def divided_difference(self, i):
"""
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,1])
sage: a.divided_difference(1)
X[2, 3, 1]
sage: a.divided_difference([3,2,1])
X[1]
"""
if isinstance(i, Integer):
return symmetrica.divdiff_schubert(i, self)
elif i in permutation.Permutations():
return symmetrica.divdiff_perm_schubert(i, self)
else:
raise TypeError, "i must either be an integer or permutation"
def divided_difference(self, i):
"""
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,1])
sage: a.divided_difference(1)
X[2, 3, 1]
sage: a.divided_difference([3,2,1])
X[1]
"""
if isinstance(i, Integer):
return symmetrica.divdiff_schubert(i, self)
elif i in permutation.Permutations():
return symmetrica.divdiff_perm_schubert(i, self)
else:
raise TypeError, "i must either be an integer or permutation"
def divided_difference(self, i, algorithm="sage"):
r"""
Return the ``i``-th divided difference operator, applied to ``self``.
Here, ``i`` can be either a permutation or a positive integer.
INPUT:
- ``i`` -- permutation or positive integer
- ``algorithm`` -- (default: ``'sage'``) either ``'sage'``
or ``'symmetrica'``; this determines which software is
called for the computation
OUTPUT:
The result of applying the ``i``-th divided difference
operator to ``self``.
If `i` is a positive integer, then the `i`-th divided
difference operator `\delta_i` is the linear operator sending
each polynomial `f = f(x_1, x_2, \ldots, x_n)` (in
`n \geq i+1` variables) to the polynomial
.. MATH::
\frac{f - f_i}{x_i - x_{i+1}}, \qquad \text{ where }
f_i = f(x_1, x_2, ..., x_{i-1}, x_{i+1}, x_i,
x_{i+1}, ..., x_n) .
If `\sigma` is a permutation in the `n`-th symmetric group,
then the `\sigma`-th divided difference operator `\delta_\sigma`
is the composition
`\delta_{i_1} \delta_{i_2} \cdots \delta_{i_k}`, where
`\sigma = s_{i_1} \circ s_{i_2} \circ \cdots \circ s_{i_k}` is
any reduced expression for `\sigma` (the precise choice of
reduced expression is immaterial).
.. NOTE::
The :meth:`expand` method results in a polynomial
in `n` variables named ``x0, x1, ..., x(n-1)`` rather than
`x_1, x_2, \ldots, x_n`.
The variable named ``xi`` corresponds to `x_{i+1}`.
Thus, ``self.divided_difference(i)`` involves the variables
``x(i-1)`` and ``xi`` getting switched (in the numerator).
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,1])
sage: a.divided_difference(1)
X[2, 3, 1]
sage: a.divided_difference([3,2,1])
X[1]
sage: a.divided_difference(5)
0
Any divided difference of `0` is `0`::
sage: X.zero().divided_difference(2)
0
This is compatible when a permutation is given as input::
sage: a = X([3,2,4,1])
sage: a.divided_difference([2,3,1])
0
sage: a.divided_difference(1).divided_difference(2)
0
::
sage: a = X([4,3,2,1])
sage: a.divided_difference([2,3,1])
X[3, 2, 4, 1]
sage: a.divided_difference(1).divided_difference(2)
X[3, 2, 4, 1]
sage: a.divided_difference([4,1,3,2])
X[1, 4, 2, 3]
sage: b = X([4, 1, 3, 2])
sage: b.divided_difference(1).divided_difference(2)
X[1, 3, 4, 2]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3)
X[1, 3, 2]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(2)
X[1]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(3)
0
sage: b.divided_difference(1).divided_difference(2).divided_difference(1)
0
TESTS:
Check that :trac:`23403` is fixed::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,4,1])
sage: a.divided_difference(2)
0
sage: a.divided_difference([3,2,1])
0
sage: a.divided_difference(0)
Traceback (most recent call last):
...
ValueError: cannot apply \delta_{0} to a (= X[3, 2, 4, 1])
"""
if not self: # if self is 0
return self
Perms = Permutations()
if i in ZZ:
if algorithm == "sage":
if i <= 0:
raise ValueError(r"cannot apply \delta_{%s} to a (= %s)" %
(i, self))
# The operator `\delta_i` sends the Schubert
# polynomial `X_\pi` (where `\pi` is a finitely supported
# permutation of `\{1, 2, 3, \ldots\}`) to:
# - the Schubert polynomial X_\sigma`, where `\sigma` is
# obtained from `\pi` by switching the values at `i` and `i+1`,
# if `i` is a descent of `\pi` (that is, `\pi(i) > \pi(i+1)`);
# - `0` otherwise.
# Notice that distinct `\pi`s lead to distinct `\sigma`s,
# so we can use `_from_dict` here.
res_dict = {}
for pi, coeff in self:
pi = pi[:]
n = len(pi)
if n <= i:
continue
if pi[i - 1] < pi[i]:
continue
pi[i - 1], pi[i] = pi[i], pi[i - 1]
pi = Perms(pi).remove_extra_fixed_points()
res_dict[pi] = coeff
return self.parent()._from_dict(res_dict)
else: # if algorithm == "symmetrica":
return symmetrica.divdiff_schubert(i, self)
elif i in Perms:
if algorithm == "sage":
i = Permutation(i)
redw = i.reduced_word()
res_dict = {}
for pi, coeff in self:
next_pi = False
pi = pi[:]
n = len(pi)
for j in redw:
if n <= j:
next_pi = True
break
if pi[j - 1] < pi[j]:
next_pi = True
break
pi[j - 1], pi[j] = pi[j], pi[j - 1]
if next_pi:
continue
pi = Perms(pi).remove_extra_fixed_points()
res_dict[pi] = coeff
return self.parent()._from_dict(res_dict)
else: # if algorithm == "symmetrica":
return symmetrica.divdiff_perm_schubert(i, self)
else:
raise TypeError("i must either be an integer or permutation")
def divided_difference(self, i, algorithm="sage"):
r"""
Return the ``i``-th divided difference operator, applied to
``self``.
