def _multiply(self, left, right):
"""
Returns the product of ``left`` and ``right``.
- ``self`` -- a monomial symmetric function basis
- ``left``, ``right`` -- instances of the Schur basis ``self``.
OUPUT:
- an element of the Schur basis, the product of ``left`` and ``right``
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SymmetricFunctions(QQx).m()
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
#Use symmetrica to do the multiplication
A = left.parent()
R = A.base_ring()
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
#if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
if left_m == [] and right_m == []:
z_elt[left_m] = left_c * right_c
continue
d = symmetrica.mult_monomial_monomial({
left_m: Integer(1)
}, {
right_m: Integer(1)
}).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[m] = z_elt[m] + left_c * right_c * d[m]
else:
z_elt[m] = left_c * right_c * d[m]
return z_elt
def _multiply(self, left, right):
"""
Return the product of ``left`` and ``right``.
- ``left``, ``right`` -- symmetric functions written in the
monomial basis ``self``.
OUTPUT:
- the product of ``left`` and ``right``, expanded in the
monomial basis, as a dictionary whose keys are partitions and
whose values are the coefficients of these partitions (more
precisely, their respective monomial symmetric functions) in the
product.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SymmetricFunctions(QQx).m()
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
#Use symmetrica to do the multiplication
#A = left.parent()
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
#if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
if left_m == [] and right_m == []:
z_elt[ left_m ] = left_c*right_c
continue
d = symmetrica.mult_monomial_monomial({left_m:Integer(1)}, {right_m:Integer(1)}).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[ m ] = z_elt[m] + left_c * right_c * d[m]
else:
z_elt[ m ] = left_c * right_c * d[m]
return z_elt
def _multiply(self, left, right):
"""
Returns the product of ``left`` and ``right``.
- ``self`` -- a monomial symmetric function basis
- ``left``, ``right`` -- instances of the Schur basis ``self``.
OUPUT:
- an element of the Schur basis, the product of ``left`` and ``right``
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SymmetricFunctions(QQx).m()
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
# Use symmetrica to do the multiplication
# A = left.parent()
# Hack due to symmetrica crashing when both of the
# partitions are the empty partition
# if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
# Hack due to symmetrica crashing when both of the
# partitions are the empty partition
if left_m == [] and right_m == []:
z_elt[left_m] = left_c * right_c
continue
d = symmetrica.mult_monomial_monomial(
{left_m: Integer(1)}, {right_m: Integer(1)}
).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[m] = z_elt[m] + left_c * right_c * d[m]
else:
z_elt[m] = left_c * right_c * d[m]
return z_elt
def _multiply(self, left, right):
"""
EXAMPLES::
sage: m = SFAMonomial(QQ)
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SFAMonomial(QQx)
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
#Use symmetrica to do the multiplication
A = left.parent()
R = A.base_ring()
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
#if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
if left_m == [] and right_m == []:
z_elt[left_m] = left_c * right_c
continue
d = symmetrica.mult_monomial_monomial({
left_m: Integer(1)
}, {
right_m: Integer(1)
}).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[m] = z_elt[m] + left_c * right_c * d[m]
else:
z_elt[m] = left_c * right_c * d[m]
return z_elt
def _multiply(self, left, right):
"""
EXAMPLES::
sage: m = SFAMonomial(QQ)
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SFAMonomial(QQx)
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
#Use symmetrica to do the multiplication
A = left.parent()
R = A.base_ring()
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
#if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
#Hack due to symmetrica crashing when both of the
#partitions are the empty partition
if left_m == [] and right_m == []:
z_elt[ left_m ] = left_c*right_c
continue
d = symmetrica.mult_monomial_monomial({left_m:Integer(1)}, {right_m:Integer(1)}).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[ m ] = z_elt[m] + left_c * right_c * d[m]
else:
z_elt[ m ] = left_c * right_c * d[m]
return z_elt
def _multiply(self, left, right):
"""
Return the product of ``left`` and ``right``.
- ``left``, ``right`` -- symmetric functions written in the
monomial basis ``self``.
OUTPUT:
- the product of ``left`` and ``right``, expanded in the
monomial basis, as a dictionary whose keys are partitions and
whose values are the coefficients of these partitions (more
precisely, their respective monomial symmetric functions) in the
product.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a^2
4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
::
sage: QQx.<x> = QQ['x']
sage: m = SymmetricFunctions(QQx).m()
sage: a = m([2,1])+x
sage: 2*a # indirect doctest
2*x*m[] + 2*m[2, 1]
sage: a^2
x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
"""
# Use symmetrica to do the multiplication
# A = left.parent()
# Hack due to symmetrica crashing when both of the
# partitions are the empty partition
# if R is ZZ or R is QQ:
# return symmetrica.mult_monomial_monomial(left, right)
z_elt = {}
for left_m, left_c in left._monomial_coefficients.items():
for right_m, right_c in right._monomial_coefficients.items():
# Hack due to symmetrica crashing when both of the
# partitions are the empty partition
if not left_m and not right_m:
z_elt[left_m] = left_c * right_c
continue
d = symmetrica.mult_monomial_monomial({
left_m: Integer(1)
}, {
right_m: Integer(1)
}).monomial_coefficients()
for m in d:
if m in z_elt:
z_elt[m] += left_c * right_c * d[m]
else:
z_elt[m] = left_c * right_c * d[m]
return z_elt
