Implementing Holt-Rees meataxe, subquotient reprs, adding duplication.

src/sage/categories/modules_with_basis.py

@@ -727,7 +727,7 @@ def echelon_form(self, elements, row_reduced=False, order=None):
 
         def submodule(self, gens, check=True, already_echelonized=False,
                       unitriangular=False, support_order=None, category=None,
-                      *args, **opts):
+                      parent_class=None, *args, **opts):
             r"""
             The submodule spanned by a finite set of elements.
 
@@ -744,6 +744,7 @@ def submodule(self, gens, check=True, already_echelonized=False,
             - ``support_order`` -- (optional) either something that can
               be converted into a tuple or a key function
             - ``category`` -- (optional) the category of the submodule
+            - ``parent_class`` -- (optional) the class of the parent to return
 
             If ``already_echelonized`` is ``False``, then the
             generators are put in reduced echelon form using
@@ -906,11 +907,12 @@ def submodule(self, gens, check=True, already_echelonized=False,
             if not already_echelonized:
                 gens = self.echelon_form(gens, unitriangular, order=support_order)
 
-            from sage.modules.with_basis.subquotient import SubmoduleWithBasis
-            return SubmoduleWithBasis(gens, ambient=self,
-                                      support_order=support_order,
-                                      unitriangular=unitriangular,
-                                      category=category, *args, **opts)
+            if parent_class is None:
+                from sage.modules.with_basis.subquotient import SubmoduleWithBasis as parent_class
+            return parent_class(gens, ambient=self,
+                                support_order=support_order,
+                                unitriangular=unitriangular,
+                                category=category, *args, **opts)
 
         def quotient_module(self, submodule, check=True, already_echelonized=False, category=None):
             r"""
@@ -1075,7 +1077,7 @@ def is_finite(self):
                 sage: GroupAlgebra(AbelianGroup(1), IntegerModRing(10)).is_finite()     # needs sage.groups sage.modules
                 False
             """
-            return (self.base_ring().is_finite() and self.group().is_finite())
+            return (self.base_ring().is_finite() and self.basis().keys().is_finite())
 
         def monomial(self, i):
             """

src/sage/combinat/partition.py

@@ -5620,6 +5620,35 @@ def specht_module(self, base_ring=None):
         R = SymmetricGroupAlgebra(base_ring, sum(self))
         return SpechtModule(R, self)
 
+    def garsia_procesi_module(self, base_ring=None):
+        r"""
+        Return the :class:`Garsia-Procesi module
+        <sage.combinat.symmetric_group_representations.GarsiaProcesiModule>`
+        corresponding to ``self``.
+
+        INPUT:
+
+        - ``base_ring`` -- (default: `\QQ`) the base ring
+
+        EXAMPLES::
+
+            sage: GP = Partition([3,2,1]).garsia_procesi_module(QQ); GP
+            Garsia-Procesi module of shape [3, 2, 1] over Rational Field
+            sage: GP.graded_frobenius_image()
+            q^4*s[3, 2, 1] + q^3*s[3, 3] + q^3*s[4, 1, 1] + (q^3+q^2)*s[4, 2]
+             + (q^2+q)*s[5, 1] + s[6]
+
+            sage: Partition([3,2,1]).garsia_procesi_module(GF(3))
+            Garsia-Procesi module of shape [3, 2, 1] over Finite Field of size 3
+        """
+        from sage.combinat.symmetric_group_representations import GarsiaProcesiModule
+        from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
+        if base_ring is None:
+            from sage.rings.rational_field import QQ
+            base_ring = QQ
+        R = SymmetricGroupAlgebra(base_ring, sum(self))
+        return GarsiaProcesiModule(R, self)
+
     def specht_module_dimension(self, base_ring=None):
         r"""
         Return the dimension of the Specht module corresponding to ``self``.

src/sage/combinat/specht_module.py

@@ -37,33 +37,22 @@
 from sage.modules.free_module_element import vector
 from sage.categories.modules_with_basis import ModulesWithBasis
 
-class SymmetricGroupRepresentation:
+class SymmetricGroupRepresentation(Representation_abstract):
     """
     Mixin class for symmetric group (algebra) representations.
     """
     def __init__(self, SGA):
         """
         Initialize ``self``.
 
