sagemathgh-36790: Combinat/sga murphy basis

    
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We implement the Murphy basis of the symmetric group algebra (as defined
by Dipper-James-Mathas) and make this the default cellular basis over
positive characteristic since it is (significantly) faster to compute
the change of basis necessary.

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### ⌛ Dependencies

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- sagemath#36718 Uses the KL basis implementation

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URL: sagemath#36790
Reported by: Travis Scrimshaw
Reviewer(s): Frédéric Chapoton

src/doc/en/reference/references/index.rst

@@ -4520,6 +4520,10 @@ REFERENCES:
 .. [Mat2002] Jiří Matousek, "Lectures on Discrete Geometry", Springer,
              2002
 
+.. [Mathas2004] Andrew Mathas.
+                *Matrix units and generic degrees for the Ariki-Koike algebras*.
+                J. Algebra. **281** (2004) pp. 695-730.
+
 .. [Mas1995] Mason, Geoffrey. *The quantum double of a finite group and its role
              in conformal field theory*. Groups '93 Galway/St. Andrews, Vol. 2,
              405-417, London Math. Soc. Lecture Note Ser., 212, Cambridge, 1995.

src/sage/algebras/cellular_basis.py

@@ -164,7 +164,7 @@ class CellularBasis(CombinatorialFreeModule):
          - C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
          + C([3], [[1, 2, 3]], [[1, 2, 3]])
     """
-    def __init__(self, A):
+    def __init__(self, A, to_algebra=None, from_algebra=None, **kwargs):
         r"""
         Initialize ``self``.
 
@@ -181,21 +181,26 @@ def __init__(self, A):
 
         # TODO: Use instead A.category().Realizations() so
         #   operations are defined by coercion?
+        prefix = kwargs.pop('prefix', 'C')
         cat = Algebras(A.category().base_ring()).FiniteDimensional().WithBasis().Cellular()
         CombinatorialFreeModule.__init__(self, A.base_ring(), I,
-                                         prefix='C', bracket=False,
-                                         category=cat)
+                                         prefix=prefix, bracket=False,
+                                         category=cat, **kwargs)
 
         # Register coercions
-        if A._to_cellular_element is not NotImplemented:
-            to_cellular = A.module_morphism(A._to_cellular_element, codomain=self,
+        if from_algebra is None:
+            from_algebra = A._to_cellular_element
+        if to_algebra is None:
+            to_algebra = A._from_cellular_index
+        if from_algebra is not NotImplemented:
+            to_cellular = A.module_morphism(from_algebra, codomain=self,
                                             category=cat)
-        if A._from_cellular_index is NotImplemented:
+        if to_algebra is NotImplemented:
             from_cellular = ~to_cellular
         else:
-            from_cellular = self.module_morphism(A._from_cellular_index, codomain=A,
+            from_cellular = self.module_morphism(to_algebra, codomain=A,
                                                  category=cat)
-            if A._to_cellular_element is NotImplemented:
+            if from_algebra is NotImplemented:
                 to_cellular = ~from_cellular
         to_cellular.register_as_coercion()
         from_cellular.register_as_coercion()