gh-38197: Column and row stabilizers for skew tableaux
    
Row and column stabilizers and the usual a, b, e elements of the
symmetric group algebra are now computable for skew standard tableaux,
not just for straight ones.

The a, b, e functions are still not exposed at class level, but this can
be done another day.

The performance cost (at least on the e function) is trivial:

```
sage: %timeit eold([[2,3],[1,4]])
1.29 ms ± 12.7 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops
each)
sage: %timeit enew([[2,3],[1,4]])
1.29 ms ± 15.5 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops
each)
sage: %timeit eold([[2,3],[1,4],[5,7]])
3.43 ms ± 141 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit enew([[2,3],[1,4],[5,7]])
3.44 ms ± 158 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
```
(Tested using factored-out code, since the actual functions are cached.)
    
URL: #38197
Reported by: Darij Grinberg
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/skew_tableau.py

@@ -50,6 +50,7 @@
 from sage.misc.persist import register_unpickle_override
 
 lazy_import('sage.matrix.special', 'zero_matrix')
+lazy_import('sage.groups.perm_gps.permgroup', 'PermutationGroup')
 
 
 class SkewTableau(ClonableList,
@@ -1139,6 +1140,81 @@ def to_list(self):
         """
         return [list(row) for row in self]
 
+    def row_stabilizer(self):
+        """
+        Return the :func:`PermutationGroup` corresponding to the row stabilizer of
+        ``self``.
+
+        This assumes that every integer from `1` to the size of ``self``
+        appears exactly once in ``self``.
+
+        EXAMPLES::
+
+            sage: # needs sage.groups
+            sage: rs = SkewTableau([[None,1,2,3],[4,5]]).row_stabilizer()
+            sage: rs.order() == factorial(3) * factorial(2)
+            True
+            sage: PermutationGroupElement([(1,3,2),(4,5)]) in rs
+            True
+            sage: PermutationGroupElement([(1,4)]) in rs
+            False
+            sage: rs = SkewTableau([[None,1,2],[3]]).row_stabilizer()
+            sage: PermutationGroupElement([(1,2),(3,)]) in rs
+            True
+            sage: rs.one().domain()
+            [1, 2, 3]
+            sage: rs = SkewTableau([[None,None,1],[None,2],[3]]).row_stabilizer()
+            sage: rs.order()
+            1
+            sage: rs = SkewTableau([[None,None,2,4,5],[1,3]]).row_stabilizer()
+            sage: rs.order()
+            12
+            sage: rs = SkewTableau([]).row_stabilizer()
+            sage: rs.order()
+            1
+        """
+        # Ensure that the permutations involve all elements of the
+        # tableau, by including the identity permutation on the set [1..k].
+        k = self.size()
+        gens = [list(range(1, k + 1))]
+        for row in self:
+            for j in range(len(row) - 1):
+                if row[j] is not None:
+                    gens.append((row[j], row[j + 1]))
+        return PermutationGroup(gens)
+
+    def column_stabilizer(self):
+        """
+        Return the :func:`PermutationGroup` corresponding to the column stabilizer
+        of ``self``.
+
+        This assumes that every integer from `1` to the size of ``self``
+        appears exactly once in ``self``.
+
+        EXAMPLES::
+
+            sage: # needs sage.groups
+            sage: cs = SkewTableau([[None,2,3],[1,5],[4]]).column_stabilizer()
+            sage: cs.order() == factorial(2) * factorial(2)
+            True
+            sage: PermutationGroupElement([(1,3,2),(4,5)]) in cs
+            False
+            sage: PermutationGroupElement([(1,4)]) in cs
+            True
+        """
+        # Ensure that the permutations involve all elements of the
+        # tableau, by including the identity permutation on the set [1..k].
+        k = self.size()
+        gens = [list(range(1, k + 1))]
+        ell = len(self)
+        while ell > 1:
+            ell -= 1
+            for i, val in enumerate(self[ell]):
+                top_neighbor = self[ell-1][i]
+                if top_neighbor is not None:
+                    gens.append((val, top_neighbor))
+        return PermutationGroup(gens)
+
     def shuffle(self, t2):
         r"""
         Shuffle the standard tableaux ``self`` and ``t2``.

src/sage/combinat/symmetric_group_algebra.py

@@ -18,6 +18,7 @@
 from sage.combinat.permutation_cython import (left_action_same_n, right_action_same_n)
 from sage.combinat.partition import _Partitions, Partitions, Partitions_n
 from sage.combinat.tableau import Tableau, StandardTableaux_size, StandardTableaux_shape, StandardTableaux
+from sage.combinat.skew_tableau import SkewTableau
 from sage.algebras.group_algebra import GroupAlgebra_class
 from sage.algebras.cellular_basis import CellularBasis
 from sage.categories.weyl_groups import WeylGroups
@@ -139,7 +140,7 @@ def SymmetricGroupAlgebra(R, W, category=None):
 
     This allows for mixed expressions::
 
-        sage: x4  = 3*QS4([3, 1, 4, 2])
+        sage: x4  = 3 * QS4([3, 1, 4, 2])
         sage: x3 + x4
         2*[2, 3, 1, 4] + [3, 1, 2, 4] + 3*[3, 1, 4, 2]
 
@@ -2608,23 +2609,32 @@ def pi_ik(itab, ktab):
     algebra.
 
     This assumes that ``itab`` and ``ktab`` are tableaux (possibly
-    given just as lists of lists) of the same shape.
+    given just as lists of lists) of the same shape. Both
+    tableaux are allowed to be skew.
 
