X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=f78af2519c15eb1aeb07eaa4a45b56cbd0a40d4f;hb=8698debba196d8746c1a32d8e6866085b6cb2161;hp=f44bee1c3f20d3040f0f77d8ff472b08c1bb5e9e;hpb=9f91faac758f0ef3a1ea7b1ad1cb506e73735d89;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index f44bee1..f78af25 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,7 +1,5 @@
 # -*- coding: utf-8 -*-
 
-from itertools import izip
-
 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
@@ -546,10 +544,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         return not (p(zero) == zero)
 
 
-    def is_minimal_idempotent(self):
+    def is_primitive_idempotent(self):
         """
-        Return whether or not this element is a minimal idempotent.
+        Return whether or not this element is a primitive (or minimal)
+        idempotent.
 
+        A primitive idempotent is a non-zero idempotent that is not
+        the sum of two other non-zero idempotents. Remark 2.7.15 in
+        Baes shows that this is what he refers to as a "minimal
+        idempotent."
 
         An element of a Euclidean Jordan algebra is a minimal idempotent
         if it :meth:`is_idempotent` and if its Peirce subalgebra
@@ -560,6 +563,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
             ....:                                  RealSymmetricEJA,
+            ....:                                  TrivialEJA,
             ....:                                  random_eja)
 
         WARNING::
@@ -575,7 +579,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x = sum(J.gens())
             sage: x.is_regular()
             False
-            sage: [ c.is_minimal_idempotent()
+            sage: [ c.is_primitive_idempotent()
             ....:   for (l,c) in x.spectral_decomposition() ]
             [False, True]
 
@@ -586,7 +590,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
             sage: x.is_regular()
             True
-            sage: [ c.is_minimal_idempotent()
+            sage: [ c.is_primitive_idempotent()
             ....:   for (l,c) in x.spectral_decomposition() ]
             [True, True]
 
@@ -596,7 +600,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja()
-            sage: J.rank() == 1 or not J.one().is_minimal_idempotent()
+            sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
             True
 
         A non-idempotent cannot be a minimal idempotent::
@@ -604,7 +608,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: set_random_seed()
             sage: J = JordanSpinEJA(4)
             sage: x = J.random_element()
-            sage: (not x.is_idempotent()) and x.is_minimal_idempotent()
+            sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
             False
 
         Proposition 2.7.19 in Baes says that an element is a minimal
@@ -615,24 +619,37 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: J = JordanSpinEJA(4)
             sage: x = J.random_element()
             sage: expected = (x.is_idempotent() and x.trace() == 1)
-            sage: actual = x.is_minimal_idempotent()
+            sage: actual = x.is_primitive_idempotent()
             sage: actual == expected
             True
 
+        Primitive idempotents must be non-zero::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: J.zero().is_idempotent()
+            True
+            sage: J.zero().is_primitive_idempotent()
+            False
+
+        As a consequence of the fact that primitive idempotents must
+        be non-zero, there are no primitive idempotents in a trivial
+        Euclidean Jordan algebra::
+
+            sage: J = TrivialEJA()
+            sage: J.one().is_idempotent()
+            True
+            sage: J.one().is_primitive_idempotent()
+            False
+
         """
-        # TODO: when the Peirce decomposition is implemented for real,
-        # we can use that instead of finding this eigenspace manually.
-        #
-        # Trivial eigenspaces don't appear in the list, so we default to the
-        # trivial one and override it if there's a nontrivial space in the
-        # list.
         if not self.is_idempotent():
             return False
 
-        J1 = VectorSpace(self.parent().base_ring(), 0)
-        for (eigval, eigspace) in self.operator().matrix().left_eigenspaces():
-            if eigval == 1:
-                J1 = eigspace
+        if self.is_zero():
+            return False
+
+        (_,_,J1) = self.parent().peirce_decomposition(self)
         return (J1.dimension() == 1)
 
 
@@ -969,7 +986,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()
-        return W.linear_combination(izip(B,self.to_vector()))
+        return W.linear_combination(zip(B,self.to_vector()))
 
 
     def norm(self):
@@ -1298,7 +1315,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         # will be minimal for some natural number s...
         s = 0
         minimal_dim = J.dimension()
-        for i in xrange(1, minimal_dim):
+        for i in range(1, minimal_dim):
             this_dim = (u**i).operator().matrix().image().dimension()
             if this_dim < minimal_dim:
                 minimal_dim = this_dim