sagemathgh-38598: Add reflection_index_set() and reflection() methods…

… for permutations

    
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->

These methods are general optional methods for complex reflections
groups. We provide an implementation for two implementations of the
symmetric group.

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [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.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->
    
URL: sagemath#38598
Reported by: Travis Scrimshaw
Reviewer(s): Amritanshu Prasad

src/sage/combinat/permutation.py

@@ -7722,6 +7722,52 @@ def simple_reflection(self, i):
         g[i] = i
         return self.element_class(self, g, check=False)
 
+    @cached_method
+    def reflection_index_set(self):
+        r"""
+        Return the index set of the reflections of ``self``.
+
+        .. SEEALSO::
+
+            - :meth:`reflection`
+            - :meth:`reflections`
+
+        EXAMPLES::
+
+            sage: P = Permutations(4)
+            sage: P.reflection_index_set()
+            ((1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4))
+        """
+        return tuple([tuple(c) for c in itertools.combinations(range(1, self.n+1), 2)])
+
+    def reflection(self, i):
+        r"""
+        Return the reflection indexed by ``i`` of ``self``.
+
+        This returns the permutation with cycle `i = (a, b)`.
+
+        .. SEEALSO::
+
+            - :meth:`reflections_index_set`
+            - :meth:`reflections`
+
+        EXAMPLES::
+
+            sage: P = Permutations(4)
+            sage: for i in P.reflection_index_set():
+            ....:     print('%s %s'%(i, P.reflection(i)))
+            (1, 2) [2, 1, 3, 4]
+            (1, 3) [3, 2, 1, 4]
+            (1, 4) [4, 2, 3, 1]
+            (2, 3) [1, 3, 2, 4]
+            (2, 4) [1, 4, 3, 2]
+            (3, 4) [1, 2, 4, 3]
+        """
+        data = list(range(1, self.n+1))
+        data[i[0]-1] = i[1]
+        data[i[1]-1] = i[0]
+        return self.element_class(self, data, check=False)
+
     class Element(Permutation):
         def has_left_descent(self, i, mult=None):
             r"""

src/sage/groups/perm_gps/permgroup_named.py

@@ -426,6 +426,50 @@ def simple_reflection(self, i):
         """
         return self([(i, self._domain[self._domain.index(i)+1])], check=False)
 
+    @cached_method
+    def reflection_index_set(self):
+        r"""
+        Return the index set of the reflections of ``self``.
+
+        .. SEEALSO::
+
+            - :meth:`reflection`
+            - :meth:`reflections`
+
+        EXAMPLES::
+
+            sage: S5 = SymmetricGroup(5)
+            sage: S5.reflection_index_set()
+            (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
+        """
+        return tuple(range(len(self.reflections())))
+
+    def reflection(self, i):
+        r"""
+        Return the `i`-th reflection of ``self``.
+
+        For `i` in `1,\dots,N`, this gives the `i`-th reflection of
+        ``self``.
+
+        .. SEEALSO::
+
+            - :meth:`reflections_index_set`
+            - :meth:`reflections`
+
+        EXAMPLES::
+
+            sage: S4 = SymmetricGroup(4)
+            sage: for i in S4.reflection_index_set():
+            ....:     print('%s %s'%(i, S4.reflection(i)))
+            0 (1,2)
+            1 (1,3)
+            2 (1,4)
+            3 (2,3)
+            4 (2,4)
+            5 (3,4)
+        """
+        return self.reflections()[i]
+
     def reflections(self):
         """
         Return the list of all reflections in ``self``.