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ArtinWedderburn.py
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ArtinWedderburn.py
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import numpy as np
import scipy.sparse as sparse
from scipy.linalg import eigh, eig
from math import sqrt
# takes an array of complex numbers and removes duplicates upto threshold
def fuzzy_filter(array, threshold):
result = []
for x in array:
already_seen = False
for v in result:
if abs(x-v) < threshold:
already_seen = True
break
if not already_seen:
result.append(x)
return result
def format_error(x):
return str.format('{0:1.0e}' ,x)
def one_sided_svd(matrix):
eigvals, eigvecs = eigh(np.dot(matrix, np.transpose(matrix.conjugate())))
u = np.flip(eigvecs, 1)
s = np.flip(eigvals, 0)
return u, s
class ArtinWedderburn:
def compute_center(self):
algebra = self.algebra
d = algebra.dimension
cm = algebra.commutator_matrix()
cm_transpose = cm.transpose()
cm_conj = cm_transpose.conj()
cm_conj_times_cm = cm_conj * cm
s, v = eigh(cm_conj_times_cm.todense())
center_dimension = len(np.where( np.abs(s) < self.threshold)[0])
center_inclusion = v[:,:center_dimension]
center_projection = np.transpose(np.conj(center_inclusion))
center_multiplication = np.array([[ np.dot(
center_projection,
algebra.multiply(center_inclusion[:,i], center_inclusion[:,j]))
for i in range(center_dimension)]
for j in range(center_dimension)])
center = Algebra(
center_dimension,
center_multiplication,
np.dot(center_projection, algebra.unit))
self.center = center
self.center_inclusion = center_inclusion
center_defect = center.algebra_defect() + center.commutative_defect()
self.log("center defect:", format_error(center_defect))
self.total_defect += center_defect
def compute_unscaled_central_idempotents(self):
algebra = self.algebra
center = self.center
center_inclusion = self.center_inclusion
v = center.random_vector()
lv = center.left_multiplication_matrix(v)
eigenvectors = eig(lv)[1]
unscaled_central_idempotents = np.dot(center_inclusion, eigenvectors)
unscaled_central_idempotent_defect = 0.0
for i in range(center.dimension):
for j in range(center.dimension):
if i != j:
unscaled_central_idempotent_defect += np.abs(np.sum(algebra.multiply(
unscaled_central_idempotents[:,i],
unscaled_central_idempotents[:,j])))
self.log("unscaled central idempotent defect:",
format_error(unscaled_central_idempotent_defect))
self.total_defect += unscaled_central_idempotent_defect
self.unscaled_central_idempotents = unscaled_central_idempotents
def compute_block(self, idempotent_index):
algebra = self.algebra
idempotent = self.unscaled_central_idempotents[:, idempotent_index]
left_multiplication = algebra.left_multiplication_matrix(idempotent)
# u is the change of basis from new basis to old basis
# u, s, v = svd(left_multiplication)
u, s = one_sided_svd(left_multiplication)
# u is unitary, so inverse is conjugate transpose
# u_inverse is change of basis from old basis to new basis
# u_inverse = np.transpose(np.conj(u))
# the basis for the block is the columns of u whose singular value
# is above the threshold
counter = 0
for i in range(len(s)):
if abs(s[i]) > self.threshold:
counter += 1
block_inclusion = u[:,:counter]
self.block_inclusions[idempotent_index] = block_inclusion
block_projection = np.conj(np.transpose(block_inclusion))
block_dimension = block_inclusion.shape[1]
block_multiplication = np.array([[ np.dot(
block_projection,
algebra.multiply(block_inclusion[:,i], block_inclusion[:,j]))
for i in range(block_dimension)]
for j in range(block_dimension)])
block_pre_unit = np.dot(block_projection, idempotent)
pre_block = Algebra(
block_dimension,
block_multiplication,
block_pre_unit)
# we have only captured the identity up to scalar multiplication
# we need the identity on the node if we want to recover the correct scale
# for the representations
factor = pre_block.left_multiplication_matrix(pre_block.unit)[0,0]