-        EXAMPLES::
-
-            sage: SM = Partition([3,1,1]).specht_module(GF(3))
-            sage: TestSuite(SM).run()
-        """
-        self._semigroup = SGA.group()
-        self._semigroup_algebra = SGA
-
-    def side(self):
-        r"""
-        Return the side of the action defining ``self``.
-
         EXAMPLES::
 
             sage: SM = Partition([3,1,1]).specht_module(GF(3))
             sage: SM.side()
             'left'
+            sage: TestSuite(SM).run()
         """
-        return "left"
+        Representation_abstract.__init__(self, SGA.group(), "left", SGA)
 
     @cached_method
     def frobenius_image(self):
@@ -127,143 +116,8 @@ def frobenius_image(self):
         return s(p.sum(QQ(sum((elt * B[k])[k] for k in B.keys())) / la.centralizer_size() * p[la]
                        for elt, la in CCR))
 
-    # TODO: Move these methods up to methods of general representations
-
-    def representation_matrix(self, elt):
-        r"""
-        Return the matrix corresponding to the left action of the symmetric
-        group (algebra) element ``elt`` on ``self``.
-
-        EXAMPLES::
-
-            sage: SM = Partition([3,1,1]).specht_module(QQ)
-            sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
-            [-1  0  0  0  0  0]
-            [ 0  0  0 -1  0  0]
-            [ 1  0  0 -1  1  0]
-            [ 0 -1  0  0  0  0]
-            [ 1 -1  1  0  0  0]
-            [ 0 -1  0  1  0 -1]
-
-            sage: SGA = SymmetricGroupAlgebra(QQ, 5)
-            sage: SM = SGA.specht_module([(0,0), (0,1), (0,2), (1,0), (2,0)])
-            sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
-            [-1  0  0  1 -1  0]
-            [ 0  0  1  0 -1  1]
-            [ 0  1  0 -1  0  1]
-            [ 0  0  0  0 -1  0]
-            [ 0  0  0 -1  0  0]
-            [ 0  0  0  0  0 -1]
-            sage: SGA = SymmetricGroupAlgebra(QQ, 5)
-            sage: SM.representation_matrix(SGA([3,1,5,2,4]))
-            [ 0 -1  0  1  0 -1]
-            [ 0  0  0  0  0 -1]
-            [ 0  0  0 -1  0  0]
-            [ 0  0 -1  0  1 -1]
-            [ 1  0  0 -1  1  0]
-            [ 0  0  0  0  1  0]
-        """
-        return matrix(self.base_ring(), [(elt * b).to_vector() for b in self.basis()])
-
-    @cached_method
-    def character(self):
-        r"""
-        Return the character of ``self``.
 
