def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time """ self._matrix = coxeter_matrix self._index_set = index_set n = ZZ(coxeter_matrix.nrows()) MS = MatrixSpace(base_ring, n, sparse=True) # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty if base_ring is UniversalCyclotomicField(): val = lambda x: base_ring.gen(2*x) + ~base_ring.gen(2*x) if x != -1 else base_ring(2) else: from sage.functions.trig import cos from sage.symbolic.constants import pi val = lambda x: base_ring(2*cos(pi / x)) if x != -1 else base_ring(2) gens = [MS.one() + MS({(i, j): val(coxeter_matrix[i, j]) for j in range(n)}) for i in range(n)] FinitelyGeneratedMatrixGroup_generic.__init__(self, n, base_ring, gens, category=CoxeterGroups())
def __init__(self, n): r""" Hecke triangle group (2, n, infinity). Namely the von Dyck group corresponding to the triangle group with angles (pi/2, pi/n, 0). INPUT: - ``n`` - ``infinity`` or an integer greater or equal to ``3``. OUTPUT: The Hecke triangle group for the given parameter ``n``. EXAMPLES:: sage: G = HeckeTriangleGroup(12) sage: G Hecke triangle group for n = 12 sage: G.category() Category of groups """ self._n = n self._T = matrix(AA, [[1,self.lam()],[0,1]]) self._S = matrix(AA, [[0,-1],[1,0]]) FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), AA, [self._S, self._T])
def __init__(self, n): r""" Hecke triangle group (2, n, infinity). Namely the von Dyck group corresponding to the triangle group with angles (pi/2, pi/n, 0). INPUT: - ``n`` - ``infinity`` or an integer greater or equal to ``3``. OUTPUT: The Hecke triangle group for the given parameter ``n``. EXAMPLES:: sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup sage: G = HeckeTriangleGroup(12) sage: G Hecke triangle group for n = 12 sage: G.category() Category of groups """ self._n = n self.element_repr_method("default") if n in [3, infinity]: self._base_ring = ZZ self._lam = ZZ(1) if n == 3 else ZZ(2) else: lam_symbolic = 2 * cos(pi / n) K = NumberField(self.lam_minpoly(), 'lam', embedding=coerce_AA(lam_symbolic)) #self._base_ring = K.order(K.gens()) self._base_ring = K.maximal_order() self._lam = self._base_ring.gen(1) T = matrix(self._base_ring, [[1, self._lam], [0, 1]]) S = matrix(self._base_ring, [[0, -1], [1, 0]]) FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), self._base_ring, [S, T])
def __init__(self, n): r""" Hecke triangle group (2, n, infinity). Namely the von Dyck group corresponding to the triangle group with angles (pi/2, pi/n, 0). INPUT: - ``n`` - ``infinity`` or an integer greater or equal to ``3``. OUTPUT: The Hecke triangle group for the given parameter ``n``. EXAMPLES:: sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup sage: G = HeckeTriangleGroup(12) sage: G Hecke triangle group for n = 12 sage: G.category() Category of groups """ self._n = n self.element_repr_method("default") if n in [3, infinity]: self._base_ring = ZZ self._lam = ZZ(1) if n==3 else ZZ(2) else: lam_symbolic = 2*cos(pi/n) K = NumberField(self.lam_minpoly(), 'lam', embedding = coerce_AA(lam_symbolic)) #self._base_ring = K.order(K.gens()) self._base_ring = K.maximal_order() self._lam = self._base_ring.gen(1) T = matrix(self._base_ring, [[1,self._lam],[0,1]]) S = matrix(self._base_ring, [[0,-1],[1,0]]) FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), self._base_ring, [S, T])
def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time We check that :trac:`16630` is fixed:: sage: CoxeterGroup(['D',4], base_ring=QQ).category() Category of finite coxeter groups sage: CoxeterGroup(['H',4], base_ring=QQbar).category() Category of finite coxeter groups sage: F = CoxeterGroups().Finite() sage: all(CoxeterGroup([letter,i]) in F ....: for i in range(2,5) for letter in ['A','B','D']) True sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9)) True sage: CoxeterGroup(['F',4]).category() Category of finite coxeter groups sage: CoxeterGroup(['G',2]).category() Category of finite coxeter groups sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5)) True sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5)) True """ self._matrix = coxeter_matrix n = coxeter_matrix.rank() # Compute the matrix with entries `2 \cos( \pi / m_{ij} )`. MS = MatrixSpace(base_ring, n, sparse=True) MC = MS._get_matrix_class() # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty if base_ring is UniversalCyclotomicField(): val = lambda x: base_ring.gen(2*x) + ~base_ring.gen(2*x) if x != -1 else base_ring(2) else: from sage.functions.trig import cos from sage.symbolic.constants import pi val = lambda x: base_ring(2*cos(pi / x)) if x != -1 else base_ring(2) gens = [MS.one() + MC(MS, entries={(i, j): val(coxeter_matrix[index_set[i], index_set[j]]) for j in range(n)}, coerce=True, copy=True) for i in range(n)] category = CoxeterGroups() # Now we shall see if the group is finite, and, if so, refine # the category to ``category.Finite()``. Otherwise the group is # infinite and we refine the category to ``category.Infinite()``. if self._matrix.is_finite(): category = category.Finite() else: category = category.Infinite() FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(n), base_ring, gens, category=category)
