def generating_series(self):
"""
Default implementation for generating series.
OUTPUT:
A series, indexed by the grading set.
EXAMPLES::
sage: N = SetsWithGrading().example(); N
Non negative integers
sage: N.generating_series()
1/(-z + 1)
"""
from sage.combinat.species.series import LazyPowerSeriesRing
from sage.rings.integer_ring import ZZ
R = LazyPowerSeriesRing(ZZ)
R(
self.graded_component(grade).cardinality()
for grade in self.grading_set())
def q_dimension(self, q=None, prec=None, use_product=False):
r"""
Return the `q`-dimension of ``self``.
Let `B(\lambda)` denote a highest weight crystal. Recall that
the degree of the `\mu`-weight space of `B(\lambda)` (under
the principal gradation) is equal to
`\langle \rho^{\vee}, \lambda - \mu \rangle` where
`\langle \rho^{\vee}, \alpha_i \rangle = 1` for all `i \in I`
(in particular, take `\rho^{\vee} = \sum_{i \in I} h_i`).
The `q`-dimension of a highest weight crystal `B(\lambda)` is
defined as
.. MATH::
\dim_q B(\lambda) := \sum_{j \geq 0} \dim(B_j) q^j,
where `B_j` denotes the degree `j` portion of `B(\lambda)`. This
can be expressed as the product
.. MATH::
\dim_q B(\lambda) = \prod_{\alpha^{\vee} \in \Delta_+^{\vee}}
\left( \frac{1 - q^{\langle \lambda + \rho, \alpha^{\vee}
\rangle}}{1 - q^{\langle \rho, \alpha^{\vee} \rangle}}
\right)^{\mathrm{mult}\, \alpha},
where `\Delta_+^{\vee}` denotes the set of positive coroots.
Taking the limit as `q \to 1` gives the dimension of `B(\lambda)`.
For more information, see [Kac]_ Section 10.10.
INPUT:
- ``q`` -- the (generic) parameter `q`
- ``prec`` -- (default: ``None``) The precision of the power
series ring to use if the crystal is not known to be finite
(i.e. the number of terms returned).
If ``None``, then the result is returned as a lazy power series.
- ``use_product`` -- (default: ``False``) if we have a finite
crystal and ``True``, use the product formula
EXAMPLES::
sage: C = crystals.Tableaux(['A',2], shape=[2,1])
sage: qdim = C.q_dimension(); qdim
q^4 + 2*q^3 + 2*q^2 + 2*q + 1
sage: qdim(1)
8
sage: len(C) == qdim(1)
True
sage: C.q_dimension(use_product=True) == qdim
True
sage: C.q_dimension(prec=20)
q^4 + 2*q^3 + 2*q^2 + 2*q + 1
sage: C.q_dimension(prec=2)
2*q + 1
sage: R.<t> = QQ[]
sage: C.q_dimension(q=t^2)
t^8 + 2*t^6 + 2*t^4 + 2*t^2 + 1
sage: C = crystals.Tableaux(['A',2], shape=[5,2])
sage: C.q_dimension()
q^10 + 2*q^9 + 4*q^8 + 5*q^7 + 6*q^6 + 6*q^5
+ 6*q^4 + 5*q^3 + 4*q^2 + 2*q + 1
sage: C = crystals.Tableaux(['B',2], shape=[2,1])
sage: qdim = C.q_dimension(); qdim
q^10 + 2*q^9 + 3*q^8 + 4*q^7 + 5*q^6 + 5*q^5
+ 5*q^4 + 4*q^3 + 3*q^2 + 2*q + 1
sage: qdim == C.q_dimension(use_product=True)
True
sage: C = crystals.Tableaux(['D',4], shape=[2,1])
sage: C.q_dimension()
q^16 + 2*q^15 + 4*q^14 + 7*q^13 + 10*q^12 + 13*q^11
+ 16*q^10 + 18*q^9 + 18*q^8 + 18*q^7 + 16*q^6 + 13*q^5
+ 10*q^4 + 7*q^3 + 4*q^2 + 2*q + 1
We check with a finite tensor product::
sage: TP = crystals.TensorProduct(C, C)
sage: TP.cardinality()
25600
sage: qdim = TP.q_dimension(use_product=True); qdim # long time
q^32 + 2*q^31 + 8*q^30 + 15*q^29 + 34*q^28 + 63*q^27 + 110*q^26
+ 175*q^25 + 276*q^24 + 389*q^23 + 550*q^22 + 725*q^21
+ 930*q^20 + 1131*q^19 + 1362*q^18 + 1548*q^17 + 1736*q^16
+ 1858*q^15 + 1947*q^14 + 1944*q^13 + 1918*q^12 + 1777*q^11
+ 1628*q^10 + 1407*q^9 + 1186*q^8 + 928*q^7 + 720*q^6
+ 498*q^5 + 342*q^4 + 201*q^3 + 117*q^2 + 48*q + 26
sage: qdim(1) # long time
25600
sage: TP.q_dimension() == qdim # long time
True
The `q`-dimensions of infinite crystals are returned
as formal power series::
sage: C = crystals.LSPaths(['A',2,1], [1,0,0])
sage: C.q_dimension(prec=5)
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + O(q^5)
sage: C.q_dimension(prec=10)
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + O(q^10)
sage: qdim = C.q_dimension(); qdim
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + 22*q^10 + O(x^11)
sage: qdim.compute_coefficients(15)
sage: qdim
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + 22*q^10 + 27*q^11
+ 36*q^12 + 44*q^13 + 57*q^14 + 70*q^15 + O(x^16)
REFERENCES:
.. [Kac] Victor G. Kac. *Infinite-dimensional Lie Algebras*.