-        EXAMPLES::
-
-            sage: SGA = SymmetricGroupAlgebra(QQ, 5)
-            sage: SM = SGA.specht_module([3,2])
-            sage: SM.character()
-            (5, 1, 1, -1, 1, -1, 0)
-            sage: matrix(SGA.specht_module(la).character() for la in Partitions(5))
-            [ 1  1  1  1  1  1  1]
-            [ 4  2  0  1 -1  0 -1]
-            [ 5  1  1 -1  1 -1  0]
-            [ 6  0 -2  0  0  0  1]
-            [ 5 -1  1 -1 -1  1  0]
-            [ 4 -2  0  1  1  0 -1]
-            [ 1 -1  1  1 -1 -1  1]
-
-            sage: SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(5))
-            sage: SM = SGA.specht_module([3,2])
-            sage: SM.character()
-            Character of Symmetric group of order 5! as a permutation group
-            sage: SM.character().values()
-            [5, 1, 1, -1, 1, -1, 0]
-            sage: matrix(SGA.specht_module(la).character().values() for la in reversed(Partitions(5)))
-            [ 1 -1  1  1 -1 -1  1]
-            [ 4 -2  0  1  1  0 -1]
-            [ 5 -1  1 -1 -1  1  0]
-            [ 6  0 -2  0  0  0  1]
-            [ 5  1  1 -1  1 -1  0]
-            [ 4  2  0  1 -1  0 -1]
-            [ 1  1  1  1  1  1  1]
-            sage: SGA.group().character_table()
-            [ 1 -1  1  1 -1 -1  1]
-            [ 4 -2  0  1  1  0 -1]
-            [ 5 -1  1 -1 -1  1  0]
-            [ 6  0 -2  0  0  0  1]
-            [ 5  1  1 -1  1 -1  0]
-            [ 4  2  0  1 -1  0 -1]
-            [ 1  1  1  1  1  1  1]
-        """
-        G = self._semigroup
-        B = self.basis()
-        chi = [sum((g * B[k])[k] for k in B.keys())
-               for g in G.conjugacy_classes_representatives()]
-        try:
-            return G.character(chi)
-        except AttributeError:
-            return vector(chi, immutable=True)
-
-    @cached_method
-    def brauer_character(self):
-        r"""
-        Return the Brauer character of ``self``.
-
-        EXAMPLES::
-
-            sage: SGA = SymmetricGroupAlgebra(GF(2), 5)
-            sage: SM = SGA.specht_module([3,2])
-            sage: SM.brauer_character()
-            (5, -1, 0)
-            sage: SM.simple_module().brauer_character()
-            (4, -2, -1)
-        """
-        from sage.rings.number_field.number_field import CyclotomicField
-        from sage.arith.functions import lcm
-        G = self._semigroup
-        p = self.base_ring().characteristic()
-        # We manually compute the order since a Permutation does not implement order()
-        chi = []
-        for g in G.conjugacy_classes_representatives():
-            if p.divides(lcm(g.cycle_type())):
-                # ignore the non-p-regular elements
-                continue
-            evals = self.representation_matrix(g).eigenvalues()
-            K = evals[0].parent()
-            val = 0
-            orders = {la: la.multiplicative_order() for la in evals if la != K.one()}
-            zetas = {o: CyclotomicField(o).gen() for o in orders.values()}
-            prims = {o: K.zeta(o) for o in orders.values()}
-            for la in evals:
-                if la == K.one():
-                    val += 1
-                    continue
-                o = la.multiplicative_order()
-                zeta = zetas[o]
-                prim = prims[o]
-                for deg in range(o):
-                    if prim ** deg == la:
-                        val += zeta ** deg
-                        break
-            chi.append(val)
-
-        return vector(chi, immutable=True)
-
-
-class SpechtModule(SubmoduleWithBasis, SymmetricGroupRepresentation, Representation_abstract):
+class SpechtModule(SymmetricGroupRepresentation, SubmoduleWithBasis):
     r"""
     A Specht module.
 
@@ -543,7 +397,7 @@ def _acted_upon_(self, x, self_on_left=False):
             return None
 
 
-class TabloidModule(SymmetricGroupRepresentation, Representation_abstract):
+class TabloidModule(SymmetricGroupRepresentation, CombinatorialFreeModule):
     r"""
     The vector space of all tabloids of a fixed shape with the natural
     symmetric group action.
@@ -800,7 +654,7 @@ def _acted_upon_(self, x, self_on_left):
                 return None
             P = self.parent()
             if x in P._semigroup_algebra:
-                return P.sum(c * (perm * self) for perm, c in x.monomial_coefficients().items())
+                return P.linear_combination((perm * self, c) for perm, c in x.monomial_coefficients().items())
             if x in P._semigroup_algebra.indices():
                 return P.element_class(P, {P._symmetric_group_action(T, x): c
                                            for T, c in self._monomial_coefficients.items()})
@@ -997,7 +851,7 @@ def simple_module(self):
         return SimpleModule(self)
 
 
-class MaximalSpechtSubmodule(SubmoduleWithBasis, SymmetricGroupRepresentation):
+class MaximalSpechtSubmodule(SymmetricGroupRepresentation, SubmoduleWithBasis):
     r"""
     The maximal submodule `U^{\lambda}` of the Specht module `S^{\lambda}`.
 