     EXAMPLES::
 
         sage: from sage.combinat.symmetric_group_algebra import pi_ik
         sage: pi_ik([[1,3],[2]], [[1,2],[3]])
         [1, 3, 2]
+
+    The same with skew tableaux::
+
+        sage: from sage.combinat.symmetric_group_algebra import pi_ik
+        sage: pi_ik([[None,1,3],[2]], [[None,1,2],[3]])
+        [1, 3, 2]
     """
-    it = Tableau(itab)
-    kt = Tableau(ktab)
+    it = SkewTableau(itab)
+    kt = SkewTableau(ktab)
+    n = kt.size()
 
-    p = [None] * kt.size()
+    p = [None] * n
     for i in range(len(kt)):
         for j in range(len(kt[i])):
-            p[it[i][j] - 1] = kt[i][j]
+            if it[i][j] is not None:
+                p[it[i][j] - 1] = kt[i][j]
 
-    QSn = SymmetricGroupAlgebra(QQ, it.size())
+    QSn = SymmetricGroupAlgebra(QQ, n)
     p = Permutation(p)
     return QSn(p)
 
@@ -2703,8 +2713,13 @@ def a(tableau, star=0, base_ring=QQ):
         [1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
         sage: a([[1,4], [2,3]], base_ring=ZZ)
         [1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]
+
+    The same with a skew tableau::
+
+        sage: a([[None,1,4], [2,3]], base_ring=ZZ)
+        [1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]
     """
-    t = Tableau(tableau)
+    t = SkewTableau(tableau)
     if star:
         t = t.restrict(t.size() - star)
 
@@ -2720,7 +2735,7 @@ def a(tableau, star=0, base_ring=QQ):
     # being [1] rather than [] (which seems to have its origins in
     # permutation group code).
     # TODO: Fix this.
-    if len(tableau) == 0:
+    if n <= 1:
         return sgalg.one()
 
     rd = dict((P(h), one) for h in rs)
@@ -2774,6 +2789,11 @@ def b(tableau, star=0, base_ring=QQ):
         sage: b([[1, 4], [2, 3]], base_ring=Integers(5))
         [1, 2, 3, 4] + 4*[1, 2, 4, 3] + 4*[2, 1, 3, 4] + [2, 1, 4, 3]
 
+    The same with a skew tableau::
+
+        sage: b([[None, 2, 4], [1, 3], [5]])
+        [1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
+
     With the ``l2r`` setting for multiplication, the unnormalized
     Young symmetrizer ``e(tableau)`` should be the product
     ``b(tableau) * a(tableau)`` for every ``tableau``. Let us check
@@ -2783,7 +2803,7 @@ def b(tableau, star=0, base_ring=QQ):
         sage: all( e(t) == b(t) * a(t) for t in StandardTableaux(5) )
         True
     """
-    t = Tableau(tableau)
+    t = SkewTableau(tableau)
     if star:
         t = t.restrict(t.size() - star)
 
@@ -2799,7 +2819,7 @@ def b(tableau, star=0, base_ring=QQ):
     # being [1] rather than [] (which seems to have its origins in
     # permutation group code).
     # TODO: Fix this.
-    if len(tableau) == 0:
+    if n <= 1:
         return sgalg.one()
 
     cd = dict((P(v), v.sign() * one) for v in cs)
@@ -2859,6 +2879,12 @@ def e(tableau, star=0):
         sage: QS3.antipode(e([[1,2],[3]]))
         [1, 2, 3] + [2, 1, 3] - [2, 3, 1] - [3, 2, 1]
 
+    And here is an example for a skew tableau::
+
+        sage: e([[None, 2, 1], [4, 3]])
+        [1, 2, 3, 4] + [1, 2, 4, 3] - [1, 3, 2, 4] - [1, 4, 2, 3]
+         + [2, 1, 3, 4] + [2, 1, 4, 3] - [2, 3, 1, 4] - [2, 4, 1, 3]
+
     .. SEEALSO::
 
         :func:`e_hat`
@@ -2868,7 +2894,7 @@ def e(tableau, star=0):
     # a way to compute them over other base rings as well. Be careful
     # with the cache.
 
-    t = Tableau(tableau)
+    t = SkewTableau(tableau)
     if star:
         t = t.restrict(t.size() - star)
 
@@ -2896,8 +2922,8 @@ def e(tableau, star=0):
         # being [1] rather than [] (which seems to have its origins in
         # permutation group code).
         # TODO: Fix this.
-        if not tableau:
-            res = QSn.one()
+        if n <= 1:
+            return QSn.one()
 
         e_cache[t] = res
 
@@ -2955,7 +2981,10 @@ def e_hat(tab, star=0):
 
         :func:`e`
     """
-    t = Tableau(tab)
+    t = SkewTableau(tab)
+    # This is for consistency's sake. This method is NOT meant
+    # to be applied to skew tableaux, since the meaning of
+    # \kappa is unclear in that case.
     if star:
         t = t.restrict(t.size() - star)
     if t in ehat_cache:
@@ -2978,8 +3007,8 @@ def e_ik(itab, ktab, star=0):
         sage: e_ik([[1,2,3]], [[1,2,3]], star=1)
         [1, 2] + [2, 1]
     """
-    it = Tableau(itab)
-    kt = Tableau(ktab)
+    it = SkewTableau(itab)
+    kt = SkewTableau(ktab)
     if star:
         it = it.restrict(it.size() - star)
         kt = kt.restrict(kt.size() - star)