Merge branch 'u/mantepse/implement_derivatives_of_lazy_series' of tra…

…c.sagemath.org:sage into t/34422/implement_functorial_composition_of_lazy_symmetric_functiosn

src/sage/categories/commutative_algebras.py

@@ -10,8 +10,11 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.misc.cachefunc import cached_method
 from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
 from sage.categories.algebras import Algebras
+from sage.categories.commutative_rings import CommutativeRings
+from sage.categories.tensor import TensorProductsCategory
 
 class CommutativeAlgebras(CategoryWithAxiom_over_base_ring):
     """
@@ -36,7 +39,7 @@ class CommutativeAlgebras(CategoryWithAxiom_over_base_ring):
         True
         sage: TestSuite(CommutativeAlgebras(ZZ)).run()
 
-    Todo:
+    .. TODO::
 
      - product   ( = Cartesian product)
      - coproduct ( = tensor product over base ring)
@@ -58,3 +61,32 @@ def __contains__(self, A):
         """
         return super().__contains__(A) or \
             (A in Algebras(self.base_ring()) and hasattr(A, "is_commutative") and A.is_commutative())
+
+    class TensorProducts(TensorProductsCategory):
+        """
+        The category of commutative algebras constructed by tensor product of commutative algebras.
+        """
+
+        @cached_method
+        def extra_super_categories(self):
+            """
+            EXAMPLES::
+
+                sage: Algebras(QQ).Commutative().TensorProducts().extra_super_categories()
+                [Category of commutative rings]
+                sage: Algebras(QQ).Commutative().TensorProducts().super_categories()
+                [Category of tensor products of algebras over Rational Field,
+                 Category of commutative algebras over Rational Field]
+
+            TESTS::
+
+                sage: X = algebras.Shuffle(QQ, 'ab')
+                sage: Y = algebras.Shuffle(QQ, 'bc')
+                sage: X in Algebras(QQ).Commutative()
+                True
+                sage: T = tensor([X, Y])
+                sage: T in CommutativeRings()
+                True
+            """
+            return [CommutativeRings()]
+

src/sage/categories/graded_algebras_with_basis.py

@@ -126,6 +126,23 @@ def free_graded_module(self, generator_degrees, names=None):
                 from sage.modules.fp_graded.free_module import FreeGradedModule
                 return FreeGradedModule(self, generator_degrees, names=names)
 
+        def formal_series_ring(self):
+            r"""
+            Return the completion of all formal linear combinations of
+            ``self`` with finite linear combinations in each homogeneous
+            degree (computed lazily).
+
+            EXAMPLES::
+
+                sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
+                sage: S = NCSF.Complete()
+                sage: L = S.formal_series_ring()
+                sage: L
+                Lazy completion of Non-Commutative Symmetric Functions over
+                 the Rational Field in the Complete basis
+            """
+            from sage.rings.lazy_series_ring import LazyCompletionGradedAlgebra
+            return LazyCompletionGradedAlgebra(self)
 
     class ElementMethods:
         pass

src/sage/combinat/partition.py

@@ -1057,6 +1057,20 @@ def __truediv__(self, p):
 
         return SkewPartition([self[:], p])
 
+    def stretch(self, k):
+        """
+        Return the partition obtained by multiplying each part with the
+        given number.
+
+        EXAMPLES::
+
+            sage: p = Partition([4,2,2,1,1])
+            sage: p.stretch(3)
+            [12,6,6,3,3]
+
+        """
+        return _Partitions([k * p for p in self])
+
     def power(self, k):
         r"""
         Return the cycle type of the `k`-th power of any permutation

src/sage/combinat/sf/powersum.py

@@ -476,8 +476,8 @@ def frobenius(self, n):
 
                 :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.plethysm`
             """
-            dct = {Partition([n * i for i in lam]): coeff
-                   for (lam, coeff) in self.monomial_coefficients().items()}
+            dct = {lam.stretch(n): coeff
+                   for lam, coeff in self.monomial_coefficients().items()}
             return self.parent()._from_dict(dct)
 
         adams_operation = frobenius