# -*- coding: utf-8 -*- from .cartan_type import CartanType from mpmath import fac from sympy.core.backend import Matrix, eye, Rational, igcd from sympy.core.basic import Atom class WeylGroup(Atom): """ For each semisimple Lie group, we have a Weyl group. It is a subgroup of the isometry group of the root system. Specifically, it's the subgroup that is generated by reflections through the hyperplanes orthogonal to the roots. Therefore, Weyl groups are reflection groups, and so a Weyl group is a finite Coxeter group. """ def __new__(cls, cartantype): obj = Atom.__new__(cls) obj.cartan_type = CartanType(cartantype) return obj def generators(self): """ This method creates the generating reflections of the Weyl group for a given Lie algebra. For a Lie algebra of rank n, there are n different generating reflections. This function returns them as a list. Examples ======== >>> from sympy.liealgebras.weyl_group import WeylGroup >>> c = WeylGroup("F4") >>> c.generators() ['r1', 'r2', 'r3', 'r4'] """ n = self.cartan_type.rank() generators = [] for i in range(1, n+1): reflection = "r"+str(i) generators.append(reflection) return generators def group_order(self): """ This method returns the order of the Weyl group. For types A, B, C, D, and E the order depends on the rank of the Lie algebra. For types F and G, the order is fixed. Examples ======== >>> from sympy.liealgebras.weyl_group import WeylGroup >>> c = WeylGroup("D4") >>> c.group_order() 192.0 """ n = self.cartan_type.rank() if self.cartan_type.series == "A": return fac(n+1) if self.cartan_type.series in ("B", "C"): return fac(n)*(2**n) if self.cartan_type.series == "D": return fac(n)*(2**(n-1)) if self.cartan_type.series == "E": if n == 6: return 51840 if n == 7: return 2903040 if n == 8: return 696729600 if self.cartan_type.series == "F": return 1152 if self.cartan_type.series == "G": return 12 def group_name(self): """ This method returns some general information about the Weyl group for a given Lie algebra. It returns the name of the group and the elements it acts on, if relevant. """ n = self.cartan_type.rank() if self.cartan_type.series == "A": return "S"+str(n+1) + ": the symmetric group acting on " + str(n+1) + " elements." if self.cartan_type.series in ("B", "C"): return "The hyperoctahedral group acting on " + str(2*n) + " elements." if self.cartan_type.series == "D": return "The symmetry group of the " + str(n) + "-dimensional demihypercube." if self.cartan_type.series == "E": if n == 6: return "The symmetry group of the 6-polytope." if n == 7: return "The symmetry group of the 7-polytope." if n == 8: return "The symmetry group of the 8-polytope." if self.cartan_type.series == "F": return "The symmetry group of the 24-cell, or icositetrachoron." if self.cartan_type.series == "G": return "D6, the dihedral group of order 12, and symmetry group of the hexagon." def element_order(self, weylelt): """ This method returns the order of a given Weyl group element, which should be specified by the user in the form of products of the generating reflections, i.e. of the form r1*r2 etc. For types A-F, this method current works by taking the matrix form of the specified element, and then finding what power of the matrix is the identity. It then returns this power. Examples ======== >>> from sympy.liealgebras.weyl_group import WeylGroup >>> b = WeylGroup("B4") >>> b.element_order('r1*r4*r2') 4 """ n = self.cartan_type.rank() if self.cartan_type.series == "A": a = self.matrix_form(weylelt) order = 1 while a != eye(n+1): a *= self.matrix_form(weylelt) order += 1 return order if self.cartan_type.series == "D": a = self.matrix_form(weylelt) order = 1 while a != eye(n): a *= self.matrix_form(weylelt) order += 1 return order if self.cartan_type.series == "E": a = self.matrix_form(weylelt) order = 1 while a != eye(8): a *= self.matrix_form(weylelt) order += 1 return order if self.cartan_type.series == "G": elts = list(weylelt) reflections = elts[1::3] m = self.delete_doubles(reflections) while self.delete_doubles(m) != m: m = self.delete_doubles(m) reflections = m if len(reflections) % 2 == 1: return 2 elif len(reflections) == 0: return 1 else: if len(reflections) == 1: return 2 else: m = len(reflections) // 2 lcm = (6 * m)/ igcd(m, 6) order = lcm / m return order if self.cartan_type.series == 'F': a = self.matrix_form(weylelt) order = 1 while a != eye(4): a *= self.matrix_form(weylelt) order += 1 return order if self.cartan_type.series in ("B", "C"): a = self.matrix_form(weylelt) order = 1 while a != eye(n): a *= self.matrix_form(weylelt) order += 1 return order def delete_doubles(self, reflections): """ This is a helper method for determining the order of an element in the Weyl group of G2. It takes a Weyl element and if repeated simple reflections in it, it deletes them. """ counter = 0 copy = list(reflections) for elt in copy: if counter < len(copy)-1: if copy[counter + 1] == elt: del copy[counter] del copy[counter] counter += 1 return copy def matrix_form(self, weylelt): """ This method takes input from the user in the form of products of the generating reflections, and returns the matrix corresponding to the element of the Weyl group. Since each element of the Weyl group is a reflection of some type, there is a corresponding matrix representation. This method uses the standard representation for all the generating reflections. Examples ======== >>> from sympy.liealgebras.weyl_group import WeylGroup >>> f = WeylGroup("F4") >>> f.matrix_form('r2*r3') Matrix([ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, -1], [0, 0, 1, 0]]) """ elts = list(weylelt) reflections = elts[1::3] n = self.cartan_type.rank() if self.cartan_type.series == 'A': matrixform = eye(n+1) for elt in reflections: a = int(elt) mat = eye(n+1) mat[a-1, a-1] = 0 mat[a-1, a] = 1 mat[a, a-1] = 1 mat[a, a] = 0 matrixform *= mat return matrixform if self.cartan_type.series == 'D': matrixform = eye(n) for elt in reflections: a = int(elt) mat = eye(n) if a < n: mat[a-1, a-1] = 0 mat[a-1, a] = 1 mat[a, a-1] = 1 mat[a, a] = 0 matrixform *= mat else: mat[n-2, n-1] = -1 mat[n-2, n-2] = 0 mat[n-1, n-2] = -1 mat[n-1, n-1] = 0 matrixform *= mat return matrixform if self.cartan_type.series == 'G': matrixform = eye(3) for elt in reflections: a = int(elt) if a == 1: gen1 = Matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]]) matrixform *= gen1 else: gen2 = Matrix([[Rational(2, 3), Rational(2, 3), Rational(-1, 3)], [Rational(2, 3), Rational(-1, 3), Rational(2, 3)], [Rational(-1, 3), Rational(2, 3), Rational(2, 3)]]) matrixform *= gen2 return matrixform if self.cartan_type.series == 'F': matrixform = eye(4) for elt in reflections: a = int(elt) if a == 1: mat = Matrix([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]) matrixform *= mat elif a == 2: mat = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]) matrixform *= mat elif a == 3: mat = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]]) matrixform *= mat else: mat = Matrix([[Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2)], [Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2)], [Rational(1, 2), Rational(-1, 2), Rational(1, 2), Rational(-1, 2)], [Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]]) matrixform *= mat return matrixform if self.cartan_type.series == 'E': matrixform = eye(8) for elt in reflections: a = int(elt) if a == 1: mat = Matrix([[Rational(3, 4), Rational(1, 4), Rational(1, 4), Rational(1, 4), Rational(1, 4), Rational(1, 4), Rational(1, 4), Rational(-1, 4)], [Rational(1, 4), Rational(3, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(1, 4), Rational(-1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(3, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(-1, 4), Rational(3, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(3, 4), Rational(-1, 4), Rational(-1, 4), Rational(1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(3, 4), Rational(-1, 4), Rational(1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-3, 4), Rational(1, 4)], [Rational(1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(-1, 4), Rational(3, 4)]]) matrixform *= mat elif a == 2: mat = eye(8) mat[0, 0] = 0 mat[0, 1] = -1 mat[1, 0] = -1 mat[1, 1] = 0 matrixform *= mat else: mat = eye(8) mat[a-3, a-3] = 0 mat[a-3, a-2] = 1 mat[a-2, a-3] = 1 mat[a-2, a-2] = 0 matrixform *= mat return matrixform if self.cartan_type.series in ("B", "C"): matrixform = eye(n) for elt in reflections: a = int(elt) mat = eye(n) if a == 1: mat[0, 0] = -1 matrixform *= mat else: mat[a - 2, a - 2] = 0 mat[a-2, a-1] = 1 mat[a - 1, a - 2] = 1 mat[a -1, a - 1] = 0 matrixform *= mat return matrixform def coxeter_diagram(self): """ This method returns the Coxeter diagram corresponding to a Weyl group. The Coxeter diagram can be obtained from a Lie algebra's Dynkin diagram by deleting all arrows; the Coxeter diagram is the undirected graph. The vertices of the Coxeter diagram represent the generating reflections of the Weyl group, $s_i$. An edge is drawn between $s_i$ and $s_j$ if the order $m(i, j)$ of $s_is_j$ is greater than two. If there is one edge, the order $m(i, j)$ is 3. If there are two edges, the order $m(i, j)$ is 4, and if there are three edges, the order $m(i, j)$ is 6. Examples ======== >>> from sympy.liealgebras.weyl_group import WeylGroup >>> c = WeylGroup("B3") >>> print(c.coxeter_diagram()) 0---0===0 1 2 3 """ n = self.cartan_type.rank() if self.cartan_type.series in ("A", "D", "E"): return self.cartan_type.dynkin_diagram() if self.cartan_type.series in ("B", "C"): diag = "---".join("0" for i in range(1, n)) + "===0\n" diag += " ".join(str(i) for i in range(1, n+1)) return diag if self.cartan_type.series == "F": diag = "0---0===0---0\n" diag += " ".join(str(i) for i in range(1, 5)) return diag if self.cartan_type.series == "G": diag = "0≡≡≡0\n1 2" return diag