Doing some refactoring and doc work.

src/sage/combinat/algebraic_combinatorics.py

@@ -39,6 +39,7 @@
 - :ref:`sage.combinat.cluster_algebra_quiver.all`
 - :class:`~sage.combinat.kazhdan_lusztig.KazhdanLusztigPolynomial`
 - :class:`~sage.combinat.symmetric_group_representations.SymmetricGroupRepresentation`
+- :class:`~sage.combinat.specht_module.SpechtModule`
 - :ref:`sage.combinat.yang_baxter_graph`
 - :ref:`sage.combinat.hall_polynomial`
 - :ref:`sage.combinat.key_polynomial`

src/sage/combinat/composition.py

@@ -1389,6 +1389,15 @@ def count(self, n):
     def specht_module(self, BR=None):
         r"""
         Return the Specht module corresponding to ``self``.
+
+        EXAMPLES::
+
+            sage: SM = Composition([1,2,2]).specht_module(QQ)
+            sage: SM
+            Free module generated by {0, 1, 2, 3, 4} over Rational Field
+            sage: s = SymmetricFunctions(QQ).s()
+            sage: s(SM.frobenius_image())
+            s[2, 2, 1]
         """
         from sage.combinat.specht_module import SpechtModule
         from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
@@ -1402,16 +1411,23 @@ def specht_module(self, BR=None):
                 cells.append((i, j))
         return SpechtModule(R, cells)
 
-    def specht_module_dimension(self):
+    def specht_module_dimension(self, BR=None):
         r"""
         Return the dimension of the Specht module corresponding to ``self``.
+
+        INPUT:
+
+        - ``BR`` -- (default: `\QQ`) the base ring
+
+        EXAMPLES::
+
+            sage: Composition([1,2,2]).specht_module_dimension()
+            5
+            sage: Composition([1,2,2]).specht_module_dimension(GF(2))
+            5
         """
         from sage.combinat.specht_module import specht_module_rank
-        cells = []
-        for i, row in enumerate(self):
-            for j in range(row):
-                cells.append((i, j))
-        return specht_module_rank(cells)
+        return specht_module_rank(self, BR)
 
 Sequence.register(Composition)
 ##############################################################

src/sage/combinat/diagram.py

@@ -511,12 +511,23 @@ def specht_module(self, BR=None):
         R = SymmetricGroupAlgebra(BR, len(self))
         return SpechtModule(R, self)
 
-    def specht_module_dimension(self):
-        """
+    def specht_module_dimension(self, BR=None):
+        r"""
         Return the dimension of the Specht module corresponding to ``self``.
+
+        INPUT:
+
+        - ``BR`` -- (default: `\QQ`) the base ring
+
+        EXAMPLES::
+
+            sage: from sage.combinat.diagram import Diagram
+            sage: D = Diagram([(0,0), (1,1), (2,2), (2,3)])
+            sage: D.specht_module_dimension()
+            sage: D.specht_module(QQ).dimension()
         """
         from sage.combinat.specht_module import specht_module_rank
-        return specht_module_rank(self)
+        return specht_module_rank(self, BR)
 
 class Diagrams(UniqueRepresentation, Parent):
     r"""

src/sage/combinat/integer_vector.py

@@ -530,22 +530,25 @@ def specht_module(self, BR=None):
             from sage.rings.rational_field import QQ
             BR = QQ
         R = SymmetricGroupAlgebra(BR, sum(self))
-        cells = []
-        for i, row in enumerate(self):
-            for j in range(row):
-                cells.append((i, j))
-        return SpechtModule(R, cells)
+        return SpechtModule(R, self)
 
