def _acted_upon_(self, scalar, self_on_left=False):
"""
Return the action of ``scalar`` on ``self``.
EXAMPLES::
sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: C = S.cellular_basis()
sage: W = S.cell_module([2,1])
sage: elt = W.an_element(); elt
2*W[[1, 2], [3]] + 2*W[[1, 3], [2]]
sage: sc = C.an_element(); sc
3*C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
+ 2*C([3], [[1, 2, 3]], [[1, 2, 3]])
sage: sc * elt
10*W[[1, 3], [2]]
sage: s = S.an_element(); s
[1, 2, 3] + 2*[1, 3, 2] + 3*[2, 1, 3] + [3, 1, 2]
sage: s * elt
11*W[[1, 2], [3]] - 3/2*W[[1, 3], [2]]
sage: 1/2 * elt
W[[1, 2], [3]] + W[[1, 3], [2]]
"""
# Check for elements coercable to the base ring first
ret = CombinatorialFreeModule.Element._acted_upon_(
self, scalar, self_on_left)
if ret is not None:
return ret
# See message in CombinatorialFreeModuleElement._acted_upon_
P = self.parent()
if isinstance(scalar,
Element) and scalar.parent() is not P._algebra:
# Temporary needed by coercion (see Polynomial/FractionField tests).
if not P._algebra.has_coerce_map_from(scalar.parent()):
return None
scalar = P._algebra(scalar)
if self_on_left:
raise NotImplementedError
#scalar = scalar.cellular_involution()
mc = self._monomial_coefficients
scalar_mc = scalar.monomial_coefficients(copy=False)
D = linear_combination([(P._action_basis(
x, k)._monomial_coefficients, scalar_mc[x] * mc[k]) for k in mc
for x in scalar_mc],
factor_on_left=False)
return P._from_dict(D, remove_zeros=False)
def _acted_upon_(self, scalar, self_on_left=False):
"""
Return the action of ``scalar`` on ``self``.
EXAMPLES::
sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: C = S.cellular_basis()
sage: W = S.cell_module([2,1])
sage: elt = W.an_element(); elt
2*W[[1, 2], [3]] + 2*W[[1, 3], [2]]
sage: sc = C.an_element(); sc
3*C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
+ 2*C([3], [[1, 2, 3]], [[1, 2, 3]])
sage: sc * elt
10*W[[1, 3], [2]]
sage: s = S.an_element(); s
[1, 2, 3] + 2*[1, 3, 2] + 3*[2, 1, 3] + [3, 1, 2]
sage: s * elt
11*W[[1, 2], [3]] - 3/2*W[[1, 3], [2]]
sage: 1/2 * elt
W[[1, 2], [3]] + W[[1, 3], [2]]
"""
# Check for elements coercable to the base ring first
ret = CombinatorialFreeModule.Element._acted_upon_(self, scalar, self_on_left)
if ret is not None:
return ret
# See message in CombinatorialFreeModuleElement._acted_upon_
P = self.parent()
if isinstance(scalar, Element) and scalar.parent() is not P._algebra:
# Temporary needed by coercion (see Polynomial/FractionField tests).
if not P._algebra.has_coerce_map_from(scalar.parent()):
return None
scalar = P._algebra( scalar )
if self_on_left:
raise NotImplementedError
#scalar = scalar.cellular_involution()
mc = self._monomial_coefficients
scalar_mc = scalar.monomial_coefficients(copy=False)
D = linear_combination([(P._action_basis(x, k)._monomial_coefficients,
scalar_mc[x] * mc[k])
for k in mc for x in scalar_mc],
factor_on_left=False)
return P._from_dict(D, remove_zeros=False)
def __truediv__(self, x):
"""
Division by coefficients.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: x / 2
1/2*x
sage: W.<x,y,z> = DifferentialWeylAlgebra(ZZ)
sage: a = 2*x + 4*y*z
sage: a / 2
2*y*z + x
"""
F = self.parent()
D = self.__monomials
if F.base_ring().is_field():
x = F.base_ring()( x )
x_inv = x**-1
D = blas.linear_combination( [ ( D, x_inv ) ] )
return self.__class__(F, D)
return self.__class__(F, {t: D[t]._divide_if_possible(x) for t in D})
def __truediv__(self, x):
"""
Division by coefficients.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: x / 2
1/2*x
sage: W.<x,y,z> = DifferentialWeylAlgebra(ZZ)
sage: a = 2*x + 4*y*z
sage: a / 2
2*y*z + x
"""
F = self.parent()
D = self.__monomials
if F.base_ring().is_field():
x = F.base_ring()(x)
x_inv = x**-1
D = blas.linear_combination([(D, x_inv)])
return self.__class__(F, D)
return self.__class__(F, {t: D[t]._divide_if_possible(x) for t in D})
def __init__(self, A, dictionary=None, dual=None):
"""
Create an element of a dual basis.
