def automatic_name_eval(s, globals, max_names=10000):
"""
Exec the string ``s`` in the scope of the ``globals``
dictionary, and if any :exc:`NameError`\ s are raised, try to
fix them by defining the variable that caused the error to be
raised, then eval again. Try up to ``max_names`` times.
INPUT:
- ``s`` -- a string
- ``globals`` -- a dictionary
- ``max_names`` -- a positive integer (default: 10000)
"""
# This entire automatic naming system really boils down to
# this bit of code below. We simply try to exec the string s
# in the globals namespace, defining undefined variables and
# functions until everything is defined.
for _ in range(max_names):
try:
exec s in globals
return
except NameError, msg:
# Determine if we hit a NameError that is probably
# caused by a variable or function not being defined:
if len(msg.args) == 0: raise # not NameError with
# specific variable name
v = msg.args[0].split("'")
if len(v) < 2: raise # also not NameError with
# specific variable name We did
# find an undefined variable: we
# simply define it and try
# again.
nm = v[1]
globals[nm] = AutomaticVariable(SR, SR.var(nm))
def automatic_name_eval(s, globals, max_names=10000):
r"""
Exec the string ``s`` in the scope of the ``globals``
dictionary, and if any :exc:`NameError`\ s are raised, try to
fix them by defining the variable that caused the error to be
raised, then eval again. Try up to ``max_names`` times.
INPUT:
- ``s`` -- a string
- ``globals`` -- a dictionary
- ``max_names`` -- a positive integer (default: 10000)
"""
# This entire automatic naming system really boils down to
# this bit of code below. We simply try to exec the string s
# in the globals namespace, defining undefined variables and
# functions until everything is defined.
for _ in range(max_names):
try:
exec(s , globals)
return
except NameError as msg:
# Determine if we hit a NameError that is probably
# caused by a variable or function not being defined:
if len(msg.args) == 0: raise # not NameError with
# specific variable name
v = msg.args[0].split("'")
if len(v) < 2: raise # also not NameError with
# specific variable name We did
# find an undefined variable: we
# simply define it and try
# again.
nm = v[1]
globals[nm] = AutomaticVariable(SR, SR.var(nm))
raise NameError("Too many automatic variable names and functions created (limit=%s)" % max_names)
def plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic():
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [
0, 1, 2
]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d / x, 0), size=pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d / x, label_offset),
color=label_color,
fontsize=label_fontsize)
p += text('', (d / x, label_offset + 0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
t = SR.var('t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt + t * w[0], (t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label, (pnt[0] + b0, pnt[1] + b1),
color=label_color,
fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
s, t = SR.var('s t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt + s * w[0] + t * w[1], (s, s0, s1),
(t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label, (pnt[0] + b0, pnt[1] + b1, pnt[2] + b2),
color=label_color,
fontsize=label_fontsize)
p += label
return p
def plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic() != 0:
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [0, 1, 2]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d/x,0), size = pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d/x,label_offset),
color=label_color,fontsize=label_fontsize)
p += text('',(d/x,label_offset+0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
x, y = hyperplane.A()
d = hyperplane.b()
t = SR.var('t')
if ranges_set:
if type(ranges) in [list,tuple]:
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt+t*w[0], (t,t0,t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label,(pnt[0]+b0,pnt[1]+b1),
color=label_color,fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
a, b, c = hyperplane.A()
d = hyperplane.b()
s,t = SR.var('s t')
if ranges_set:
if type(ranges) in [list,tuple]:
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt+s*w[0]+t*w[1],(s,s0,s1),(t,t0,t1),**kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label,(pnt[0]+b0,pnt[1]+b1,pnt[2]+b2),
color=label_color,fontsize=label_fontsize)
p += label
return p
def demazure_character(self, w, f=None):
r"""
Return the Demazure character associated to ``w``.
INPUT:
- ``w`` -- an element of the ambient weight lattice
realization of the crystal, or a reduced word, or an element
in the associated Weyl group
OPTIONAL:
- ``f`` -- a function from the crystal to a module
This is currently only supported for crystals whose underlying
weight space is the ambient space.
The Demazure character is obtained by applying the Demazure operator
`D_w` (see :meth:`sage.categories.regular_crystals.RegularCrystals.ParentMethods.demazure_operator`)
to the highest weight element of the classical crystal. The simple
Demazure operators `D_i` (see
:meth:`sage.categories.regular_crystals.RegularCrystals.ElementMethods.demazure_operator_simple`)
do not braid on the level of crystals, but on the level of characters they do.
That is why it makes sense to input ``w`` either as a weight, a reduced word,
or as an element of the underlying Weyl group.
