Some extra doctests and explanations about the e=0 case.

src/sage/combinat/partition.py

@@ -3025,8 +3025,18 @@ def ladders(self, e):
             sage: Partition([3, 2]).ladders(3)
             {0: [(0, 0)], 1: [(0, 1)], 2: [(0, 2), (1, 0)], 3: [(1, 1)]}
 
+        When ``e`` is ``0``, the cells are in bijection with the ladders,
+        but the index of the ladder depends on the size of the partition::
+
             sage: Partition([3, 2]).ladders(0)
             {0: [(0, 0)], 1: [(0, 1)], 2: [(0, 2)], 5: [(1, 0)], 6: [(1, 1)]}
+            sage: Partition([3, 2, 1]).ladders(0)
+            {0: [(0, 0)], 1: [(0, 1)], 2: [(0, 2)], 6: [(1, 0)], 7: [(1, 1)],
+             12: [(2, 0)]}
+            sage: Partition([3, 1, 1]).ladders(0)
+            {0: [(0, 0)], 1: [(0, 1)], 2: [(0, 2)], 5: [(1, 0)], 10: [(2, 0)]}
+            sage: Partition([1, 1, 1]).ladders(0)
+            {0: [(0, 0)], 3: [(1, 0)], 6: [(2, 0)]}
         """
         if e == 0:
             e = sum(self) + 1

src/sage/combinat/symmetric_group_algebra.py

@@ -1248,7 +1248,7 @@ def ladder_idemponent(self, la):
         r"""
         Return the ladder idempontent of ``self``.
 
-        Let `F` be a field of characteristic `p > 0`. The *ladder idempotent*
+        Let `F` be a field of characteristic `p`. The *ladder idempotent*
         of shape `\lambda` is the idempotent of `F[S_n]` defined as follows.
         Let `T` be the :meth:`ladder tableau
         <sage.combinat.partition.Partition.ladder_tableau>` of shape `\lambda`.
@@ -1264,7 +1264,7 @@ def ladder_idemponent(self, la):
 
         where `E_{UU}` is the :meth:`seminormal_basis` element over `\QQ`
         and we project the sum to `F`, `S_{\alpha}` is the Young subgroup
-        corresponding to `\alpha`, and `\alpha! = \alpha_1 ! \cdots \alpha_k !`.
+        corresponding to `\alpha`, and `\alpha! = \alpha_1! \cdots \alpha_k!`.
 
         EXAMPLES::
 
@@ -1293,6 +1293,25 @@ def ladder_idemponent(self, la):
             2*[1, 2, 3, 4] + [1, 2, 4, 3] + [2, 1, 3, 4] + 2*[2, 1, 4, 3]
              + 2*[3, 4, 1, 2] + [3, 4, 2, 1] + [4, 3, 1, 2] + 2*[4, 3, 2, 1]
 
+        When `p = 0`, these idempotents will generate all of the simple
+        modules (which are the :meth:`specht_modules` and also projective)::
+
+            sage: SGA = SymmetricGroupAlgebra(QQ, 5)
+            sage: for la in Partitions(SGA.n):
+            ....:     idem = SGA.ladder_idemponent(la)
+            ....:     assert idem^2 == idem
+            ....:     print(la, SGA.principal_ideal(idem).dimension())
+            ....:     
+            [5] 1
+            [4, 1] 4
+            [3, 2] 5
+            [3, 1, 1] 6
+            [2, 2, 1] 5
+            [2, 1, 1, 1] 4
+            [1, 1, 1, 1, 1] 1
+            sage: [StandardTableaux(la).cardinality() for la in Partitions(SGA.n)]
+            [1, 4, 5, 6, 5, 4, 1]
+
         REFERENCES:
 
         - [Ryo2015]_