Trac #31276: Tensor Product Method for FiniteRankFreeModule

We introduce the methods `tensor_product` and `tensor_power` as per
#30373.

Things that are probably nice to get to work:

{{{
sage: M = FiniteRankFreeModule(QQ, 2)
sage: M.tensor_product(M)
sage: M.tensor_product(M.tensor_module(1,2))
sage: M.tensor_module(1,2).tensor_product(M)
sage: M.tensor_module(1,1).tensor_product(M.tensor_module(1,2))
sage: M.tensor_power(3)
sage: M.tensor_module(1,2).tensor_power(3)
}}}

Not in this ticket: products between different underlying modules such
as:
{{{
sage: M = FiniteRankFreeModule(QQ, 2)
sage: N = FiniteRankFreeModule(QQ, 3)
sage: M.tensor_product(N)
sage: N.tensor_product(M.tensor_module(1,2))
sage: N.tensor_module(1,2).tensor_product(M)
sage: N.tensor_module(1,1).tensor_product(M.tensor_module(1,2))
}}}

URL: https://trac.sagemath.org/31276
Reported by: gh-mjungmath
Ticket author(s): Matthias Koeppe
Reviewer(s): Eric Gourgoulhon

src/sage/tensor/modules/finite_rank_free_module.py

@@ -2857,3 +2857,68 @@ def identity_map(self, name='Id', latex_name=None):
                     latex_name = name
                 self._identity_map.set_name(name=name, latex_name=latex_name)
         return self._identity_map
+
+    def base_module(self):
+        r"""
+        Return the free module on which ``self`` is constructed, namely ``self`` itself.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: M.base_module() is M
+            True
+
+        """
+        return self
+
+    def tensor_type(self):
+        r"""
+        Return the tensor type of ``self``, the pair `(1, 0)`.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3)
+            sage: M.tensor_type()
+            (1, 0)
+
+        """
+        return (1, 0)
+
+    def tensor_power(self, n):
+        r"""
+        Return the ``n``-fold tensor product of ``self``.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(QQ, 2)
+            sage: M.tensor_power(3)
+            Free module of type-(3,0) tensors on the 2-dimensional vector space over the Rational Field
+            sage: M.tensor_module(1,2).tensor_power(3)
+            Free module of type-(3,6) tensors on the 2-dimensional vector space over the Rational Field
+        """
+        tensor_type = self.tensor_type()
+        return self.base_module().tensor_module(n * tensor_type[0], n * tensor_type[1])
+
+    def tensor_product(self, *others):
+        r"""
+        Return the tensor product of ``self`` and ``others``.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(QQ, 2)
+            sage: M.tensor_product(M)
+            Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
+            sage: M.tensor_product(M.tensor_module(1,2))
+            Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
+            sage: M.tensor_module(1,2).tensor_product(M)
+            Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
+            sage: M.tensor_module(1,1).tensor_product(M.tensor_module(1,2))
+            Free module of type-(2,3) tensors on the 2-dimensional vector space over the Rational Field
+
+        """
+        from sage.modules.free_module_element import vector
+        base_module = self.base_module()
+        if not all(module.base_module() == base_module for module in others):
+            raise NotImplementedError('all factors must be tensor modules over the same base module')
+        tensor_type = sum(vector(module.tensor_type()) for module in [self] + list(others))
+        return base_module.tensor_module(*tensor_type)