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)