def weyl_group_representation(self):
r"""
Transforms the weights in the LS path ``self`` to elements in the Weyl group.
Each LS path can be written as the piecewise linear map:
.. MATH::
\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}
for `0<\sigma_1<\sigma_2<\cdots<\sigma_s=1` and `\sigma_{u-1} \le t \le \sigma_{u}` and `1 \le u \le s`.
Each weight `\nu_u` is also associated to a Weyl group element. This method returns the list
of Weyl group elements associated to the `\nu_u` for `1\le u\le s`.
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.weyl_group_representation()
[s2*s3*s1, s3*s1]
"""
cartan = self.parent().weight.parent().cartan_type().classical()
I = cartan.index_set()
W = WeylGroup(cartan,prefix='s')
return [W.from_reduced_word(x.to_dominant_chamber(index_set=I, reduced_word=True)[1]) for x in self.value]
def __classcall_private__(cls, w, n, x=None, k=None):
r"""
Classcall to mend the input.
TESTS::
sage: A = crystals.AffineFactorization([[3,1],[1]], 4, k=3); A
Crystal on affine factorizations of type A3 associated to s3*s2*s1
sage: AC = crystals.AffineFactorization([Core([4,1],4),Core([1],4)], 4, k=3)
sage: AC is A
True
"""
if k is not None:
from sage.combinat.core import Core
from sage.combinat.partition import Partition
W = WeylGroup(['A', k, 1], prefix='s')
if isinstance(w[0], Core):
w = [w[0].to_bounded_partition(), w[1].to_bounded_partition()]
else:
w = [Partition(w[0]), Partition(w[1])]
w0 = W.from_reduced_word(w[0].from_kbounded_to_reduced_word(k))
w1 = W.from_reduced_word(w[1].from_kbounded_to_reduced_word(k))
w = w0 * (w1.inverse())
return super(AffineFactorizationCrystal,
cls).__classcall__(cls, w, n, x)
def weyl_group_representation(self):
r"""
Transforms the weights in the LS path ``self`` to elements in the Weyl group.
Each LS path can be written as the piecewise linear map:
.. MATH::
\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}
for `0<\sigma_1<\sigma_2<\cdots<\sigma_s=1` and `\sigma_{u-1} \le t \le \sigma_{u}` and `1 \le u \le s`.
Each weight `\nu_u` is also associated to a Weyl group element. This method returns the list
of Weyl group elements associated to the `\nu_u` for `1\le u\le s`.
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.weyl_group_representation()
[s2*s3*s1, s3*s1]
"""
weight = self.parent().weight
level = -weight.coefficient(0)
La = weight.parent().fundamental_weights()
cartan = weight.parent().cartan_type().classical()
W = WeylGroup(cartan, prefix="s")
scalars = [0] + self.scalar_factors()
l = [
(self.value[i] / (scalars[i + 1] - scalars[i]) + level * La[0]).to_dominant_chamber(reduced_word=True)
for i in range(len(scalars) - 1)
]
return [W.from_reduced_word(a[1]) for a in l]
def __classcall_private__(cls, w, n, x = None, k = None):
r"""
Classcall to mend the input.
TESTS::
sage: A = crystals.AffineFactorization([[3,1],[1]], 4, k=3); A
Crystal on affine factorizations of type A3 associated to s3*s2*s1
sage: AC = crystals.AffineFactorization([Core([4,1],4),Core([1],4)], 4, k=3)
sage: AC is A
True
"""
if k is not None:
from sage.combinat.core import Core
from sage.combinat.partition import Partition
W = WeylGroup(['A',k,1], prefix='s')
if isinstance(w[0], Core):
w = [w[0].to_bounded_partition(), w[1].to_bounded_partition()]
else:
w = [Partition(w[0]), Partition(w[1])]
w0 = W.from_reduced_word(w[0].from_kbounded_to_reduced_word(k))
w1 = W.from_reduced_word(w[1].from_kbounded_to_reduced_word(k))
w = w0*(w1.inverse())
return super(AffineFactorizationCrystal, cls).__classcall__(cls, w, n, x)
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