From 2e4211deee2c4556f2c2b4dccc857106ce6bc89a Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sat, 6 Mar 2021 12:09:11 -0500
Subject: [PATCH] eja: fix matrix_space() with underlying MatrixAlgebras.

---
 mjo/eja/eja_algebra.py | 50 ++++++++++++++++++++++++++++++++++++++----
 1 file changed, 46 insertions(+), 4 deletions(-)

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index aef11ac..f2d7ba7 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -3380,9 +3380,17 @@ class CartesianProductEJA(FiniteDimensionalEJA):
         Return the space that our matrix basis lives in as a Cartesian
         product.
 
+        We don't simply use the ``cartesian_product()`` functor here
+        because it acts differently on SageMath MatrixSpaces and our
+        custom MatrixAlgebras, which are CombinatorialFreeModules. We
+        always want the result to be represented (and indexed) as
+        an ordered tuple.
+
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (HadamardEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  HadamardEJA,
+            ....:                                  OctonionHermitianEJA,
             ....:                                  RealSymmetricEJA)
 
         EXAMPLES::
@@ -3395,10 +3403,44 @@ class CartesianProductEJA(FiniteDimensionalEJA):
             matrices over Algebraic Real Field, Full MatrixSpace of 2
             by 2 dense matrices over Algebraic Real Field)
 
+        ::
+
+            sage: J1 = ComplexHermitianEJA(1)
+            sage: J2 = ComplexHermitianEJA(1)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.one().to_matrix()[0]
+            [1 0]
+            [0 1]
+            sage: J.one().to_matrix()[1]
+            [1 0]
+            [0 1]
+
+        ::
+
+            sage: J1 = OctonionHermitianEJA(1)
+            sage: J2 = OctonionHermitianEJA(1)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.one().to_matrix()[0]
+            +----+
+            | e0 |
+            +----+
+            sage: J.one().to_matrix()[1]
+            +----+
+            | e0 |
+            +----+
+
         """
-        from sage.categories.cartesian_product import cartesian_product
-        return cartesian_product( [J.matrix_space()
-                                   for J in self.cartesian_factors()] )
+        scalars = self.cartesian_factor(0).base_ring()
+
+        # This category isn't perfect, but is good enough for what we
+        # need to do.
+        cat = MagmaticAlgebras(scalars).FiniteDimensional().WithBasis()
+        cat = cat.Unital().CartesianProducts()
+        factors = tuple( J.matrix_space() for J in self.cartesian_factors() )
+
+        from sage.sets.cartesian_product import CartesianProduct
+        return CartesianProduct(factors, cat)
+
 
     @cached_method
     def cartesian_projection(self, i):
-- 
2.44.2