TESTS::
sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(scalar=zee, prefix='h')
sage: a = h([2])
sage: ec = h._element_class
sage: ec(h, dual=m([2]))
-h[1, 1] + 2*h[2]
sage: h(m([2]))
-h[1, 1] + 2*h[2]
sage: h([2])
h[2]
sage: h([2])._dual
m[1, 1] + m[2]
sage: m(h([2]))
m[1, 1] + m[2]
"""
if dictionary is None and dual is None:
raise ValueError("you must specify either x or dual")
parent = A
base_ring = parent.base_ring()
zero = base_ring.zero()
if dual is None:
# We need to compute the dual
dual_dict = {}
from_self_cache = parent._from_self_cache
# Get the underlying dictionary for self
s_mcs = dictionary
# Make sure all the conversions from self to
# to the dual basis have been precomputed
for part in s_mcs:
if part not in from_self_cache:
parent._precompute(sum(part))
# Create the monomial coefficient dictionary from the
# the monomial coefficient dictionary of dual
for s_part in s_mcs:
from_dictionary = from_self_cache[s_part]
for part in from_dictionary:
dual_dict[ part ] = dual_dict.get(part, zero) + base_ring(s_mcs[s_part]*from_dictionary[part])
dual = parent._dual_basis._from_dict(dual_dict)
if dictionary is None:
# We need to compute the monomial coefficients dictionary
dictionary = {}
to_self_cache = parent._to_self_cache
# Get the underlying dictionary for the dual
d_mcs = dual._monomial_coefficients
# Make sure all the conversions from the dual basis
# to self have been precomputed
for part in d_mcs:
if part not in to_self_cache:
parent._precompute(sum(part))
# Create the monomial coefficient dictionary from the
# the monomial coefficient dictionary of dual
dictionary = blas.linear_combination( (to_self_cache[d_part], d_mcs[d_part]) for d_part in d_mcs)
# Initialize self
self._dual = dual
classical.SymmetricFunctionAlgebra_classical.Element.__init__(self, A, dictionary)
def __init__(self, A, dictionary=None, dual=None):
"""
Create an element of a dual basis.
TESTS::
sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(scalar=zee, prefix='h')
sage: a = h([2])
sage: ec = h._element_class
sage: ec(h, dual=m([2]))
-h[1, 1] + 2*h[2]
sage: h(m([2]))
-h[1, 1] + 2*h[2]
sage: h([2])
h[2]
sage: h([2])._dual
m[1, 1] + m[2]
sage: m(h([2]))
m[1, 1] + m[2]
"""
if dictionary is None and dual is None:
raise ValueError("you must specify either x or dual")
parent = A
base_ring = parent.base_ring()
zero = base_ring.zero()
if dual is None:
# We need to compute the dual
dual_dict = {}
from_self_cache = parent._from_self_cache
# Get the underlying dictionary for self
s_mcs = dictionary
# Make sure all the conversions from self to
# to the dual basis have been precomputed
for part in s_mcs:
if part not in from_self_cache:
parent._precompute(sum(part))
# Create the monomial coefficient dictionary from the
# the monomial coefficient dictionary of dual
for s_part in s_mcs:
from_dictionary = from_self_cache[s_part]
for part in from_dictionary:
dual_dict[ part ] = dual_dict.get(part, zero) + base_ring(s_mcs[s_part]*from_dictionary[part])
dual = parent._dual_basis._from_dict(dual_dict)
if dictionary is None:
# We need to compute the monomial coefficients dictionary
dictionary = {}
to_self_cache = parent._to_self_cache
# Get the underlying dictionary for the dual
d_mcs = dual._monomial_coefficients
# Make sure all the conversions from the dual basis
# to self have been precomputed
for part in d_mcs:
if part not in to_self_cache:
parent._precompute(sum(part))
# Create the monomial coefficient dictionary from the
# the monomial coefficient dictionary of dual
dictionary = blas.linear_combination( (to_self_cache[d_part], d_mcs[d_part]) for d_part in d_mcs)
# Initialize self
self._dual = dual
classical.SymmetricFunctionAlgebra_classical.Element.__init__(self, A, dictionary)