-    def specht_module_dimension(self):
+    def specht_module_dimension(self, BR=None):
         r"""
         Return the dimension of the Specht module corresponding to ``self``.
+
+        INPUT:
+
+        - ``BR`` -- (default: `\QQ`) the base ring
+
+        EXAMPLES::
+
+            sage: IntegerVectors()([2,0,1,0,2]).specht_module_dimension()
+            5
+            sage: IntegerVectors()([2,0,1,0,2]).specht_module_dimension(GF(2))
+            5
         """
         from sage.combinat.specht_module import specht_module_rank
-        cells = []
-        for i, row in enumerate(self):
-            for j in range(row):
-                cells.append((i, j))
-        return specht_module_rank(cells)
+        return specht_module_rank(self, BR)
 
 
 class IntegerVectors(Parent, metaclass=ClasscallMetaclass):

src/sage/combinat/partition.py

@@ -5484,25 +5484,51 @@ def coloring(i):
         return self.dual_equivalence_graph(directed, coloring)
 
     def specht_module(self, BR=None):
-        """
-        Return the Specht module corresponding to ``self``.
+        r"""
+        Return the Specht module corresponding to ``self``.      
+
+        EXAMPLES::
+
+            sage: SM = Partition([2,2,1]).specht_module(QQ)
+            sage: SM
+            Free module generated by {0, 1, 2, 3, 4} over Rational Field
+            sage: s = SymmetricFunctions(QQ).s()
+            sage: s(SM.frobenius_image())
+            s[2, 2, 1]
         """
         from sage.combinat.specht_module import SpechtModule
         from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
         if BR is None:
             from sage.rings.rational_field import QQ
             BR = QQ
         R = SymmetricGroupAlgebra(BR, sum(self))
-        return SpechtModule(R, self.cells())
+        return SpechtModule(R, self)
 
-    def specht_module_dimension(self):
-        """
+    def specht_module_dimension(self, BR=None):
+        r"""
         Return the dimension of the Specht module corresponding to ``self``.
+
+        This is equal to the number of standard tableaux of shape ``self`` when
+        over a field of characteristic `0`.
+
+        INPUT:
+
+        - ``BR`` -- (default: `\QQ`) the base ring
+
+        EXAMPLES::
 
-        This is equal to the number of standard tableaux of shape ``self``.
+            sage: Partition([2,2,1]).specht_module_dimension()
+            5
+            sage: Partition([2,2,1]).specht_module_dimension(GF(2))
+            5
         """
-        from sage.combinat.tableau import StandardTableaux
-        return StandardTableaux(self).cardinality()
+        from sage.categories.fields import Fields
+        if BR is None or (BR in Fields() and BR.characteristic() == 0):
+            from sage.combinat.tableau import StandardTableaux
+            return StandardTableaux(self).cardinality()
+        from sage.combinat.specht_module import specht_module_rank
+        return specht_module_rank(self, BR)
+
 
 ##############
 # Partitions #

src/sage/combinat/skew_partition.py

@@ -1323,8 +1323,12 @@ def specht_module_dimension(self):
             sage: SM = mu.specht_module_dimension(QQ)
             8
         """
-        from sage.combinat.skew_tableau import StandardSkewTableaux
-        return StandardSkewTableaux(self).cardinality()
+        from sage.categories.fields import Fields
+        if BR is None or (BR in Fields() and BR.characteristic() == 0):
+            from sage.combinat.skew_tableau import StandardSkewTableaux
+            return StandardSkewTableaux(self).cardinality()
+        from sage.combinat.specht_module import specht_module_rank
+        return specht_module_rank(self, BR)
 
 
 def row_lengths_aux(skp):

src/sage/combinat/specht_module.py

@@ -51,6 +51,11 @@ class SpechtModule(SubmoduleWithBasis):
 
     - ``SGA`` -- a symmetric group algebra
     - ``D`` -- a diagram
+
+    .. SEEALSO::
+
+        :class:`~sage.combinat.symmetric_group_representations.SpechtRepresentation`
+        for an implementation of the representation by matrices.
     """
     @staticmethod
     def __classcall_private__(cls, SGA, D):
@@ -73,8 +78,8 @@ def __classcall_private__(cls, SGA, D):
             ...
             the rank (=3) does not match the size (=2) of the diagram
         """
-        if not isinstance(D, Diagram):
-            D = Diagram(D)
+        D = _to_diagram(D)
+        D = Diagram(D)
         n = len(D)
         if SGA.group().rank() != n - 1:
             rk = SGA.group().rank()
@@ -105,6 +110,11 @@ def representation_matrix(self, elt):
         Return the matrix corresponding to the left action of the symmetric
         group (algebra) element ``elt`` on ``self``.
 