EXAMPLES::
sage: T = crystals.Tableaux(['A',2], shape = [2,1])
sage: e = T.weight_lattice_realization().basis()
sage: weight = e[0] + 2*e[2]
sage: weight.reduced_word()
[2, 1]
sage: T.demazure_character(weight)
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x1*x3^2
sage: T = crystals.Tableaux(['A',3],shape=[2,1])
sage: T.demazure_character([1,2,3])
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([1,2,3])
sage: T.demazure_character(w)
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
sage: T = crystals.Tableaux(['B',2], shape = [2])
sage: e = T.weight_lattice_realization().basis()
sage: weight = -2*e[1]
sage: T.demazure_character(weight)
x1^2 + x1*x2 + x2^2 + x1 + x2 + x1/x2 + 1/x2 + 1/x2^2 + 1
sage: T = crystals.Tableaux("B2",shape=[1/2,1/2])
sage: b2=WeylCharacterRing("B2",base_ring=QQ).ambient()
sage: T.demazure_character([1,2],f=lambda x:b2(x.weight()))
b2(-1/2,1/2) + b2(1/2,-1/2) + b2(1/2,1/2)
REFERENCES:
- [De1974]_
- [Ma2009]_
"""
from sage.misc.misc_c import prod
from sage.rings.integer_ring import ZZ
if hasattr(w, 'reduced_word'):
word = w.reduced_word()
else:
word = w
n = self.weight_lattice_realization().n
u = self.algebra(ZZ).sum_of_monomials(self.module_generators)
u = self.demazure_operator(u, word)
if f is None:
from sage.symbolic.all import SR as P
x = [P.var('x%s' % (i + 1)) for i in range(n)]
# TODO: use P.linear_combination when PolynomialRing will be a ModulesWithBasis
return sum((coeff * prod((x[i]**(c.weight()[i])
for i in range(n)), P.one())
for c, coeff in u), P.zero())
else:
return sum(coeff * f(c) for c, coeff in u)
def demazure_character(self, w, f = None):
r"""
Return the Demazure character associated to ``w``.
INPUT:
- ``w`` -- an element of the ambient weight lattice
realization of the crystal, or a reduced word, or an element
in the associated Weyl group
OPTIONAL:
- ``f`` -- a function from the crystal to a module
This is currently only supported for crystals whose underlying
weight space is the ambient space.
The Demazure character is obtained by applying the Demazure operator
`D_w` (see :meth:`sage.categories.regular_crystals.RegularCrystals.ParentMethods.demazure_operator`)
to the highest weight element of the classical crystal. The simple
Demazure operators `D_i` (see
:meth:`sage.categories.regular_crystals.RegularCrystals.ElementMethods.demazure_operator_simple`)
do not braid on the level of crystals, but on the level of characters they do.
That is why it makes sense to input ``w`` either as a weight, a reduced word,
or as an element of the underlying Weyl group.
EXAMPLES::
sage: T = crystals.Tableaux(['A',2], shape = [2,1])
sage: e = T.weight_lattice_realization().basis()
sage: weight = e[0] + 2*e[2]
sage: weight.reduced_word()
[2, 1]
sage: T.demazure_character(weight)
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x1*x3^2
sage: T = crystals.Tableaux(['A',3],shape=[2,1])
sage: T.demazure_character([1,2,3])
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([1,2,3])
sage: T.demazure_character(w)
x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
sage: T = crystals.Tableaux(['B',2], shape = [2])
sage: e = T.weight_lattice_realization().basis()
sage: weight = -2*e[1]
sage: T.demazure_character(weight)
x1^2 + x1*x2 + x2^2 + x1 + x2 + x1/x2 + 1/x2 + 1/x2^2 + 1
sage: T = crystals.Tableaux("B2",shape=[1/2,1/2])
sage: b2=WeylCharacterRing("B2",base_ring=QQ).ambient()
sage: T.demazure_character([1,2],f=lambda x:b2(x.weight()))
b2(-1/2,1/2) + b2(1/2,-1/2) + b2(1/2,1/2)
REFERENCES:
- [De1974]_
- [Ma2009]_
"""
from sage.misc.misc_c import prod
from sage.rings.integer_ring import ZZ
if hasattr(w, 'reduced_word'):
word = w.reduced_word()
else:
word = w
n = self.weight_lattice_realization().n
u = self.algebra(ZZ).sum_of_monomials(self.module_generators)
u = self.demazure_operator(u, word)
if f is None:
from sage.symbolic.all import SR as P
x = [P.var('x%s' % (i+1)) for i in range(n)]
# TODO: use P.linear_combination when PolynomialRing will be a ModulesWithBasis
return sum((coeff*prod((x[i]**(c.weight()[i]) for i in range(n)), P.one()) for c, coeff in u), P.zero())
else:
return sum(coeff * f(c) for c, coeff in u)
