(K2,K1)
sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
True
sage: R = RootSystem(['D',4,2])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
sage: K1 = crystals.KirillovReshetikhin(['D',4,2],1,1)
sage: K2 = crystals.KirillovReshetikhin(['D',4,2],2,1)
sage: T = crystals.TensorProduct(K2,K1)
sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
True
sage: R = RootSystem(['A',5,2])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(3*La[1])
sage: K1 = crystals.KirillovReshetikhin(['A',5,2],1,1)
sage: T = crystals.TensorProduct(K1,K1,K1)
sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
True
"""
if q is None:
from sage.rings.all import QQ
q = QQ['q'].gens()[0]
#P0 = self.weight_lattice_realization().classical()
P0 = RootSystem(self.cartan_type().classical()).weight_lattice()
B = P0.algebra(q.parent())
def weight(x):
w = x.weight()
return P0.sum(
int(c) * P0.basis()[i] for i, c in w if i in P0.index_set())
if group_components:
G = self.digraph(
index_set=self.cartan_type().classical().index_set())
C = G.connected_components()
return sum(q**(c[0].energy_function()) *
B.sum(B(weight(b)) for b in c) for c in C)
return B.sum(q**(b.energy_function()) * B(weight(b)) for b in self)
def __init__(self, starting_weight, starting_weight_parent):
"""
EXAMPLES::
sage: C = crystals.LSPaths(['A',2,1],[-1,0,1]); C
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]
sage: C.R
Root system of type ['A', 2, 1]
sage: C.weight
-Lambda[0] + Lambda[2]
sage: C.weight.parent()
Extended weight space over the Rational Field of the Root system of type ['A', 2, 1]
sage: C.module_generators
[(-Lambda[0] + Lambda[2],)]
TESTS::
sage: C = crystals.LSPaths(['A',2,1], [-1,0,1])
sage: TestSuite(C).run() # long time
sage: C = crystals.LSPaths(['E',6], [1,0,0,0,0,0])
sage: TestSuite(C).run()
sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LaE = R.weight_space(extended=True).basis()
sage: B = crystals.LSPaths(La[0])
sage: BE = crystals.LSPaths(LaE[0])
sage: B is BE
False
sage: B.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type ['C', 3, 1]
sage: BE.weight_lattice_realization()
Extended weight space over the Rational Field of the Root system of type ['C', 3, 1]
"""
cartan_type = starting_weight.parent().cartan_type()
self.R = RootSystem(cartan_type)
self.weight = starting_weight
if not self.weight.parent().base_ring().has_coerce_map_from(QQ):
raise ValueError(
"Please use the weight space, rather than weight lattice for your weights"
)
self._cartan_type = cartan_type
self._name = "The crystal of LS paths of type %s and weight %s" % (
cartan_type, starting_weight)
if cartan_type.is_affine():
if all(i >= 0 for i in starting_weight.coefficients()):
Parent.__init__(self,
category=(RegularCrystals(),
HighestWeightCrystals(),
InfiniteEnumeratedSets()))
elif starting_weight.parent().is_extended():
Parent.__init__(self,
category=(RegularCrystals(),
InfiniteEnumeratedSets()))
else:
Parent.__init__(self,
category=(RegularCrystals(), FiniteCrystals()))
else:
Parent.__init__(self, category=ClassicalCrystals())
if starting_weight == starting_weight.parent().zero():
initial_element = self(tuple([]))
else:
initial_element = self(tuple([starting_weight]))
self.module_generators = [initial_element]