def scalar(self, x):
r"""
Returns the standard scalar product of self and x.
Note that the power-sum symmetric functions are orthogonal under
this scalar product. The value of `\langle p_\lambda, p_\lambda \rangle`
is given by the size of the centralizer in `S_n` of a
permutation of cycle type `\lambda`.
EXAMPLES::
sage: p = SFAPower(QQ)
sage: p4 = Partitions(4)
sage: matrix([ [p(a).scalar(p(b)) for a in p4] for b in p4])
[ 4 0 0 0 0]
[ 0 3 0 0 0]
[ 0 0 8 0 0]
[ 0 0 0 4 0]
[ 0 0 0 0 24]
"""
parent = self.parent()
R = parent.base_ring()
x = parent(x)
f = lambda part1, part2: sfa.zee(part1)
return parent._apply_multi_module_morphism(self,
x,
f,
orthogonal=True)
def scalar(self, x):
r"""
Returns the standard scalar product of self and x.
Note that the power-sum symmetric functions are orthogonal under
this scalar product. The value of `\langle p_\lambda, p_\lambda \rangle`
is given by the size of the centralizer in `S_n` of a
permutation of cycle type `\lambda`.
EXAMPLES::
sage: p = SFAPower(QQ)
sage: p4 = Partitions(4)
sage: matrix([ [p(a).scalar(p(b)) for a in p4] for b in p4])
[ 4 0 0 0 0]
[ 0 3 0 0 0]
[ 0 0 8 0 0]
[ 0 0 0 4 0]
[ 0 0 0 0 24]
"""
parent = self.parent()
R = parent.base_ring()
x = parent(x)
f = lambda part1, part2: sfa.zee(part1)
return parent._apply_multi_module_morphism(self, x, f, orthogonal=True)
def scalar(self, x, zee=None):
r"""
Return the standard scalar product of ``self`` and ``x``.
INPUT:
- ``x`` -- a power sum symmetric function
- ``zee`` -- (default: uses standard ``zee`` function) optional
input specifying the scalar product on the power sum basis with
normalization `\langle p_{\mu}, p_{\mu} \rangle =
\mathrm{zee}(\mu)`. ``zee`` should be a function on partitions.
Note that the power-sum symmetric functions are orthogonal under
this scalar product. With the default value of ``zee``, the value
of `\langle p_{\lambda}, p_{\lambda} \rangle` is given by the
size of the centralizer in `S_n` of a permutation of cycle
type `\lambda`.
OUTPUT:
- the standard scalar product between ``self`` and ``x``, or, if
the optional parameter ``zee`` is specified, then the scalar
product with respect to the normalization `\langle p_{\mu},
p_{\mu} \rangle = \mathrm{zee}(\mu)` with the power sum basis
elements being orthogonal
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p4 = Partitions(4)
sage: matrix([ [p(a).scalar(p(b)) for a in p4] for b in p4])
[ 4 0 0 0 0]
[ 0 3 0 0 0]
[ 0 0 8 0 0]
[ 0 0 0 4 0]
[ 0 0 0 0 24]
sage: p(0).scalar(p(1))
0
sage: p(1).scalar(p(2))
2
sage: zee = lambda x : 1
sage: matrix( [[p[la].scalar(p[mu], zee) for la in Partitions(3)] for mu in Partitions(3)])
[1 0 0]
[0 1 0]
[0 0 1]
"""
parent = self.parent()
x = parent(x)
if zee is None:
f = lambda part1, part2: sfa.zee(part1)
else:
f = lambda part1, part2: zee(part1)
return parent._apply_multi_module_morphism(self,
x,
f,
orthogonal=True)
def scalar(self, x, zee=None):
r"""
Return the standard scalar product of ``self`` and ``x``.
INPUT:
- ``x`` -- an power sum symmetric function
- ``zee`` -- (default: uses standard ``zee`` function) optional
input specifying the scalar product on the power sum basis with
normalization `\langle p_{\mu}, p_{\mu} \rangle =
\mathrm{zee}(\mu)`. ``zee`` should be a function on partitions.
Note that the power-sum symmetric functions are orthogonal under
this scalar product. With the default value of ``zee``, the value
of `\langle p_{\lambda}, p_{\lambda} \rangle` is given by the
size of the centralizer in `S_n` of a permutation of cycle
type `\lambda`.
OUTPUT:
- the standard scalar product between ``self`` and ``x``, or, if
the optional parameter ``zee`` is specified, then the scalar
product with respect to the normalization `\langle p_{\mu},
p_{\mu} \rangle = \mathrm{zee}(\mu)` with the power sum basis
elements being orthogonal
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p4 = Partitions(4)
sage: matrix([ [p(a).scalar(p(b)) for a in p4] for b in p4])
[ 4 0 0 0 0]
[ 0 3 0 0 0]
[ 0 0 8 0 0]
[ 0 0 0 4 0]
[ 0 0 0 0 24]
sage: p(0).scalar(p(1))
0
sage: p(1).scalar(p(2))
2
sage: zee = lambda x : 1
sage: matrix( [[p[la].scalar(p[mu], zee) for la in Partitions(3)] for mu in Partitions(3)])
[1 0 0]
[0 1 0]
[0 0 1]
"""
parent = self.parent()
x = parent(x)
if zee is None:
f = lambda part1, part2: sfa.zee(part1)
else:
f = lambda part1, part2: zee(part1)
return parent._apply_multi_module_morphism(self, x, f, orthogonal=True)
