X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=3124132b61e5f196268cb027ce1bd26028aaa942;hb=4d6b173a3ef90c7243e15febf60b0dc45282fd14;hp=e2d644a903d8eca09a585064f96e31278197e4d6;hpb=2a3d7383e72864ac42a532c6c4250cdc45b4c363;p=sage.d.git

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index e2d644a..3124132 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -20,7 +20,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                               names='e',
                               assume_associative=False,
                               category=None,
-                              rank=None):
+                              rank=None,
+                              natural_basis=None):
         n = len(mult_table)
         mult_table = [b.base_extend(field) for b in mult_table]
         for b in mult_table:
@@ -43,7 +44,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                                  assume_associative=assume_associative,
                                  names=names,
                                  category=cat,
-                                 rank=rank)
+                                 rank=rank,
+                                 natural_basis=natural_basis)
 
 
     def __init__(self, field,
@@ -51,7 +53,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                  names='e',
                  assume_associative=False,
                  category=None,
-                 rank=None):
+                 rank=None,
+                 natural_basis=None):
         """
         EXAMPLES:
 
@@ -66,6 +69,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
         """
         self._rank = rank
+        self._natural_basis = natural_basis
         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
         fda.__init__(field,
                      mult_table,
@@ -80,6 +84,49 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         fmt = "Euclidean Jordan algebra of degree {} over {}"
         return fmt.format(self.degree(), self.base_ring())
 
+
+    def natural_basis(self):
+        """
+        Return a more-natural representation of this algebra's basis.
+
+        Every finite-dimensional Euclidean Jordan Algebra is a direct
+        sum of five simple algebras, four of which comprise Hermitian
+        matrices. This method returns the original "natural" basis
+        for our underlying vector space. (Typically, the natural basis
+        is used to construct the multiplication table in the first place.)
+
+        Note that this will always return a matrix. The standard basis
+        in `R^n` will be returned as `n`-by-`1` column matrices.
+
+        EXAMPLES::
+
+            sage: J = RealSymmetricSimpleEJA(2)
+            sage: J.basis()
+            Family (e0, e1, e2)
+            sage: J.natural_basis()
+            (
+            [1 0]  [0 1]  [0 0]
+            [0 0], [1 0], [0 1]
+            )
+
+        ::
+
+            sage: J = JordanSpinSimpleEJA(2)
+            sage: J.basis()
+            Family (e0, e1)
+            sage: J.natural_basis()
+            (
+            [1]  [0]
+            [0], [1]
+            )
+
+        """
+        if self._natural_basis is None:
+            return tuple( b.vector().column() for b in self.basis() )
+        else:
+            return self._natural_basis
+
+
     def rank(self):
         """
         Return the rank of this EJA.
@@ -865,7 +912,7 @@ def _real_symmetric_basis(n, field=QQ):
                 # Beware, orthogonal but not normalized!
                 Sij = Eij + Eij.transpose()
             S.append(Sij)
-    return S
+    return tuple(S)
 
 
 def _complex_hermitian_basis(n, field=QQ):
@@ -902,7 +949,7 @@ def _complex_hermitian_basis(n, field=QQ):
                 S.append(Sij_real)
                 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
                 S.append(Sij_imag)
-    return S
+    return tuple(S)
 
 
 def _multiplication_table_from_matrix_basis(basis):
@@ -912,7 +959,10 @@ def _multiplication_table_from_matrix_basis(basis):
     multiplication on the right is matrix multiplication. Given a basis
     for the underlying matrix space, this function returns a
     multiplication table (obtained by looping through the basis
-    elements) for an algebra of those matrices.
+    elements) for an algebra of those matrices. A reordered copy
+    of the basis is also returned to work around the fact that
+    the ``span()`` in this function will change the order of the basis
+    from what we think it is, to... something else.
     """
     # In S^2, for example, we nominally have four coordinates even
     # though the space is of dimension three only. The vector space V
@@ -934,7 +984,7 @@ def _multiplication_table_from_matrix_basis(basis):
     # Taking the span above reorders our basis (thanks, jerk!) so we
     # need to put our "matrix basis" in the same order as the
     # (reordered) vector basis.
-    S = [ vec2mat(b) for b in W.basis() ]
+    S = tuple( vec2mat(b) for b in W.basis() )
 
     Qs = []
     for s in S:
@@ -952,7 +1002,7 @@ def _multiplication_table_from_matrix_basis(basis):
         Q = matrix(field, W.dimension(), Q_rows)
         Qs.append(Q)
 
-    return Qs
+    return (Qs, S)
 
 
 def _embed_complex_matrix(M):
@@ -1059,9 +1109,12 @@ def RealSymmetricSimpleEJA(n, field=QQ):
 
     """
     S = _real_symmetric_basis(n, field=field)
-    Qs = _multiplication_table_from_matrix_basis(S)
+    (Qs, T) = _multiplication_table_from_matrix_basis(S)
 
-    return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n)
+    return FiniteDimensionalEuclideanJordanAlgebra(field,
+                                                   Qs,
+                                                   rank=n,
+                                                   natural_basis=T)
 
 
 def ComplexHermitianSimpleEJA(n, field=QQ):
@@ -1083,8 +1136,11 @@ def ComplexHermitianSimpleEJA(n, field=QQ):
 
     """
     S = _complex_hermitian_basis(n)
-    Qs = _multiplication_table_from_matrix_basis(S)
-    return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n)
+    (Qs, T) = _multiplication_table_from_matrix_basis(S)
+    return FiniteDimensionalEuclideanJordanAlgebra(field,
+                                                   Qs,
+                                                   rank=n,
+                                                   natural_basis=T)
 
 
 def QuaternionHermitianSimpleEJA(n):