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 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)
