X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=8ab1381665711ee3c09f1a0e3387c9e4dd85bcf3;hb=ed093213197d2ffa48849b0089ebe391b7574b93;hp=51973d14822bdc5dd49065de77c1927883161ae9;hpb=19909f0fa338d79db1c003528eee3d1123713810;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 51973d1..8ab1381 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -61,9 +61,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): self._rank = rank self._natural_basis = natural_basis - # TODO: HACK for the charpoly.. needs redesign badly. - self._basis_normalizers = None - if category is None: category = MagmaticAlgebras(field).FiniteDimensional() category = category.WithBasis().Unital() @@ -259,19 +256,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): store the trace/determinant (a_{r-1} and a_{0} respectively) separate from the entire characteristic polynomial. """ - if self._basis_normalizers is not None: - # Must be a matrix class? - # WARNING/TODO: this whole mess is mis-designed. - n = self.natural_basis_space().nrows() - field = self.base_ring().base_ring() # yeeeeaaaahhh - J = self.__class__(n, field, False) - (_,x,_,_) = J._charpoly_matrix_system() - p = J._charpoly_coeff(i) - # p might be missing some vars, have to substitute "optionally" - pairs = zip(x.base_ring().gens(), self._basis_normalizers) - substitutions = { v: v*c for (v,c) in pairs } - return p.subs(substitutions) - (A_of_x, x, xr, detA) = self._charpoly_matrix_system() R = A_of_x.base_ring() if i >= self.rank(): @@ -694,7 +678,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # not worry about it. raise NotImplementedError - n = ZZ.random_element(1, cls._max_test_case_size()) + n = ZZ.random_element(cls._max_test_case_size()) + 1 return cls(n, field, **kwargs) @@ -899,7 +883,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): def _max_test_case_size(): # Play it safe, since this will be squared and the underlying # field can have dimension 4 (quaternions) too. - return 3 + return 2 @classmethod def _denormalized_basis(cls, n, field): @@ -908,6 +892,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): S = self._denormalized_basis(n, field) + # Used in this class's fast _charpoly_coeff() override. + self._basis_normalizers = None + if n > 1 and normalize_basis: # We'll need sqrt(2) to normalize the basis, and this # winds up in the multiplication table, so the whole @@ -932,6 +919,30 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) + @cached_method + def _charpoly_coeff(self, i): + """ + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self._basis_normalizers is None: + # We didn't normalize, so assume that the basis we started + # with had entries in a nice field. + return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i) + else: + # If we didn't unembed first, this number would be wrong + # by a power-of-two factor for complex/quaternion matrices. + n = self.real_unembed(self.natural_basis_space().zero()).nrows() + field = self.base_ring().base_ring() # yeeeeaaaahhh + J = self.__class__(n, field, False) + (_,x,_,_) = J._charpoly_matrix_system() + p = J._charpoly_coeff(i) + # p might be missing some vars, have to substitute "optionally" + pairs = zip(x.base_ring().gens(), self._basis_normalizers) + substitutions = { v: v*c for (v,c) in pairs } + return p.subs(substitutions) + + @staticmethod def multiplication_table_from_matrix_basis(basis): """ @@ -1138,7 +1149,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): @staticmethod def _max_test_case_size(): - return 5 # Dimension 10 + return 4 # Dimension 10 @@ -1261,6 +1272,37 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix(F, n/2, elements) + @classmethod + def natural_inner_product(cls,X,Y): + """ + Compute a natural inner product in this algebra directly from + its real embedding. + + SETUP:: + + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + TESTS: + + This gives the same answer as the slow, default method implemented + in :class:`MatrixEuclideanJordanAlgebra`:: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.natural_representation() + sage: Ye = y.natural_representation() + sage: X = ComplexHermitianEJA.real_unembed(Xe) + sage: Y = ComplexHermitianEJA.real_unembed(Ye) + sage: expected = (X*Y).trace().vector()[0] + sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual == expected + True + + """ + return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + + class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n @@ -1509,6 +1551,36 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix(Q, n/4, elements) + @classmethod + def natural_inner_product(cls,X,Y): + """ + Compute a natural inner product in this algebra directly from + its real embedding. + + SETUP:: + + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + + TESTS: + + This gives the same answer as the slow, default method implemented + in :class:`MatrixEuclideanJordanAlgebra`:: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.natural_representation() + sage: Ye = y.natural_representation() + sage: X = QuaternionHermitianEJA.real_unembed(Xe) + sage: Y = QuaternionHermitianEJA.real_unembed(Ye) + sage: expected = (X*Y).trace().coefficient_tuple()[0] + sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual == expected + True + + """ + return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 + class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): """