Third edition. Cambridge University Press, Cambridge, 1990.
"""
from sage.rings.all import ZZ
WLR = self.weight_lattice_realization()
I = self.index_set()
mg = self.highest_weight_vectors()
max_deg = float('inf') if prec is None else prec - 1
def iter_by_deg(gens):
next = set(gens)
deg = -1
while next and deg < max_deg:
deg += 1
yield len(next)
todo = next
next = set([])
while todo:
x = todo.pop()
for i in I:
y = x.f(i)
if y is not None:
next.add(y)
# def iter_by_deg
from sage.categories.finite_crystals import FiniteCrystals
if self in FiniteCrystals():
if q is None:
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
q = PolynomialRing(ZZ, 'q').gen(0)
if use_product:
# Since we are in the classical case, all roots occur with multiplicity 1
pos_coroots = map(lambda x: x.associated_coroot(), WLR.positive_roots())
rho = WLR.rho()
P = q.parent()
ret = P.zero()
for v in self.highest_weight_vectors():
hw = v.weight()
ret += P.prod((1 - q**(rho+hw).scalar(ac)) / (1 - q**rho.scalar(ac))
for ac in pos_coroots)
# We do a cast since the result would otherwise live in the fraction field
return P(ret)
elif prec is None:
# If we're here, we may not be a finite crystal.
# In fact, we're probably infinite.
from sage.combinat.species.series import LazyPowerSeriesRing
if q is None:
P = LazyPowerSeriesRing(ZZ, names='q')
else:
P = q.parent()
if not isinstance(P, LazyPowerSeriesRing):
raise TypeError("the parent of q must be a lazy power series ring")
ret = P(iter_by_deg(mg))
ret.compute_coefficients(10)
return ret
from sage.rings.power_series_ring import PowerSeriesRing, PowerSeriesRing_generic
if q is None:
q = PowerSeriesRing(ZZ, 'q', default_prec=prec).gen(0)
P = q.parent()
ret = P.sum(c * q**deg for deg,c in enumerate(iter_by_deg(mg)))
if ret.degree() == max_deg and isinstance(P, PowerSeriesRing_generic):
ret = P(ret, prec)
return ret
def q_dimension(self, q=None, prec=None, use_product=False):
r"""
Return the `q`-dimension of ``self``.
Let `B(\lambda)` denote a highest weight crystal. Recall that
the degree of the `\mu`-weight space of `B(\lambda)` (under
the principal gradation) is equal to
`\langle \rho^{\vee}, \lambda - \mu \rangle` where
`\langle \rho^{\vee}, \alpha_i \rangle = 1` for all `i \in I`
(in particular, take `\rho^{\vee} = \sum_{i \in I} h_i`).
The `q`-dimension of a highest weight crystal `B(\lambda)` is
defined as
.. MATH::
\dim_q B(\lambda) := \sum_{j \geq 0} \dim(B_j) q^j,
where `B_j` denotes the degree `j` portion of `B(\lambda)`. This
can be expressed as the product
.. MATH::
\dim_q B(\lambda) = \prod_{\alpha^{\vee} \in \Delta_+^{\vee}}
\left( \frac{1 - q^{\langle \lambda + \rho, \alpha^{\vee}
\rangle}}{1 - q^{\langle \rho, \alpha^{\vee} \rangle}}
\right)^{\mathrm{mult}\, \alpha},
where `\Delta_+^{\vee}` denotes the set of positive coroots.
Taking the limit as `q \to 1` gives the dimension of `B(\lambda)`.
For more information, see [Ka1990]_ Section 10.10.
INPUT:
- ``q`` -- the (generic) parameter `q`
- ``prec`` -- (default: ``None``) The precision of the power
series ring to use if the crystal is not known to be finite
(i.e. the number of terms returned).
If ``None``, then the result is returned as a lazy power series.
