def _multiply_basis(
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`` -- partitions
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]
Examples failing with three different messages in symmetrica::
sage: s[123,1]*s[1,1]
s[123, 1, 1, 1] + s[123, 2, 1] + s[124, 1, 1] + s[124, 2]
sage: s[123]*s[2,1]
s[123, 2, 1] + s[124, 1, 1] + s[124, 2] + s[125, 1]
sage: s[125]*s[3]
s[125, 3] + s[126, 2] + s[127, 1] + s[128]
::
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
"""
import sage.libs.lrcalc.lrcalc as lrcalc
return lrcalc.mult(left, right)
def _multiply_basis(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`` -- partitions
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]
Examples failing with three different messages in symmetrica::
sage: s[123,1]*s[1,1]
s[123, 1, 1, 1] + s[123, 2, 1] + s[124, 1, 1] + s[124, 2]
sage: s[123]*s[2,1]
s[123, 2, 1] + s[124, 1, 1] + s[124, 2] + s[125, 1]
sage: s[125]*s[3]
s[125, 3] + s[126, 2] + s[127, 1] + s[128]
::
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
"""
import sage.libs.lrcalc.lrcalc as lrcalc
return lrcalc.mult(left, right)
def product_on_basis(self, left, right):
"""
Return the product of ``left`` and ``right``.
INPUT:
- ``self`` -- a Schur symmetric function basis
- ``left``, ``right`` -- partitions
OUTPUT:
- 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]
Examples failing with three different messages in symmetrica::
sage: s[123,1]*s[1,1]
s[123, 1, 1, 1] + s[123, 2, 1] + s[124, 1, 1] + s[124, 2]
sage: s[123]*s[2,1]
s[123, 2, 1] + s[124, 1, 1] + s[124, 2] + s[125, 1]
sage: s[125]*s[3]
s[125, 3] + s[126, 2] + s[127, 1] + s[128]
::
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
"""
return self._from_dict(lrcalc.mult(left, right))