def _multiply(self, left, right): # TODO: factor out this code for all bases (as is done for coercions)
"""
Returns the product of ``left`` and ``right``.
INPUT:
- ``self`` -- a Schur symmetric function basis
- ``left``, ``right`` -- symmetric functions written in the Schur basis
``self``.
OUPUT:
- an element of the Schur basis, the product of ``left`` and ``right``
TESTS::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1]) + 1; a
s[] + s[2, 1]
sage: a^2 # indirect doctest
s[] + 2*s[2, 1] + s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]
::
sage: QQx.<x> = QQ[]
sage: s = SymmetricFunctions(QQx).s()
sage: a = x^2*s([2,1]) + 2*x; a
2*x*s[] + x^2*s[2, 1]
sage: a^2
4*x^2*s[] + 4*x^3*s[2, 1] + x^4*s[2, 2, 1, 1] + x^4*s[2, 2, 2] + x^4*s[3, 1, 1, 1] + 2*x^4*s[3, 2, 1] + x^4*s[3, 3] + x^4*s[4, 1, 1] + x^4*s[4, 2]
::
::
sage: 0*s([2,1])
0
"""
#Use symmetrica to do the multiplication
A = left.parent()
R = A.base_ring()
if R is ZZ or R is QQ:
if left and right:
return symmetrica.mult_schur_schur(left, right)
else:
return A._from_dict({})
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
c = left_c * right_c
d = symmetrica.mult_schur_schur({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] + c * d[m]
else:
z_elt[ m ] = c * d[m]
return A._from_dict(z_elt)
def _multiply(self, left, right): # TODO: factor out this code for all bases (as is done for coercions)
"""
Returns the product of ``left`` and ``right``.
INPUT:
- ``self`` -- a Schur symmetric function basis
- ``left``, ``right`` -- instances of the Schur basis ``self``.
OUPUT:
- an element of the Schur basis, the product of ``left`` and ``right``
TESTS::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1]) + 1; a
s[] + s[2, 1]
sage: a^2 # indirect doctest
s[] + 2*s[2, 1] + s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]
::
sage: QQx.<x> = QQ[]
sage: s = SymmetricFunctions(QQx).s()
sage: a = x^2*s([2,1]) + 2*x; a
2*x*s[] + x^2*s[2, 1]
sage: a^2
4*x^2*s[] + 4*x^3*s[2, 1] + x^4*s[2, 2, 1, 1] + x^4*s[2, 2, 2] + x^4*s[3, 1, 1, 1] + 2*x^4*s[3, 2, 1] + x^4*s[3, 3] + x^4*s[4, 1, 1] + x^4*s[4, 2]
::
::
sage: 0*s([2,1])
0
"""
#Use symmetrica to do the multiplication
A = left.parent()
R = A.base_ring()
if R is ZZ or R is QQ:
if left and right:
return symmetrica.mult_schur_schur(left, right)
else:
return A._from_dict({})
z_elt = {}
for (left_m, left_c) in left._monomial_coefficients.iteritems():
for (right_m, right_c) in right._monomial_coefficients.iteritems():
c = left_c * right_c
d = symmetrica.mult_schur_schur({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] + c * d[m]
else:
z_elt[ m ] = c * d[m]
return A._from_dict(z_elt)