def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time We check that :trac:`16630` is fixed:: sage: CoxeterGroup(['D',4], base_ring=QQ).category() Category of finite coxeter groups sage: CoxeterGroup(['H',4], base_ring=QQbar).category() Category of finite coxeter groups sage: F = CoxeterGroups().Finite() sage: all(CoxeterGroup([letter,i]) in F ....: for i in range(2,5) for letter in ['A','B','D']) True sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9)) True sage: CoxeterGroup(['F',4]).category() Category of finite coxeter groups sage: CoxeterGroup(['G',2]).category() Category of finite coxeter groups sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5)) True sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5)) True """ self._matrix = coxeter_matrix n = coxeter_matrix.rank() # Compute the matrix with entries `2 \cos( \pi / m_{ij} )`. MS = MatrixSpace(base_ring, n, sparse=True) MC = MS._get_matrix_class() # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty E = UniversalCyclotomicField().gen if base_ring is UniversalCyclotomicField(): def val(x): if x == -1: return 2 else: return E(2 * x) + ~E(2 * x) elif is_QuadraticField(base_ring): def val(x): if x == -1: return 2 else: return base_ring( (E(2 * x) + ~E(2 * x)).to_cyclotomic_field()) else: from sage.functions.trig import cos from sage.symbolic.constants import pi def val(x): if x == -1: return 2 else: return base_ring(2 * cos(pi / x)) gens = [ MS.one() + MC(MS, entries={(i, j): val(coxeter_matrix[index_set[i], index_set[j]]) for j in range(n)}, coerce=True, copy=True) for i in range(n) ] # Make the generators dense matrices for consistency and speed gens = [g.dense_matrix() for g in gens] category = CoxeterGroups() # Now we shall see if the group is finite, and, if so, refine # the category to ``category.Finite()``. Otherwise the group is # infinite and we refine the category to ``category.Infinite()``. if self._matrix.is_finite(): category = category.Finite() else: category = category.Infinite() self._index_set_inverse = { i: ii for ii, i in enumerate(self._matrix.index_set()) } FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(n), base_ring, gens, category=category)
def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time We check that :trac:`16630` is fixed:: sage: CoxeterGroup(['D',4], base_ring=QQ).category() Category of finite coxeter groups sage: CoxeterGroup(['H',4], base_ring=QQbar).category() Category of finite coxeter groups sage: F = CoxeterGroups().Finite() sage: all(CoxeterGroup([letter,i]) in F ....: for i in range(2,5) for letter in ['A','B','D']) True sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9)) True sage: CoxeterGroup(['F',4]).category() Category of finite coxeter groups sage: CoxeterGroup(['G',2]).category() Category of finite coxeter groups sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5)) True sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5)) True """ self._matrix = coxeter_matrix self._index_set = index_set n = ZZ(coxeter_matrix.nrows()) # Compute the matrix with entries `2 \cos( \pi / m_{ij} )`. MS = MatrixSpace(base_ring, n, sparse=True) MC = MS._get_matrix_class() # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty if base_ring is UniversalCyclotomicField(): val = lambda x: base_ring.gen(2 * x) + ~base_ring.gen(2 * x) if x != -1 else base_ring(2) else: from sage.functions.trig import cos from sage.symbolic.constants import pi val = lambda x: base_ring(2 * cos(pi / x)) if x != -1 else base_ring(2) gens = [ MS.one() + MC(MS, entries={(i, j): val(coxeter_matrix[i, j]) for j in range(n)}, coerce=True, copy=True) for i in range(n) ] # Compute the matrix with entries `- \cos( \pi / m_{ij} )`. # This describes the bilinear form corresponding to this # Coxeter system, and might lead us out of our base ring. base_field = base_ring.fraction_field() MS2 = MatrixSpace(base_field, n, sparse=True) MC2 = MS2._get_matrix_class() self._bilinear = MC2( MS2, entries={ (i, j): val(coxeter_matrix[i, j]) / base_field(-2) for i in range(n) for j in range(n) if coxeter_matrix[i, j] != 2 }, coerce=True, copy=True, ) self._bilinear.set_immutable() category = CoxeterGroups() # Now we shall see if the group is finite, and, if so, refine # the category to ``category.Finite()``. Otherwise the group is # infinite and we refine the category to ``category.Infinite()``. is_finite = self._finite_recognition() if is_finite: category = category.Finite() else: category = category.Infinite() FinitelyGeneratedMatrixGroup_generic.__init__(self, n, base_ring, gens, category=category)
