diff --git a/src/sage/algebras/fusion_rings/fusion_ring.py b/src/sage/algebras/fusion_rings/fusion_ring.py
index 915ca62e73b..8f4ed4b3cb7 100644
--- a/src/sage/algebras/fusion_rings/fusion_ring.py
+++ b/src/sage/algebras/fusion_rings/fusion_ring.py
@@ -401,7 +401,7 @@ def test_braid_representation(self, max_strands=6, anyon=None):
             sage: A21 = FusionRing("A2", 1)
             sage: A21.test_braid_representation(max_strands=4)
             True
-            sage: F41 = FusionRing("F4", 1)             # long time
+            sage: F41 = FusionRing("F4", 1)            # long time
             sage: F41.test_braid_representation()      # long time
             True
         """
diff --git a/src/sage/combinat/root_system/weyl_characters.py b/src/sage/combinat/root_system/weyl_characters.py
index f306df7eddf..b65e3eff5e5 100644
--- a/src/sage/combinat/root_system/weyl_characters.py
+++ b/src/sage/combinat/root_system/weyl_characters.py
@@ -154,9 +154,11 @@ def next_level(wt):
         else:
             B = self._space
 
-        cat = AlgebrasWithBasis(base_ring).Subobjects()
+        cat = AlgebrasWithBasis(base_ring).Commutative()
         if k is None:
-            cat = cat.Graded()
+            cat = cat.Subobjects().Graded()
+        else:
+            cat = cat.FiniteDimensional()
         CombinatorialFreeModule.__init__(self, base_ring, B, category=cat)
 
         # Register the embedding of self into ambient as a coercion
@@ -537,7 +539,7 @@ def _product_helper(self, d1, b):
                 d[g] = d.get(g,0) + d1[k]
             elif epsilon == -1:
                 d[g] = d.get(g,0) - d1[k]
-        return self._from_dict(d)
+        return self._from_dict(d, coerce=True)
 
     def dot_reduce(self, a):
         r"""
@@ -715,8 +717,15 @@ def _demazure_helper(self, dd, word="long", debug=False):
         alpha = self._space.simple_roots()
         r = self.rank()
         cm = {}
+        supp = []
         for i in index_set:
-            cm[i] = tuple(int(alpha[i].inner_product(alphacheck[j])) for j in index_set)
+            temp = []
+            cm[i] = [0] * r
+            for ind,j in enumerate(index_set):
+                cm[i][ind] = int(alpha[i].inner_product(alphacheck[j]))
+                if cm[i][ind]:
+                    temp.append(ind)
+            supp.append(temp)
             if debug:
                 print("cm[%s]=%s" % (i, cm[i]))
         accum = dd
@@ -733,15 +742,21 @@ def _demazure_helper(self, dd, word="long", debug=False):
                 if coroot >= 0:
                     mu = v
                     for j in range(coroot+1):
-                        next[mu] = next.get(mu,0)+accum[v]
+                        next[mu] = next.get(mu,0) + accum[v]
                         if debug:
                             print("     mu=%s, next[mu]=%s" % (mu, next[mu]))
-                        mu = tuple(mu[k] - cm[i][k] for k in range(r))
+                        mu = list(mu)
+                        for k in supp[i-1]:
+                            mu[k] -= cm[i][k]
+                        mu = tuple(mu)
                 else:
                     mu = v
                     for j in range(-1-coroot):
-                        mu = tuple(mu[k] + cm[i][k] for k in range(r))
-                        next[mu] = next.get(mu,0)-accum[v]
+                        mu = list(mu)
+                        for k in supp[i-1]:
+                            mu[k] += cm[i][k]
+                        mu = tuple(mu)
+                        next[mu] = next.get(mu,0) - accum[v]
                         if debug:
                             print("     mu=%s, next[mu]=%s" % (mu, next[mu]))
             accum = {}
@@ -1779,7 +1794,7 @@ def __init__(self, parent, prefix):
                 # TODO: this only works for irreducible Cartan types!
                 prefix = (self._cartan_type[0].lower() + str(self._rank))
         self._prefix = prefix
-        category = AlgebrasWithBasis(self._base_ring)
+        category = AlgebrasWithBasis(self._base_ring).Commutative()
         CombinatorialFreeModule.__init__(self, self._base_ring, self._space, category=category)
 
     def _repr_(self):