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