Here, ``i`` can be either a permutation or a positive integer.
INPUT:
- ``i`` -- permutation or positive integer
- ``algorithm`` -- (default: ``'sage'``) either ``'sage'``
or ``'symmetrica'``; this determines which software is
called for the computation
OUTPUT:
The result of applying the ``i``-th divided difference
operator to ``self``.
If `i` is a positive integer, then the `i`-th divided
difference operator `\delta_i` is the linear operator sending
each polynomial `f = f(x_1, x_2, \ldots, x_n)` (in
`n \geq i+1` variables) to the polynomial
.. MATH::
\frac{f - f_i}{x_i - x_{i+1}}, \qquad \text{ where }
f_i = f(x_1, x_2, ..., x_{i-1}, x_{i+1}, x_i,
x_{i+1}, ..., x_n) .
If `\sigma` is a permutation in the `n`-th symmetric group,
then the `\sigma`-th divided difference operator `\delta_\sigma`
is the composition
`\delta_{i_1} \delta_{i_2} \cdots \delta_{i_k}`, where
`\sigma = s_{i_1} \circ s_{i_2} \circ \cdots \circ s_{i_k}` is
any reduced expression for `\sigma` (the precise choice of
reduced expression is immaterial).
.. NOTE::
The :meth:`expand` method results in a polynomial
in `n` variables named ``x0, x1, ..., x(n-1)`` rather than
`x_1, x_2, \ldots, x_n`.
The variable named ``xi`` corresponds to `x_{i+1}`.
Thus, ``self.divided_difference(i)`` involves the variables
``x(i-1)`` and ``xi`` getting switched (in the numerator).
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,1])
sage: a.divided_difference(1)
X[2, 3, 1]
sage: a.divided_difference([3,2,1])
X[1]
sage: a.divided_difference(5)
0
Any divided difference of `0` is `0`::
sage: X.zero().divided_difference(2)
0
This is compatible when a permutation is given as input::
sage: a = X([3,2,4,1])
sage: a.divided_difference([2,3,1])
0
sage: a.divided_difference(1).divided_difference(2)
0
::
sage: a = X([4,3,2,1])
sage: a.divided_difference([2,3,1])
X[3, 2, 4, 1]
sage: a.divided_difference(1).divided_difference(2)
X[3, 2, 4, 1]
sage: a.divided_difference([4,1,3,2])
X[1, 4, 2, 3]
sage: b = X([4, 1, 3, 2])
sage: b.divided_difference(1).divided_difference(2)
X[1, 3, 4, 2]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3)
X[1, 3, 2]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(2)
X[1]
sage: b.divided_difference(1).divided_difference(2).divided_difference(3).divided_difference(3)
0
sage: b.divided_difference(1).divided_difference(2).divided_difference(1)
0
TESTS:
Check that :trac:`23403` is fixed::
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,4,1])
sage: a.divided_difference(2)
0
sage: a.divided_difference([3,2,1])
0
sage: a.divided_difference(0)
Traceback (most recent call last):
...
ValueError: cannot apply \delta_{0} to a (= X[3, 2, 4, 1])
"""
if not self: # if self is 0
return self
Perms = Permutations()
if i in ZZ:
if algorithm == "sage":
if i <= 0:
raise ValueError(r"cannot apply \delta_{%s} to a (= %s)" % (i, self))
# The operator `\delta_i` sends the Schubert
# polynomial `X_\pi` (where `\pi` is a finitely supported
# permutation of `\{1, 2, 3, \ldots\}`) to:
# - the Schubert polynomial X_\sigma`, where `\sigma` is
# obtained from `\pi` by switching the values at `i` and `i+1`,
# if `i` is a descent of `\pi` (that is, `\pi(i) > \pi(i+1)`);
# - `0` otherwise.
# Notice that distinct `\pi`s lead to distinct `\sigma`s,
# so we can use `_from_dict` here.
res_dict = {}
for pi, coeff in self:
pi = pi[:]
n = len(pi)
if n <= i:
continue
if pi[i-1] < pi[i]:
continue
pi[i-1], pi[i] = pi[i], pi[i-1]
pi = Perms(pi).remove_extra_fixed_points()
res_dict[pi] = coeff
return self.parent()._from_dict(res_dict)
else: # if algorithm == "symmetrica":
return symmetrica.divdiff_schubert(i, self)
elif i in Perms:
if algorithm == "sage":
i = Permutation(i)
redw = i.reduced_word()
res_dict = {}
for pi, coeff in self:
next_pi = False
pi = pi[:]
n = len(pi)
for j in redw:
if n <= j:
next_pi = True
break
if pi[j-1] < pi[j]:
next_pi = True
break
pi[j-1], pi[j] = pi[j], pi[j-1]
if next_pi:
continue
pi = Perms(pi).remove_extra_fixed_points()
res_dict[pi] = coeff
return self.parent()._from_dict(res_dict)
else: # if algorithm == "symmetrica":
return symmetrica.divdiff_perm_schubert(i, self)
else:
raise TypeError("i must either be an integer or permutation")