- ``use_product`` -- (default: ``False``) if we have a finite
crystal and ``True``, use the product formula
EXAMPLES::
sage: C = crystals.Tableaux(['A',2], shape=[2,1])
sage: qdim = C.q_dimension(); qdim
q^4 + 2*q^3 + 2*q^2 + 2*q + 1
sage: qdim(1)
8
sage: len(C) == qdim(1)
True
sage: C.q_dimension(use_product=True) == qdim
True
sage: C.q_dimension(prec=20)
q^4 + 2*q^3 + 2*q^2 + 2*q + 1
sage: C.q_dimension(prec=2)
2*q + 1
sage: R.<t> = QQ[]
sage: C.q_dimension(q=t^2)
t^8 + 2*t^6 + 2*t^4 + 2*t^2 + 1
sage: C = crystals.Tableaux(['A',2], shape=[5,2])
sage: C.q_dimension()
q^10 + 2*q^9 + 4*q^8 + 5*q^7 + 6*q^6 + 6*q^5
+ 6*q^4 + 5*q^3 + 4*q^2 + 2*q + 1
sage: C = crystals.Tableaux(['B',2], shape=[2,1])
sage: qdim = C.q_dimension(); qdim
q^10 + 2*q^9 + 3*q^8 + 4*q^7 + 5*q^6 + 5*q^5
+ 5*q^4 + 4*q^3 + 3*q^2 + 2*q + 1
sage: qdim == C.q_dimension(use_product=True)
True
sage: C = crystals.Tableaux(['D',4], shape=[2,1])
sage: C.q_dimension()
q^16 + 2*q^15 + 4*q^14 + 7*q^13 + 10*q^12 + 13*q^11
+ 16*q^10 + 18*q^9 + 18*q^8 + 18*q^7 + 16*q^6 + 13*q^5
+ 10*q^4 + 7*q^3 + 4*q^2 + 2*q + 1
We check with a finite tensor product::
sage: TP = crystals.TensorProduct(C, C)
sage: TP.cardinality()
25600
sage: qdim = TP.q_dimension(use_product=True); qdim # long time
q^32 + 2*q^31 + 8*q^30 + 15*q^29 + 34*q^28 + 63*q^27 + 110*q^26
+ 175*q^25 + 276*q^24 + 389*q^23 + 550*q^22 + 725*q^21
+ 930*q^20 + 1131*q^19 + 1362*q^18 + 1548*q^17 + 1736*q^16
+ 1858*q^15 + 1947*q^14 + 1944*q^13 + 1918*q^12 + 1777*q^11
+ 1628*q^10 + 1407*q^9 + 1186*q^8 + 928*q^7 + 720*q^6
+ 498*q^5 + 342*q^4 + 201*q^3 + 117*q^2 + 48*q + 26
sage: qdim(1) # long time
25600
sage: TP.q_dimension() == qdim # long time
True
The `q`-dimensions of infinite crystals are returned
as formal power series::
sage: C = crystals.LSPaths(['A',2,1], [1,0,0])
sage: C.q_dimension(prec=5)
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + O(q^5)
sage: C.q_dimension(prec=10)
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + O(q^10)
sage: qdim = C.q_dimension(); qdim
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + 22*q^10 + O(x^11)
sage: qdim.compute_coefficients(15)
sage: qdim
1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 7*q^6
+ 9*q^7 + 13*q^8 + 16*q^9 + 22*q^10 + 27*q^11
+ 36*q^12 + 44*q^13 + 57*q^14 + 70*q^15 + O(x^16)
"""
from sage.rings.all import ZZ
WLR = self.weight_lattice_realization()
I = self.index_set()
mg = self.highest_weight_vectors()
max_deg = float('inf') if prec is None else prec - 1
def iter_by_deg(gens):
next = set(gens)
deg = -1
while next and deg < max_deg:
deg += 1
yield len(next)
todo = next
next = set([])
while todo:
x = todo.pop()
for i in I:
y = x.f(i)
if y is not None:
next.add(y)
# def iter_by_deg
from sage.categories.finite_crystals import FiniteCrystals
if self in FiniteCrystals():
if q is None:
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
q = PolynomialRing(ZZ, 'q').gen(0)
if use_product:
# Since we are in the classical case, all roots occur with multiplicity 1
pos_coroots = [
x.associated_coroot() for x in WLR.positive_roots()
]
rho = WLR.rho()
P = q.parent()
ret = P.zero()
for v in self.highest_weight_vectors():
hw = v.weight()
ret += P.prod((1 - q**(rho + hw).scalar(ac)) /
(1 - q**rho.scalar(ac))
for ac in pos_coroots)
# We do a cast since the result would otherwise live in the fraction field
return P(ret)
elif prec is None:
# If we're here, we may not be a finite crystal.
# In fact, we're probably infinite.
from sage.combinat.species.series import LazyPowerSeriesRing
if q is None:
P = LazyPowerSeriesRing(ZZ, names='q')
else:
P = q.parent()
if not isinstance(P, LazyPowerSeriesRing):
raise TypeError(
"the parent of q must be a lazy power series ring")
ret = P(iter_by_deg(mg))
ret.compute_coefficients(10)
return ret
from sage.rings.power_series_ring import PowerSeriesRing, PowerSeriesRing_generic
if q is None:
q = PowerSeriesRing(ZZ, 'q', default_prec=prec).gen(0)
P = q.parent()
ret = P.sum(c * q**deg for deg, c in enumerate(iter_by_deg(mg)))
if ret.degree() == max_deg and isinstance(P,
PowerSeriesRing_generic):
ret = P(ret, prec)
return ret
