Fixing doctest failures in the thematic tutorials.

src/doc/en/thematic_tutorials/lie/weyl_groups.rst

@@ -77,39 +77,44 @@ for the simple reflections, that is, the case where `\alpha` is a simple
 root. The reflections are just the conjugates of the simple reflections.
 
 The reflections are the keys in a finite family, which is a wrapper
-around a python dictionary. The values are the positive roots, so
+around a python dictionary. The keys are the positive roots, so
 given a reflection, you can look up the corresponding root. If you
 want a list of all reflections, use the method ``keys`` for the
 family of reflections::
 
     sage: W = WeylGroup("B3",prefix="s")
-    sage: [s1,s2,s3] = W.simple_reflections()
     sage: ref = W.reflections(); ref
-    Finite family {s1*s2*s1: (1, 0, -1), s2: (0, 1, -1), s3*s2*s3: (0, 1, 1),
-    s3*s1*s2*s3*s1: (1, 0, 1), s1: (1, -1, 0), s2*s3*s1*s2*s3*s1*s2: (1, 1, 0),
-    s1*s2*s3*s2*s1: (1, 0, 0), s2*s3*s2: (0, 1, 0), s3: (0, 0, 1)}
-    sage: ref[s3*s2*s3]
-    (0, 1, 1)
-    sage: ref.keys()
-    [s1*s2*s1, s2, s3*s2*s3, s2*s3*s1*s2*s3*s1*s2, s1, s3*s1*s2*s3*s1, s1*s2*s3*s2*s1, s2*s3*s2, s3]
-
-If instead you want a dictionary whose keys are the roots and whose
-values are the reflections, you may use the inverse family::
-
-    sage: from pprint import pprint
-    sage: W = WeylGroup("B3",prefix="s")
-    sage: [s1,s2,s3] = W.simple_reflections()
-    sage: altref = W.reflections().inverse_family()
-    sage: pprint(altref)
     Finite family {(1, 0, 0): s1*s2*s3*s2*s1, (0, 1, 1): s3*s2*s3,
                    (0, 1, -1): s2, (0, 0, 1): s3, (1, -1, 0): s1,
                    (1, 1, 0): s2*s3*s1*s2*s3*s1*s2, (1, 0, -1): s1*s2*s1,
                    (1, 0, 1): s3*s1*s2*s3*s1, (0, 1, 0): s2*s3*s2}
     sage: [a1,a2,a3] = W.domain().simple_roots()
     sage: a1+a2+a3
     (1, 0, 0)
-    sage: altref[a1+a2+a3]
+    sage: ref[a1+a2+a3]
     s1*s2*s3*s2*s1
+    sage: list(ref)
+    [s1*s2*s3*s2*s1, s3*s2*s3, s2, s3, s1, s2*s3*s1*s2*s3*s1*s2,
+     s1*s2*s1, s3*s1*s2*s3*s1, s2*s3*s2]
+
+If instead you want a dictionary whose keys are the reflections
+and whose values are the roots, you may use the inverse family::
+
+    sage: from pprint import pprint
+    sage: W = WeylGroup("B3",prefix="s")
+    sage: [s1,s2,s3] = W.simple_reflections()
+    sage: altref = W.reflections().inverse_family()
+    sage: pprint(altref)
+    Finite family {s1*s2*s1: (1, 0, -1), s2: (0, 1, -1), s3*s2*s3: (0, 1, 1),
+                   s1*s2*s3*s2*s1: (1, 0, 0), s1: (1, -1, 0),
+                   s2*s3*s1*s2*s3*s1*s2: (1, 1, 0), s3*s1*s2*s3*s1: (1, 0, 1),
+                   s2*s3*s2: (0, 1, 0), s3: (0, 0, 1)}
+    sage: altref[s3*s2*s3]
+    (0, 1, 1)
+
+.. NOTE::
+
+    The behaviour of this function was changed in :trac:`20027`.
 
 The Weyl group is implemented as a GAP matrix group. You therefore can
 display its character table. The character table is returned as a