sagemathgh-38446: Implement basic multivariate polynomial species

    
Related to sagemath#30727.

We implement basic functionality for multivariate polynomial species,
using its representation as a pair of a permutation group and a mapping
between the domain of the permutation group and some variables. We
provide addition, multiplication, and (partitional) composition (for
some special cases). We also allow it to be constructed as a group
action (or a sequence thereof). Atomic and molecular decompositions are
automatically computed thanks to sagemath#38371.

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#38446
Reported by: Mainak Roy
Reviewer(s): Mainak Roy, Martin Rubey, Travis Scrimshaw

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

@@ -104,6 +104,10 @@ REFERENCES:
              graphs and isoperimetric inequalities*, The Annals of Probability
              32 (2004), no.  3A, 1727-1745.
 
+.. [ALL2002] P. Auger, G. Labelle and P. Leroux, *Combinatorial
+             addition formulas and applications*, Advances in Applied
+             Mathematics 28 (2002) 302-342.
+
 .. [ASV2020] Federico Ardila, Mariel Supina, and Andrés R. Vindas-Meléndez,
          *The Equivariant Ehrhart Theory of the Permutahedron*,
          Proc. Amer. Math. Soc. Volume 148, Number 12, 2020, pp. 5091--5107.

src/sage/categories/filtered_modules_with_basis.py

@@ -852,7 +852,7 @@ def homogeneous_degree(self):
                 raise ValueError("the zero element does not have a well-defined degree")
             if not self.is_homogeneous():
                 raise ValueError("element is not homogeneous")
-            return self.parent().degree_on_basis(self.leading_support())
+            return self.parent().degree_on_basis(next(iter(self.support())))
 
         # default choice for degree; will be overridden as necessary
         degree = homogeneous_degree

src/sage/groups/perm_gps/permgroup.py

@@ -2733,6 +2733,25 @@ def conjugate(self, g):
             raise TypeError("{0} does not convert to a permutation group element".format(g))
         return PermutationGroup(gap_group=libgap.ConjugateGroup(self, g))
 
+    def are_conjugate(self, H1, H2):
+        r"""
+        Return whether ``H1`` and ``H2`` are conjugate subgroups in ``G``.
+
+        EXAMPLES::
+
+            sage: G = SymmetricGroup(3)
+            sage: H1 = PermutationGroup([(1,2)])
+            sage: H2 = PermutationGroup([(2,3)])
+            sage: G.are_conjugate(H1, H2)
+            True
+            sage: G = SymmetricGroup(4)
+            sage: H1 = PermutationGroup([[(1,3),(2,4)], [(1,2),(3,4)]])
+            sage: H2 = PermutationGroup([[(1,2)], [(1,2),(3,4)]])
+            sage: G.are_conjugate(H1, H2)
+            False
+        """
+        return libgap.IsConjugate(self, H1, H2).sage()
+
     def direct_product(self, other, maps=True):
         """
         Wraps GAP's ``DirectProduct``, ``Embedding``, and ``Projection``.