@@ -1082,7 +936,7 @@ def _repr_(self):
     Element = SpechtModule.Element
 
 
-class SimpleModule(QuotientModuleWithBasis, SymmetricGroupRepresentation):
+class SimpleModule(SymmetricGroupRepresentation, QuotientModuleWithBasis):
     r"""
     The simgle `S_n`-module associated with a partition `\lambda`.
 

src/sage/combinat/symmetric_group_algebra.py

@@ -16,7 +16,7 @@
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.combinat.permutation import Permutation, Permutations, from_permutation_group_element
 from sage.combinat.permutation_cython import (left_action_same_n, right_action_same_n)
-from sage.combinat.partition import _Partitions, Partitions_n
+from sage.combinat.partition import _Partitions, Partitions, Partitions_n
 from sage.combinat.tableau import Tableau, StandardTableaux_size, StandardTableaux_shape, StandardTableaux
 from sage.algebras.group_algebra import GroupAlgebra_class
 from sage.algebras.cellular_basis import CellularBasis
@@ -294,6 +294,10 @@ def __init__(self, R, W, category):
         GroupAlgebra_class.__init__(self, R, W, prefix='',
                                     latex_prefix='', category=category)
 
+        # Mixin class for extra methods for representations
+        from sage.combinat.specht_module import SymmetricGroupRepresentation
+        self._representation_mixin_class = SymmetricGroupRepresentation
+
     def _repr_(self):
         """
         Return a string representation of ``self``.
@@ -1623,6 +1627,28 @@ def specht_module_dimension(self, D):
         span_set = specht_module_spanning_set(D, self)
         return matrix(self.base_ring(), [v.to_vector() for v in span_set]).rank()
 
+    def simple_module_parameterization(self):
+        r"""
+        Return a parameterization of the simple modules of ``self``.
+
+        The symmetric group algebra of `S_n` over a field of characteristic `p`
+        has its simple modules indexed by all `p`-regular partitions of `n`.
+
+        EXAMPLES::
+
+            sage: SGA = SymmetricGroupAlgebra(QQ, 6)
+            sage: SGA.simple_module_parameterization()
+            Partitions of the integer 6
+
+            sage: SGA = SymmetricGroupAlgebra(GF(2), 6)
+            sage: SGA.simple_module_parameterization()
+            2-Regular Partitions of the integer 6
+        """
+        p = self.base_ring().characteristic()
+        if p > 0:
+            return Partitions(self.n, regular=p)
+        return Partitions_n(self.n)
+
     def simple_module(self, la):
         r"""
         Return the simple module of ``self`` indexed by the partition ``la``.
@@ -1671,11 +1697,17 @@ def simple_module_dimension(self, la):
         return simple_module_rank(la, self.base_ring())
 
     def garsia_procesi_module(self, la):
+        r"""
+        Return the :class:`Garsia-Procesi module
+        <sage.combinat.symmetric_group_representations.GarsiaProcesiModule>`
+        of ``self`` indexed by ``la``.
+
+        EXAMPLES::
+
+            sage: SGA = SymmetricGroupAlgebra(GF(2), 6)
+            sage: SGA.garsia_procesi_module(Partition([2,2,1,1]))
+            Garsia-Procesi module of shape [2, 2, 1, 1] over Finite Field of size 2
         """
-        Return the Garsia-Procesi module of ``self`` indexed by ``la``.
-        """
-        if sum(la) != self.n:
-            raise ValueError(f"{la} is not a partition of {self.n}")
         from sage.combinat.symmetric_group_representations import GarsiaProcesiModule
         return GarsiaProcesiModule(self, la)