From 0fd07263cc543e345f3cd7668938f8a0de70641f Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 9 Aug 2019 19:39:15 -0400 Subject: [PATCH] eja: fix the characteristic polynomial in subalgebras. --- mjo/eja/eja_algebra.py | 19 +++++++++++++++---- mjo/eja/eja_element.py | 25 +++++++++++++++++++++++++ 2 files changed, 40 insertions(+), 4 deletions(-) diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 832e7a1..06f6f53 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -12,7 +12,7 @@ from sage.matrix.constructor import matrix from sage.misc.cachefunc import cached_method from sage.misc.prandom import choice from sage.misc.table import table -from sage.modules.free_module import VectorSpace +from sage.modules.free_module import FreeModule, VectorSpace from sage.rings.integer_ring import ZZ from sage.rings.number_field.number_field import QuadraticField from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing @@ -207,8 +207,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): determinant). """ z = self._a_regular_element() - V = self.vector_space() - V1 = V.span_of_basis( (z**k).to_vector() for k in range(self.rank()) ) + # Don't use the parent vector space directly here in case this + # happens to be a subalgebra. In that case, we would be e.g. + # two-dimensional but span_of_basis() would expect three + # coordinates. + V = VectorSpace(self.base_ring(), self.vector_space().dimension()) + basis = [ (z**k).to_vector() for k in range(self.rank()) ] + V1 = V.span_of_basis( basis ) b = (V1.basis() + V1.complement().basis()) return V.span_of_basis(b) @@ -263,7 +268,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # have multivatiate polynomial entries. names = tuple('X' + str(i) for i in range(1,n+1)) R = PolynomialRing(self.base_ring(), names) - V = self.vector_space().change_ring(R) + + # Using change_ring() on the parent's vector space doesn't work + # here because, in a subalgebra, that vector space has a basis + # and change_ring() tries to bring the basis along with it. And + # that doesn't work unless the new ring is a PID, which it usually + # won't be. + V = FreeModule(R,n) # Now let x = (X1,X2,...,Xn) be the vector whose entries are # indeterminates... diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index c47156d..e1f7563 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -165,6 +165,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.apply_univariate_polynomial(p) 0 + The characteristic polynomials of the zero and unit elements + should be what we think they are in a subalgebra, too:: + + sage: J = RealCartesianProductEJA(3) + sage: p1 = J.one().characteristic_polynomial() + sage: q1 = J.zero().characteristic_polynomial() + sage: e0,e1,e2 = J.gens() + sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3 + sage: p2 = A.one().characteristic_polynomial() + sage: q2 = A.zero().characteristic_polynomial() + sage: p1 == p2 + True + sage: q1 == q2 + True + """ p = self.parent().characteristic_polynomial() return p(*self.to_vector()) @@ -368,6 +383,16 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.is_invertible() == (x.det() != 0) True + Ensure that the determinant is multiplicative on an associative + subalgebra as in Faraut and Koranyi's Proposition II.2.2:: + + sage: set_random_seed() + sage: J = random_eja().random_element().subalgebra_generated_by() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (x*y).det() == x.det()*y.det() + True + """ P = self.parent() r = P.rank() -- 2.44.2