Merge branch 'public/ticket/11726' into raise-plus

Conflicts:
	src/sage/rings/polynomial/laurent_polynomial_ring.py
resolve merge conflicts with #11726

Author: wluebbe

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

@@ -15,9 +15,9 @@ Kazhdan-Lusztig polynomials as follows::
     sage: P.<q> = LaurentPolynomialRing(QQ)
     sage: KL = KazhdanLusztigPolynomial(W,q)
     sage: KL.R(s2, s2*s1*s3*s2)
-    q^3 - 3*q^2 + 3*q - 1
+    -1 + 3*q - 3*q^2 + q^3
     sage: KL.P(s2, s2*s1*s3*s2)
-    q + 1
+    1 + q
 
 Thus we have the Kazhdan-Lusztig R and P polynomials.
 

src/sage/algebras/hall_algebra.py

@@ -194,11 +194,11 @@ class HallAlgebra(CombinatorialFreeModule):
         sage: H = HallAlgebra(R, q)
         sage: I = H.monomial_basis()
         sage: hi = H(I[2,1]); hi
-        H[2, 1] + (q^2+q+1)*H[1, 1, 1]
+        H[2, 1] + (1+q+q^2)*H[1, 1, 1]
         sage: hi.parent() is H
         True
         sage: h22 = H[2]*H[2]; h22
-        H[4] + (q-1)*H[3, 1] + (q^2+q)*H[2, 2]
+        H[4] - (1-q)*H[3, 1] + (q+q^2)*H[2, 2]
         sage: h22.parent() is H
         True
         sage: e = SymmetricFunctions(R).e()
@@ -339,10 +339,10 @@ def coproduct_on_basis(self, la):
             sage: R.<q> = LaurentPolynomialRing(ZZ)
             sage: H = HallAlgebra(R, q)
             sage: H.coproduct_on_basis(Partition([2]))
-            H[] # H[2] + (1-q^-1)*H[1] # H[1] + H[2] # H[]
+            H[] # H[2] - (q^-1-1)*H[1] # H[1] + H[2] # H[]
             sage: H.coproduct_on_basis(Partition([2,1]))
-            H[] # H[2, 1] + (1-q^-2)*H[1] # H[1, 1] + (q^-1)*H[1] # H[2]
-             + (1-q^-2)*H[1, 1] # H[1] + (q^-1)*H[2] # H[1] + H[2, 1] # H[]
+            H[] # H[2, 1] - (q^-2-1)*H[1] # H[1, 1] + (q^-1)*H[1] # H[2]
+             - (q^-2-1)*H[1, 1] # H[1] + (q^-1)*H[2] # H[1] + H[2, 1] # H[]
         """
         S = self.tensor_square()
         if all(x == 1 for x in la):
@@ -373,7 +373,7 @@ def antipode_on_basis(self, la):
             sage: H.antipode_on_basis(Partition([1,1]))
             (q^-1)*H[2] + (q^-1)*H[1, 1]
             sage: H.antipode_on_basis(Partition([2]))
-            (-q^-1)*H[2] + (q-q^-1)*H[1, 1]
+            -(q^-1)*H[2] - (q^-1-q)*H[1, 1]
         """
         if all(x == 1 for x in la):
             r = len(la)
@@ -541,9 +541,9 @@ class HallAlgebraMonomials(CombinatorialFreeModule):
         (q^-8)*e[4, 2, 2, 1]
         sage: HLP = Sym.hall_littlewood(q).P()
         sage: H(I[2,1])
-        H[2, 1] + (q^2+q+1)*H[1, 1, 1]
+        H[2, 1] + (1+q+q^2)*H[1, 1, 1]
         sage: HLP(e[2,1])
-        (q^2+q+1)*HLP[1, 1, 1] + HLP[2, 1]
+        (1+q+q^2)*HLP[1, 1, 1] + HLP[2, 1]
         sage: all( e(H[lam]) == q**-sum([i * x for i, x in enumerate(lam)])
         ....:          * e(HLP[lam]).map_coefficients(lambda p: p(q**(-1)))
         ....:      for lam in Partitions(4) )
@@ -706,7 +706,7 @@ def antipode_on_basis(self, a):
             sage: R.<q> = LaurentPolynomialRing(ZZ)
             sage: I = HallAlgebra(R, q).monomial_basis()
             sage: I.antipode_on_basis(Partition([2,1]))
-            (-q^-1)*I[1, 1, 1] + I[2, 1]
+            -(q^-1)*I[1, 1, 1] + I[2, 1]
         """
         H = HallAlgebra(self.base_ring(), self._q)
         cur = self.one()