block = Algebra(
pre_block.dimension,
pre_block.multiplication,
pre_block.unit / factor)
block_defect = block.algebra_defect()
self.log("block defect:", format_error(block_defect))
self.total_defect += block_defect
central_idempotent = np.dot(block_inclusion,block.unit)
self.blocks[idempotent_index] = block
self.central_idempotents[idempotent_index] = central_idempotent
def compute_irrep(self,block_index):
block = self.blocks[block_index]
sqrt_dimension = int(sqrt(block.dimension))
self.irrep_dimensions[block_index] = sqrt_dimension
inclusion = self.block_inclusions[block_index]
projection = np.transpose(np.conj(inclusion))
if block.dimension == 1:
block_irrep = block.multiplication
m_defect = 0.0
else:
v = block.random_vector()
vr = block.right_multiplication_matrix(v)
eigenvalues_with_repeats = eig(vr)[0]
eigenvalues = fuzzy_filter(eigenvalues_with_repeats,self.threshold)
accumulator = block.unit
for i in range(sqrt_dimension - 1):
accumulator = block.multiply(accumulator, v - eigenvalues[i] * block.unit)
proj = block.right_multiplication_matrix(accumulator)
# u goes from new basis to old basis
# u, s, v = svd(proj)
u, s = one_sided_svd(proj)
# u_inverse goes from old basis to new basis
u_inverse = np.transpose(np.conj(u))
m = block.multiplication
change_codomain_basis = np.tensordot(m,u_inverse, (2,1))
m_rotated = np.transpose(np.tensordot(u,change_codomain_basis,(0,1)),(1,0,2))
m_defect = np.sum(np.abs(m_rotated[:,
0:sqrt_dimension,
sqrt_dimension:block.dimension]))
block_irrep = m_rotated[:,0:sqrt_dimension,0:sqrt_dimension]
self.log("SVD defect:", format_error(m_defect))
self.total_defect += m_defect
block_irrep_defect = block.irrep_defect(block_irrep)
self.log("block irred defect:", format_error(block_irrep_defect))
self.total_defect += block_irrep_defect
irrep = np.tensordot(projection, block_irrep, (0,0))
self.irreps[block_index] = irrep
irrep_defect = self.algebra.irrep_defect(irrep)
self.log("irrep defect:", format_error(irrep_defect))
self.total_defect += irrep_defect
def central_idempotent_defect(self):
center = self.center
algebra = self.algebra
central_idempotents = self.central_idempotents
defect_mat = np.zeros((center.dimension, center.dimension))
for i in range(center.dimension):
for j in range(center.dimension):
if i != j:
defect_mat[i,j] = np.sum(np.abs(algebra.multiply(
central_idempotents[i],
central_idempotents[j])))
else:
defect_mat[i,j] = np.sum(np.abs(algebra.multiply(
central_idempotents[i],
central_idempotents[j]) - central_idempotents[i]))
id_defect = np.copy(algebra.unit)
for i in range(center.dimension):
id_defect -= central_idempotents[i]
return np.sum(defect_mat) + np.sum(np.abs(id_defect))
def central_idempotent_irrep_defect(self):
center = self.center
central_idempotents = self.central_idempotents
irreps = self.irreps
defect_mat = np.zeros((center.dimension, center.dimension))
for i in range(center.dimension):
for j in range(center.dimension):
idempotent = central_idempotents[i]
irrep = irreps[j]
if i != j:
defect_mat[i,j] = np.sum(np.abs(np.tensordot(
idempotent,
irrep,
(0,0))))
else:
mat = np.tensordot(
central_idempotents[i],
irreps[j],
(0,0))
mat -= np.identity(irrep.shape[-1])
defect_mat[i,j] = np.sum(np.abs(mat))
return np.sum(defect_mat)
def log(self,*args, **kwargs):
if self.logging:
print(*args, **kwargs)
def __init__(self, sparse_algebra, threshold = 1.0e-5, logging = False):
self.algebra = sparse_algebra
self.threshold = threshold
self.logging = logging
self.total_defect = sparse_algebra.algebra_defect()
self.log("algebra defect:", self.total_defect)
self.log("")
self.log("computing center...")
self.compute_center()
self.log("")
self.log("computing unscaled central idempotents...")