+
+        .. SEEALSO::
+
+            :class:`~sage.combinat.symmetric_group_representations.SpechtRepresentation`
+
         EXAMPLES::
 
             sage: SM = Partition([3,1,1]).specht_module(QQ)
@@ -192,6 +202,27 @@ def _acted_upon_(self, x, self_on_left=False):
                     return P.retract(P._ambient(x) * self.lift())
             return super()._acted_upon_(self, x, self_on_left)
 
+def _to_diagram(D):
+    r"""
+    Convert ``D`` to a list of cells representing a diagram.
+    """
+    from sage.combinat.integer_vector import IntegerVectors
+    from sage.combinat.skew_partition import SkewPartitions
+    from sage.combinat.partition import _Partitions
+    if isinstance(D, Diagram):
+        return D
+    if D in _Partitions:
+        D = _Partitions(D).cells()
+    elif D in SkewPartitions():
+        D = SkewPartitions()(D).cells()
+    elif D in IntegerVectors():
+        cells = []
+        for i, row in enumerate(D):
+            for j in range(row):
+                cells.append((i, j))
+        D = cells
+    return D
+
 def specht_module_spanning_set(D, SGA=None):
     r"""
     Return a spanning set of the Specht module of diagram ``D``.
@@ -214,8 +245,6 @@ def specht_module_spanning_set(D, SGA=None):
         (() - (2,3), -(1,2) + (1,3,2), (1,2,3) - (1,3),
          -() + (2,3), -(1,2,3) + (1,3), (1,2) - (1,3,2))
     """
-    if not isinstance(D, Diagram):
-        D = Diagram(D)
     n = len(D)
     if SGA is None:
         from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
@@ -247,7 +276,7 @@ def specht_module_spanning_set(D, SGA=None):
     gen = col_stab * row_stab
     return tuple([b * gen for b in B])
 
-def specht_module_rank(D):
+def specht_module_rank(D, BR=None):
     r"""
     Return the rank of the Specht module of diagram ``D``.
 
@@ -257,5 +286,9 @@ def specht_module_rank(D):
         sage: specht_module_rank([(0,0), (1,1), (2,2)])
         6
     """
+    D = _to_diagram(D)
     span_set = specht_module_spanning_set(D)
-    return matrix(QQ, [v.to_vector() for v in span_set]).rank()
+    if BR is None:
+        BR = QQ
+    return matrix(BR, [v.to_vector() for v in span_set]).rank()
+

src/sage/combinat/symmetric_group_algebra.py

@@ -1544,6 +1544,22 @@ def complement(xs):
         return self.sum_of_monomials([self._indices(P(list(q) + complement(q)))
                                       for q in itertools.combinations(range(1, n + 1), int(k))])
 
+    def specht_module(self, D):
+        r"""
+        Return the Specht module of ``self`` indexed by the diagram ``D``.
+        """
+        from sage.combinat.specht_module import SpechtModule
+        return SpechtModule(self, D)
+
+    def specht_module_dimension(self, D):
+        r"""
+        Return the dimension of the Specht module of ``self`` indexed by ``D``.
+        """
+        from sage.combinat.specht_module import specht_module_spanning_set, _to_diagram
+        D = _to_diagram(D)
+        span_set = specht_module_spanning_set(D, self)
+        return matrix(self.base_ring(), [v.to_vector() for v in span_set]).rank()
+
     def jucys_murphy(self, k):
         r"""
         Return the Jucys-Murphy element `J_k` (also known as a