gh-36748: Implement Specht modules in the tabloid basis

    
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This provides an implementation of the Specht modules over arbitrary
fields in the standard tableaux basis. To do so, we also implement the
tabloid module, where the tabloids are implemented as
`OrderedSetsPartitions`. Using this, we can construct the maximal
submodule and simple submodule from a Specht module.

We also implement a method to compute the Brauer character of the
representations.

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### ⌛ Dependencies

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- #36724: Trivial conflict with working on the same files.
- #36985: So the 0 dimensional tests will pass.

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URL: #36748
Reported by: Travis Scrimshaw
Reviewer(s): Darij Grinberg, Frédéric Chapoton, Martin Rubey, Travis Scrimshaw

src/sage/combinat/algebraic_combinatorics.py

@@ -12,7 +12,8 @@
 ----------------------------------------
 
 - :ref:`sage.combinat.catalog_partitions`
-- :class:`~sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPattern`, :class:`~sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns`
+- :class:`~sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPattern`,
+  :class:`~sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns`
 - :class:`~sage.combinat.knutson_tao_puzzles.KnutsonTaoPuzzleSolver`
 
 Groups and Algebras
@@ -39,7 +40,7 @@
 - :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.specht_module`
 - :ref:`sage.combinat.yang_baxter_graph`
 - :ref:`sage.combinat.hall_polynomial`
 - :ref:`sage.combinat.key_polynomial`

src/sage/combinat/catalog_partitions.py

@@ -8,6 +8,7 @@
 - :ref:`sage.combinat.skew_partition`
 - :ref:`sage.combinat.partition_tuple`
 - :ref:`sage.combinat.superpartition`
+- :ref:`sage.combinat.tableau`
 - :ref:`sage.combinat.tableau_tuple`
 - :ref:`sage.combinat.skew_tableau`
 - :ref:`sage.combinat.ribbon`

src/sage/combinat/composition.py

@@ -45,6 +45,9 @@
 from sage.combinat.combinatorial_map import combinatorial_map
 from sage.misc.persist import register_unpickle_override
 
+from sage.misc.lazy_import import lazy_import
+lazy_import("sage.combinat.partition", "Partition")
+
 
 class Composition(CombinatorialElement):
     r"""
@@ -1186,7 +1189,6 @@ def to_partition(self):
             sage: Composition([]).to_partition()                                        # needs sage.combinat
             []
         """
-        from sage.combinat.partition import Partition
         return Partition(sorted(self, reverse=True))
 
     def to_skew_partition(self, overlap=1):
@@ -1753,8 +1755,10 @@ def _element_constructor_(self, lst) -> Composition:
             sage: P = Compositions()
             sage: P([3,3,1]) # indirect doctest
             [3, 3, 1]
+            sage: P(Partition([5,2,1]))
+            [5, 2, 1]
         """
-        if isinstance(lst, Composition):
+        if isinstance(lst, (Composition, Partition)):
             lst = list(lst)
         elt = self.element_class(self, lst)
         if elt not in self:
@@ -1774,7 +1778,7 @@ def __contains__(self, x) -> bool:
             sage: [0,0] in Compositions()
             True
         """
-        if isinstance(x, Composition):
+        if isinstance(x, (Composition, Partition)):
             return True
         elif isinstance(x, list):
             for i in x:

src/sage/combinat/partition.py

@@ -5495,9 +5495,8 @@ def specht_module(self, base_ring=None):
         EXAMPLES::
 
             sage: SM = Partition([2,2,1]).specht_module(QQ); SM
-            Specht module of [(0, 0), (0, 1), (1, 0), (1, 1), (2, 0)] over Rational Field
-            sage: s = SymmetricFunctions(QQ).s()
-            sage: s(SM.frobenius_image())                                               # needs sage.modules
+            Specht module of [2, 2, 1] over Rational Field
+            sage: SM.frobenius_image()                                               # needs sage.modules
             s[2, 2, 1]
         """
         from sage.combinat.specht_module import SpechtModule
@@ -5570,6 +5569,24 @@ def simple_module_dimension(self, base_ring=None):
         from sage.combinat.specht_module import simple_module_rank
         return simple_module_rank(self, base_ring)
 
+    def tabloid_module(self, base_ring=None):
+        r"""
+        Return the tabloid module corresponding to ``self``.
+
+        EXAMPLES::
+
+            sage: TM = Partition([2,2,1]).tabloid_module(QQ); TM
+            Tabloid module of [2, 2, 1] over Rational Field
+            sage: TM.frobenius_image()
+            s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1] + s[5]
+        """
+        from sage.combinat.specht_module import TabloidModule
+        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 TabloidModule(R, self)
 
 ##############
 # Partitions #

src/sage/combinat/permutation.py

@@ -1263,7 +1263,7 @@ def __mul__(self, rp):
             sage: p213 * SGA.an_element()
             3*[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + 2*[3, 1, 2]
             sage: p213 * SM.an_element()
-            2*B[0] - 4*B[1]
+            2*S[[1, 2], [3]] - 4*S[[1, 3], [2]]
         """
         if not isinstance(rp, Permutation) and isinstance(rp, Element):
             return get_coercion_model().bin_op(self, rp, operator.mul)
@@ -7342,6 +7342,19 @@ def cardinality(self):
         """
         return factorial(self.n)
 
+    @cached_method
+    def gens(self) -> tuple:
+        r"""
+        Return a set of generators for ``self`` as a group.
+
+        EXAMPLES::
+
+            sage: P4 = Permutations(4)
+            sage: P4.gens()
+            ([2, 1, 3, 4], [1, 3, 2, 4], [1, 2, 4, 3])
+        """
+        return tuple(self.group_generators())
+
     def degree(self):
         """
         Return the degree of ``self``.

src/sage/combinat/skew_tableau.py

@@ -6,11 +6,11 @@
 - Mike Hansen: Initial version
 - Travis Scrimshaw, Arthur Lubovsky (2013-02-11):
   Factored out ``CombinatorialClass``
-- Trevor K. Karn (2022-08-03): added `backward_slide`
+- Trevor K. Karn (2022-08-03): added ``backward_slide``
 """
 # ****************************************************************************
 #       Copyright (C) 2007 Mike Hansen <[email protected]>,
-#       Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu>
+#       Copyright (C) 2013 Travis Scrimshaw <tcscrims at gmail.com>
 #       Copyright (C) 2013 Arthur Lubovsky
 #
 #  Distributed under the terms of the GNU General Public License (GPL)