Trac #23443: More Schubert polynomial shenanigans

Title stolen from Stanley, but this is, alas, a bug, not a determinant
conjecture:
{{{
sage: X = SchubertPolynomialRing(ZZ)
sage: X([]).expand()
1
sage: X([1]).expand()
1
sage: X([]) == X([1])
False
}}}

Normally, `X(perm)` reduces the permutation `perm` by removing all fixed
points attached to its right end (i.e., if a permutation in `S_n` sends
`n` to `n`, then it reduces it to a permutation in `S_{n-1}` and so on,
until this reduction is no longer possible). And rightfully so, since
the Schubert basis is indexed by reduced permutations.
{{{
sage: X = SchubertPolynomialRing(ZZ)
sage: X([1])
X[1]
sage: X([1,2])
X[1]
sage: X([1,2,3])
X[1]
}}}
However, it fails to reduce `[1]` to `[]` due to the behavior of
`Permutation.remove_extra_fixed_points`.

Can we just fix it, or does symmetrica break on contact with the empty
list? In the latter case, should we reduce `[]` to `[1]` instead?

Related: #23403.

URL: https://trac.sagemath.org/23443
Reported by: darij
Ticket author(s): Darij Grinberg
Reviewer(s): Travis Scrimshaw

src/sage/combinat/permutation.py

@@ -4756,18 +4756,31 @@ def remove_extra_fixed_points(self):
         """
         Return the permutation obtained by removing any fixed points at
         the end of ``self``.
+        However, return ``[1]`` rather than ``[]`` if ``self`` is the
+        identity permutation.
+
+        This is mostly a helper method for
+        :mod:`sage.combinat.schubert_polynomial`, where it is
+        used to normalize finitary permutations of
+        `\{1,2,3,\ldots\}`.
 
         EXAMPLES::
 
             sage: Permutation([2,1,3]).remove_extra_fixed_points()
             [2, 1]
             sage: Permutation([1,2,3,4]).remove_extra_fixed_points()
             [1]
+            sage: Permutation([2,1]).remove_extra_fixed_points()
+            [2, 1]
+            sage: Permutation([]).remove_extra_fixed_points()
+            [1]
 
         .. SEEALSO::
 
             :meth:`retract_plain`
         """
+        if not self:
+            return Permutations()([1])
         #Strip off all extra fixed points at the end of
         #the permutation.
         i = len(self)-1

src/sage/combinat/schubert_polynomial.py

@@ -66,9 +66,7 @@ def expand(self):
         TESTS:
 
         Calling .expand() should always return an element of an
-        MPolynomialRing
-
-        ::
+        MPolynomialRing::
 
             sage: X = SchubertPolynomialRing(ZZ)
             sage: f = X([1]); f
@@ -83,10 +81,16 @@ def expand(self):
             sage: f = X([1,3,2,4])
             sage: type(f.expand())
             <class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
+
+        Now we check for correct handling of the empty
+        permutation (:trac:`23443`)::
+
+            sage: X([1]).expand() * X([2,1]).expand()
+            x0
         """
         p = symmetrica.t_SCHUBERT_POLYNOM(self)
         if not is_MPolynomial(p):
-            R = PolynomialRing(self.parent().base_ring(), 1, 'x')
+            R = PolynomialRing(self.parent().base_ring(), 1, 'x0')
             p = R(p)
         return p
 
@@ -361,6 +365,12 @@ def _element_constructor_(self, x):
             Traceback (most recent call last):
             ...
             ValueError: The input [1, 2, 1] is not a valid permutation
+
+        Now we check for correct handling of the empty
+        permutation (:trac:`23443`)::
+
+            sage: X([])
+            X[1]
         """
         if isinstance(x, list):
             # checking the input to avoid symmetrica crashing Sage, see trac 12924