Adding some systematic doctests and one detail.

Author: tscrim

src/sage/combinat/root_system/reflection_group_complex.py

@@ -299,8 +299,6 @@ def __init__(self, W_types, index_set=None, hyperplane_index_set=None, reflectio
                                           canonicalize=False,
                                           category=category)
 
-        self._length_of_permutation_representation = len(Permutation(self.gens()[0]))
-
         l_set = list(range(1, len(self.gens()) + 1))
         if self._index_set is None:
             self._index_set = tuple(l_set)

src/sage/combinat/root_system/reflection_group_element.pyx

@@ -175,6 +175,11 @@ cdef class ComplexReflectionGroupElement(PermutationGroupElement):
             [1, 2, 2, 1, 1] 5
             [1, 1, 2, 1, 1, 2] 6
             [1, 1, 2, 2, 1, 1] 6
+
+            sage: data = {w: (len(w.reduced_word()), w.length())    # optional - gap3
+            ....:         for w in W.iteration_tracking_words()}
+            sage: for w in W:                                       # optional - gap3
+            ....:     assert data[w] == (w.length(), w.length()), w
         """
         return ZZ(len(self.reduced_word()))
 
@@ -192,9 +197,12 @@ cdef class ComplexReflectionGroupElement(PermutationGroupElement):
         EXAMPLES::
 
             sage: W = ReflectionGroup((3,1,2))            # optional - gap3
+            sage: data = {w: [w.to_matrix(), w.to_matrix(on_space="dual")] for w in W}  # optional - gap3
             sage: for w in W.iteration_tracking_words():  # optional - gap3
             ....:     w.reduced_word()                    # optional - gap3
-            ....:     [w.to_matrix(), w.to_matrix(on_space="dual")] # optional - gap3
+            ....:     mats = [w.to_matrix(), w.to_matrix(on_space="dual")]  # optional - gap3
+            ....:     mats
+            ....:     assert data[w] == mats
             []
             [
             [1 0]  [1 0]
@@ -564,6 +572,10 @@ cdef class ComplexReflectionGroupElement(PermutationGroupElement):
             [2/3, 0]
             [1/2, 1/2]
             [1/4, 3/4]
+
+            sage: data = {w: w.reflection_eigenvalues() for w in W}  # optional - gap3
+            sage: all(w.reflection_eigenvalues() == data[w] for w in W.iteration_tracking_words())  # optional - gap3
+            True
         """
         return self._parent.reflection_eigenvalues(self, is_class_representative=is_class_representative)
 
@@ -665,6 +677,10 @@ cdef class ComplexReflectionGroupElement(PermutationGroupElement):
              [ 2/3*E(3) - 2/3*E(3)^2  1/3*E(3) - 1/3*E(3)^2],
              [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2]
              [-2/3*E(3) + 2/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2]]
+
+            sage: data = {w: w.galois_conjugates() for w in W}      # optional - gap3
+            sage: all(w.galois_conjugates() == data[w] for w in W.iteration_tracking_words())  # optional - gap3
+            True
         """
         rk = self._parent.rank()
         M = self.to_matrix().list()