self.compute_unscaled_central_idempotents()
self.log("")
self.blocks = {}
self.central_idempotents = {}
# the block inclusions are unitary, so to project onto a block
# take the conjugate transpose of the inclusion
self.block_inclusions = {}
self.log("computing blocks...")
self.log("")
for i in range(self.center.dimension):
self.log("computing block ", i)
self.compute_block(i)
self.log("")
self.irreps = {}
self.irrep_dimensions = {}
self.log("computing irreps...")
self.log("")
for i in range(self.center.dimension):
self.log("computing irrep", i)
self.compute_irrep(i)
self.log("")
self.log("all done!")
central_idempotent_defect = self.central_idempotent_defect()
self.log("central idempotent defect:", format_error(central_idempotent_defect))
self.total_defect += central_idempotent_defect
central_idempotent_irrep_cross_defect = self.central_idempotent_irrep_defect()
self.log("central idempotent irrep cross defect:",
format_error(central_idempotent_irrep_cross_defect))
self.total_defect += central_idempotent_irrep_cross_defect
self.log("total defect:", format_error(self.total_defect))
self.log("irrep tensors are stored in the attribute self.irreps")
self.log(self.center.dimension, "irreducible representations with dimensions")
for index, dim in self.irrep_dimensions.items():
self.log(index, ":", dim)
class SparseAlgebra:
def associative_defect(self):
x = self.random_vector()
y = self.random_vector()
z = self.random_vector()
l = self.multiply(self.multiply(x,y), z)
r = self.multiply(x, self.multiply(y,z))
return np.sum(np.abs(l - r))
def left_identity_defect(self):
lm = self.left_multiplication_matrix(self.unit)
return np.sum(np.abs(lm - np.identity(self.dimension)))
def right_identity_defect(self):
rm = self.right_multiplication_matrix(self.unit)
return np.sum(np.abs(rm - np.identity(self.dimension)))
def algebra_defect(self):
return sum([self.associative_defect(),
self.associative_defect(),
self.associative_defect(),
self.left_identity_defect(),
self.right_identity_defect()])
def irrep_defect_multiplication(self,irrep):
x = self.random_vector()
y = self.random_vector()
xy = self.multiply(x,y)
# irrep[i,j] is the vector e_i b_j
l = np.tensordot(xy, irrep, (0,0))
r_uneval = np.tensordot(irrep, irrep, (2,1))
# this is backwards because irrep[i] is the transpose of
# what would conventionally be called the action matrix
r = np.tensordot(y, np.tensordot(x, r_uneval, (0,0)), (0,1))
result = np.sum(np.abs(l - r))
return result
def irrep_defect_identity(self,irrep):
return np.sum(np.abs(np.tensordot(self.unit, irrep, (0,0)) - np.identity(irrep.shape[1])))
def irrep_defect(self,irrep):
return sum([ self.irrep_defect_multiplication(irrep),
self.irrep_defect_multiplication(irrep),
self.irrep_defect_multiplication(irrep),
self.irrep_defect_identity(irrep)])
def multiply(self, x, y):
return self.multiplication_matrix * np.kron(x,y)
def random_vector(self):
d = self.dimension
return (np.random.rand(d) - 0.5 ) + 1j * (np.random.rand(d) - 0.5 )
def commutator_matrix(self):
d = self.dimension
l = sparse.hstack(
[lm.reshape(d * d, 1)
for lm in self.left_multiplication_matrices])
r = sparse.hstack(
[rm.reshape(d * d, 1)
for rm in self.right_multiplication_matrices])
return l - r
def mult_helper(self,v, ms):
# ms is a list of sparse matrices
d = self.dimension
accumulator = np.zeros((d,d), dtype=complex)
for (c, m) in zip(v, ms):
accumulator += c * m.todense()
return accumulator
def left_multiplication_matrix(self,v):
return self.mult_helper(v, self.left_multiplication_matrices)
def right_multiplication_matrix(self,v):
return self.mult_helper(v, self.right_multiplication_matrices)
def __init__(
self,
dimension,
left_multiplication_matrices,
right_multiplication_matrices,
unit):
# dimension of algebra
# we denote the basis vectors as e_1, e_2, ...., e_dimension
self.dimension = dimension
