eja: fix element powers.

We were using row-vector multiplication for powers (taken from the
superclass), but our vectors are column vectors. Oops. This broke
things when we assumed column vectors were being used, like when
we constructed a multiplication table. This commit fixes the powers
and adds/updates some tests.

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index a4e2ad04e097454ffabd36c9ccdcb3b83e936f8e..ef5249bc6abe4f9d4dd62bf0f5de07caa85fed26 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -101,6 +101,21 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             Jordan algebras are always power-associative; see for
             example Faraut and Koranyi, Proposition II.1.2 (ii).
+
+            .. WARNING:
+
+                We have to override this because our superclass uses row vectors
+                instead of column vectors! We, on the other hand, assume column
+                vectors everywhere.
+
+            EXAMPLES:
+
+                sage: set_random_seed()
+                sage: J = eja_ln(5)
+                sage: x = J.random_element()
+                sage: x.matrix()*x.vector() == (x**2).vector()
+                True
+
             """
             A = self.parent()
             if n == 0:
@@ -108,7 +123,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             elif n == 1:
                 return self
             else:
-                return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+                return A.element_class(A, (self.matrix()**(n-1))*self.vector())
 
 
         def span_of_powers(self):
@@ -171,14 +186,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 sage: x.subalgebra_generated_by().is_associative()
                 True
 
-            This is buggy right now::
+            Squaring in the subalgebra should be the same thing as
+            squaring in the superalgebra::
 
                 sage: J = eja_ln(5)
                 sage: x = J.random_element()
-                sage: x.matrix()*x.vector() == (x**2).vector() # works
-                True
                 sage: u = x.subalgebra_generated_by().random_element()
-                sage: u.matrix()*u.vector() == (u**2).vector() # busted
+                sage: u.matrix()*u.vector() == (u**2).vector()
                 True
 
             """
@@ -197,6 +211,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 # b1 is what we get if we apply that matrix to b1. The
                 # second row of the right multiplication matrix by b1
                 # is what we get when we apply that matrix to b2...
+                #
+                # IMPORTANT: this assumes that all vectors are COLUMN
+                # vectors, unlike our superclass (which uses row vectors).
                 for b_left in V.basis():
                     eja_b_left = self.parent()(b_left)
                     # Multiply in the original EJA, but then get the
@@ -363,6 +380,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             # subspace... or do we? Can't we just solve, knowing that
             # A(c) = u^(s+1) should have a solution in the big space,
             # too?
+            #
+            # Beware, solve_right() means that we're using COLUMN vectors.
+            # Our FiniteDimensionalAlgebraElement superclass uses rows.
             u_next = u**(s+1)
             A = u_next.matrix()
             c_coordinates = A.solve_right(u_next.vector())
@@ -372,10 +392,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             #
             # We need the basis for J, but as elements of the parent algebra.
             #
-            #
-            # TODO: this is buggy, but it's probably because the
-            # multiplication table for the subalgebra is wrong! The
-            # matrices should be symmetric I bet.
             basis = [self.parent(v) for v in V.basis()]
             return self.parent().linear_combination(zip(c_coordinates, basis))