def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time We check that :trac:`16630` is fixed:: sage: CoxeterGroup(['D',4], base_ring=QQ).category() Category of finite irreducible coxeter groups sage: CoxeterGroup(['H',4], base_ring=QQbar).category() Category of finite irreducible coxeter groups sage: F = CoxeterGroups().Finite() sage: all(CoxeterGroup([letter,i]) in F ....: for i in range(2,5) for letter in ['A','B','D']) True sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9)) True sage: CoxeterGroup(['F',4]).category() Category of finite irreducible coxeter groups sage: CoxeterGroup(['G',2]).category() Category of finite irreducible coxeter groups sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5)) True sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5)) True """ self._matrix = coxeter_matrix n = coxeter_matrix.rank() # Compute the matrix with entries `2 \cos( \pi / m_{ij} )`. MS = MatrixSpace(base_ring, n, sparse=True) one = MS.one() # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty E = UniversalCyclotomicField().gen if base_ring is UniversalCyclotomicField(): def val(x): if x == -1: return 2 else: return E(2 * x) + ~E(2 * x) elif is_QuadraticField(base_ring): def val(x): if x == -1: return 2 else: return base_ring((E(2 * x) + ~E(2 * x)).to_cyclotomic_field()) else: from sage.functions.trig import cos from sage.symbolic.constants import pi def val(x): if x == -1: return 2 else: return base_ring(2 * cos(pi / x)) gens = [one + MS([SparseEntry(i, j, val(coxeter_matrix[index_set[i], index_set[j]])) for j in range(n)]) for i in range(n)] # Make the generators dense matrices for consistency and speed gens = [g.dense_matrix() for g in gens] category = CoxeterGroups() # Now we shall see if the group is finite, and, if so, refine # the category to ``category.Finite()``. Otherwise the group is # infinite and we refine the category to ``category.Infinite()``. if self._matrix.is_finite(): category = category.Finite() else: category = category.Infinite() if all(self._matrix._matrix[i, j] == 2 for i in range(n) for j in range(i)): category = category.Commutative() if self._matrix.is_irreducible(): category = category.Irreducible() self._index_set_inverse = {i: ii for ii, i in enumerate(self._matrix.index_set())} FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(n), base_ring, gens, category=category)
def __init__(self, coxeter_matrix, base_ring, index_set): """ Initialize ``self``. EXAMPLES:: sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]]) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar) sage: TestSuite(W).run() # long time sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]]) sage: TestSuite(W).run(max_runs=30) # long time sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]]) sage: TestSuite(W).run(max_runs=30) # long time We check that :trac:`16630` is fixed:: sage: CoxeterGroup(['D',4], base_ring=QQ).category() Category of finite coxeter groups sage: CoxeterGroup(['H',4], base_ring=QQbar).category() Category of finite coxeter groups sage: F = CoxeterGroups().Finite() sage: all(CoxeterGroup([letter,i]) in F ....: for i in range(2,5) for letter in ['A','B','D']) True sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9)) True sage: CoxeterGroup(['F',4]).category() Category of finite coxeter groups sage: CoxeterGroup(['G',2]).category() Category of finite coxeter groups sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5)) True sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5)) True """ self._matrix = coxeter_matrix self._index_set = index_set n = ZZ(coxeter_matrix.nrows()) # Compute the matrix with entries `2 \cos( \pi / m_{ij} )`. MS = MatrixSpace(base_ring, n, sparse=True) MC = MS._get_matrix_class() # FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty if base_ring is UniversalCyclotomicField(): val = lambda x: base_ring.gen(2 * x) + ~base_ring.gen( 2 * x) if x != -1 else base_ring(2) else: from sage.functions.trig import cos from sage.symbolic.constants import pi val = lambda x: base_ring(2 * cos(pi / x) ) if x != -1 else base_ring(2) gens = [ MS.one() + MC(MS, entries={(i, j): val(coxeter_matrix[i, j]) for j in range(n)}, coerce=True, copy=True) for i in range(n) ] # Compute the matrix with entries `- \cos( \pi / m_{ij} )`. # This describes the bilinear form corresponding to this # Coxeter system, and might lead us out of our base ring. base_field = base_ring.fraction_field() MS2 = MatrixSpace(base_field, n, sparse=True) MC2 = MS2._get_matrix_class() self._bilinear = MC2(MS2, entries={ (i, j): val(coxeter_matrix[i, j]) / base_field(-2) for i in range(n) for j in range(n) if coxeter_matrix[i, j] != 2 }, coerce=True, copy=True) self._bilinear.set_immutable() category = CoxeterGroups() # Now we shall see if the group is finite, and, if so, refine # the category to ``category.Finite()``. Otherwise the group is # infinite and we refine the category to ``category.Infinite()``. is_finite = self._finite_recognition() if is_finite: category = category.Finite() else: category = category.Infinite() FinitelyGeneratedMatrixGroup_generic.__init__(self, n, base_ring, gens, category=category)