# list of matrices
# jth column of ith matrix is e_i e_j
self.left_multiplication_matrices = left_multiplication_matrices
# list of matrices
# jth column of ith matrix is e_j e_i
self.right_multiplication_matrices = right_multiplication_matrices
# useful optimization for implementing multiply
self.multiplication_matrix = sparse.hstack(left_multiplication_matrices)
# unit of algebra. Dense numpy array
self.unit = unit
class Algebra:
def associative_defect(self):
x = self.random_vector()
y = self.random_vector()
z = self.random_vector()
l = self.multiply(self.multiply(x,y), z)
r = self.multiply(x, self.multiply(y,z))
return np.sum(np.abs(l - r))
def left_identity_defect(self):
m = self.multiplication
return np.sum(
np.abs(np.tensordot(self.unit, m, (0,0)) -
np.identity(self.dimension)))
def right_identity_defect(self):
m = self.multiplication
return np.sum(
np.abs(np.tensordot(self.unit, m, (0,1)) -
np.identity(self.dimension)))
def commutative_defect(self):
m = self.multiplication
return np.sum(np.abs(m - np.transpose(m,(1,0,2))))
def algebra_defect(self):
return sum([self.associative_defect(),
self.associative_defect(),
self.associative_defect(),
self.left_identity_defect(),
self.right_identity_defect()])
def irrep_defect_multiplication(self,irrep):
m = self.multiplication
result = np.sum(np.abs(np.tensordot(m, irrep, (2,0)) -
np.transpose(np.tensordot(irrep, irrep, (2,1)),(2,0,1,3))))
return result
def irrep_defect_identity(self,irrep):
m = self.multiplication
result = np.sum(np.abs(np.tensordot(self.unit, irrep, (0,0)) - np.identity(irrep.shape[1])))
return result
def irrep_defect(self,irrep):
return self.irrep_defect_multiplication(irrep) + self.irrep_defect_identity(irrep)
def multiply(self,x,y):
m = self.multiplication
return np.tensordot(np.tensordot(x,m,(0,0)),y,(0,0))
def random_vector(self):
d = self.dimension
return (np.random.rand(d) - 0.5 ) + 1j * (np.random.rand(d) - 0.5 )
def commutator_matrix(self):
d = self.dimension
m = self.multiplication
return np.transpose(np.reshape(m - np.transpose(m,(1,0,2)),(d,d*d)))
def left_multiplication_matrix(self,v):
m = self.multiplication
return np.transpose(np.tensordot(v, m, (0,0)))
def right_multiplication_matrix(self,v):
m = self.multiplication
return np.transpose(np.tensordot(v,m,(0,1)))
def __init__(
self,
dimension,
multiplication,
unit
):
# dimension of algebra
# we denote the basis vectors as e_1, e_2, ...., e_dimension
self.dimension = dimension
# multiplication tensor (3D numpy array)
# e_i e_j = sum over k multiplication[i,j,k] e_k
self.multiplication = multiplication
# unit of the algebra in the given basis
self.unit = unit
def load_algebra_from_file(path):
with open(path,'r') as f:
lines = f.readlines()
unit = np.array([complex(x) for x in lines[0].split(' ')])
dimension = len(unit)
multiplication_lines = lines[1:]
left_multiplication_data = [([],([],[])) for _ in range(dimension)]
right_multiplication_data = [([],([],[])) for _ in range(dimension)]
for line in multiplication_lines:
parts = line.split(' ')
i = int(parts[0])
j = int(parts[1])
k = int(parts[2])
val_real = float(parts[3])
val_imaginary = float(parts[4])
val = complex(val_real, val_imaginary)
left_multiplication_data[i][0].append(val)
left_multiplication_data[i][1][0].append(k)
left_multiplication_data[i][1][1].append(j)
right_multiplication_data[j][0].append(val)
right_multiplication_data[j][1][0].append(k)
right_multiplication_data[j][1][1].append(i)
left_multiplication_matrices = [
sparse.coo_matrix(m, dtype=complex, shape=(dimension,dimension))
for m in left_multiplication_data]
right_multiplication_matrices = [
sparse.coo_matrix(m, dtype=complex, shape=(dimension,dimension))
for m in right_multiplication_data]
return SparseAlgebra(
dimension,
left_multiplication_matrices,
right_multiplication_matrices,
unit)