def asym_bsgs(self, hermitian): """ Return minimal base and strong generating set for antisymmetric indices. :param hermitian: upper and lower indices can be swapped if True :return: a tuple of (base, strong generating set) """ if not hermitian: u_base, u_gens = get_symmetric_group_sgs(self.n_upper, 1) l_base, l_gens = get_symmetric_group_sgs(self.n_lower, 1) return bsgs_direct_product(u_base, u_gens, l_base, l_gens) if self.n_upper != self.n_lower: raise ValueError(f"{self} cannot be Hermitian.") upper = list(range(self.n_upper)) lower = list(range(self.n_upper, self.size)) sign = [self.size, self.size + 1] perms = [ Permutation(i, j)(sign[0], sign[1]) for i, j in zip(upper[:-1], upper[1:]) ] perms += [ Permutation(i, j)(sign[0], sign[1]) for i, j in zip(lower[:-1], lower[1:]) ] p = list(range(self.size + 2)) for i, j in zip(upper, lower): p[i] = j p[j] = i perms.append(Permutation(p)) asymmetric = PermutationGroup(*perms) asymmetric.schreier_sims() return get_minimal_bsgs(asymmetric.base, asymmetric.strong_gens)
def test_riemann_products(): baser, gensr = riemann_bsgs base1, gens1 = get_symmetric_group_sgs(1) base2, gens2 = get_symmetric_group_sgs(2) base2a, gens2a = get_symmetric_group_sgs(2, 1) # R^{a b d0}_d0 = 0 g = Permutation([0, 1, 2, 3, 4, 5]) can = canonicalize(g, list(range(2, 4)), 0, (baser, gensr, 1, 0)) assert can == 0 # R^{d0 b a}_d0 ; ord = [a,b,d0,-d0}; g = [2,1,0,3,4,5] # T_c = -R^{a d0 b}_d0; can = [0,2,1,3,5,4] g = Permutation([2, 1, 0, 3, 4, 5]) can = canonicalize(g, list(range(2, 4)), 0, (baser, gensr, 1, 0)) assert can == [0, 2, 1, 3, 5, 4] # R^d1_d2^b_d0 * R^{d0 a}_d1^d2; ord=[a,b,d0,-d0,d1,-d1,d2,-d2] # g = [4,7,1,3,2,0,5,6,8,9] # T_c = -R^{a d0 d1 d2}* R^b_{d0 d1 d2} # can = [0,2,4,6,1,3,5,7,9,8] g = Permutation([4, 7, 1, 3, 2, 0, 5, 6, 8, 9]) can = canonicalize(g, list(range(2, 8)), 0, (baser, gensr, 2, 0)) assert can == [0, 2, 4, 6, 1, 3, 5, 7, 9, 8] can1 = canonicalize_naive(g, list(range(2, 8)), 0, (baser, gensr, 2, 0)) assert can == can1 # A symmetric commuting # R^{d6 d5}_d2^d1 * R^{d4 d0 d2 d3} * A_{d6 d0} A_{d3 d1} * A_{d4 d5} # g = [12,10,5,2, 8,0,4,6, 13,1, 7,3, 9,11,14,15] # T_c = -R^{d0 d1 d2 d3} * R_d0^{d4 d5 d6} * A_{d1 d4}*A_{d2 d5}*A_{d3 d6} g = Permutation([12, 10, 5, 2, 8, 0, 4, 6, 13, 1, 7, 3, 9, 11, 14, 15]) can = canonicalize(g, list(range(14)), 0, ((baser, gensr, 2, 0)), (base2, gens2, 3, 0)) assert can == [0, 2, 4, 6, 1, 8, 10, 12, 3, 9, 5, 11, 7, 13, 15, 14] # R^{d2 a0 a2 d0} * R^d1_d2^{a1 a3} * R^{a4 a5}_{d0 d1} # ord = [a0,a1,a2,a3,a4,a5,d0,-d0,d1,-d1,d2,-d2] # 0 1 2 3 4 5 6 7 8 9 10 11 # can = [0, 6, 2, 8, 1, 3, 7, 10, 4, 5, 9, 11, 12, 13] # T_c = R^{a0 d0 a2 d1}*R^{a1 a3}_d0^d2*R^{a4 a5}_{d1 d2} g = Permutation([10, 0, 2, 6, 8, 11, 1, 3, 4, 5, 7, 9, 12, 13]) can = canonicalize(g, list(range(6, 12)), 0, (baser, gensr, 3, 0)) assert can == [0, 6, 2, 8, 1, 3, 7, 10, 4, 5, 9, 11, 12, 13] #can1 = canonicalize_naive(g, list(range(6,12)), 0, (baser, gensr, 3, 0)) #assert can == can1 # A^n_{i, j} antisymmetric in i,j # A_m0^d0_a1 * A_m1^a0_d0; ord = [m0,m1,a0,a1,d0,-d0] # g = [0,4,3,1,2,5,6,7] # T_c = -A_{m a1}^d0 * A_m1^a0_d0 # can = [0,3,4,1,2,5,7,6] base, gens = bsgs_direct_product(base1, gens1, base2a, gens2a) dummies = list(range(4, 6)) g = Permutation([0, 4, 3, 1, 2, 5, 6, 7]) can = canonicalize(g, dummies, 0, (base, gens, 2, 0)) assert can == [0, 3, 4, 1, 2, 5, 7, 6] # A^n_{i, j} symmetric in i,j # A^m0_a0^d2 * A^n0_d2^d1 * A^n1_d1^d0 * A_{m0 d0}^a1 # ordering: first the free indices; then first n, then d # ord=[n0,n1,a0,a1, m0,-m0,d0,-d0,d1,-d1,d2,-d2] # 0 1 2 3 4 5 6 7 8 9 10 11] # g = [4,2,10, 0,11,8, 1,9,6, 5,7,3, 12,13] # if the dummy indices m_i and d_i were separated, # one gets # T_c = A^{n0 d0 d1} * A^n1_d0^d2 * A^m0^a0_d1 * A_m0^a1_d2 # can = [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 12, 13] # If they are not, so can is # T_c = A^{n0 m0 d0} A^n1_m0^d1 A^{d2 a0}_d0 A_d2^a1_d1 # can = [0, 4, 6, 1, 5, 8, 10, 2, 7, 11, 3, 9, 12, 13] # case with single type of indices base, gens = bsgs_direct_product(base1, gens1, base2, gens2) dummies = list(range(4, 12)) g = Permutation([4, 2, 10, 0, 11, 8, 1, 9, 6, 5, 7, 3, 12, 13]) can = canonicalize(g, dummies, 0, (base, gens, 4, 0)) assert can == [0, 4, 6, 1, 5, 8, 10, 2, 7, 11, 3, 9, 12, 13] # case with separated indices dummies = [list(range(4, 6)), list(range(6, 12))] sym = [0, 0] can = canonicalize(g, dummies, sym, (base, gens, 4, 0)) assert can == [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 12, 13] # case with separated indices with the second type of index # with antisymmetric metric: there is a sign change sym = [0, 1] can = canonicalize(g, dummies, sym, (base, gens, 4, 0)) assert can == [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 13, 12]
def test_canonicalize1(): base1, gens1 = get_symmetric_group_sgs(1) base1a, gens1a = get_symmetric_group_sgs(1, 1) base2, gens2 = get_symmetric_group_sgs(2) base3, gens3 = get_symmetric_group_sgs(3) base2a, gens2a = get_symmetric_group_sgs(2, 1) base3a, gens3a = get_symmetric_group_sgs(3, 1) # A_d0*A^d0; ord = [d0,-d0]; g = [1,0,2,3] # T_c = A^d0*A_d0; can = [0,1,2,3] g = Permutation([1, 0, 2, 3]) can = canonicalize(g, [0, 1], 0, (base1, gens1, 2, 0)) assert can == list(range(4)) # A commuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0; ord=[d0,-d0,d1,-d1,d2,-d2] # g = [1,3,5,4,2,0,6,7] # T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2; can = list(range(8)) g = Permutation([1, 3, 5, 4, 2, 0, 6, 7]) can = canonicalize(g, list(range(6)), 0, (base1, gens1, 6, 0)) assert can == list(range(8)) # A anticommuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0; ord=[d0,-d0,d1,-d1,d2,-d2] # g = [1,3,5,4,2,0,6,7] # T_c 0; can = 0 g = Permutation([1, 3, 5, 4, 2, 0, 6, 7]) can = canonicalize(g, list(range(6)), 0, (base1, gens1, 6, 1)) assert can == 0 can1 = canonicalize_naive(g, list(range(6)), 0, (base1, gens1, 6, 1)) assert can1 == 0 # A commuting symmetric # A^{d0 b}*A^a_d1*A^d1_d0; ord=[a,b,d0,-d0,d1,-d1] # g = [2,1,0,5,4,3,6,7] # T_c = A^{a d0}*A^{b d1}*A_{d0 d1}; can = [0,2,1,4,3,5,6,7] g = Permutation([2, 1, 0, 5, 4, 3, 6, 7]) can = canonicalize(g, list(range(2, 6)), 0, (base2, gens2, 3, 0)) assert can == [0, 2, 1, 4, 3, 5, 6, 7] # A, B commuting symmetric # A^{d0 b}*A^d1_d0*B^a_d1; ord=[a,b,d0,-d0,d1,-d1] # g = [2,1,4,3,0,5,6,7] # T_c = A^{b d0}*A_d0^d1*B^a_d1; can = [1,2,3,4,0,5,6,7] g = Permutation([2, 1, 4, 3, 0, 5, 6, 7]) can = canonicalize(g, list(range(2, 6)), 0, (base2, gens2, 2, 0), (base2, gens2, 1, 0)) assert can == [1, 2, 3, 4, 0, 5, 6, 7] # A commuting symmetric # A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1] # g = [4,2,1,0,5,3,6,7] # T_c = A^{a d0 d1}*A^{b}_{d0 d1}; can = [0,2,4,1,3,5,6,7] g = Permutation([4, 2, 1, 0, 5, 3, 6, 7]) can = canonicalize(g, list(range(2, 6)), 0, (base3, gens3, 2, 0)) assert can == [0, 2, 4, 1, 3, 5, 6, 7] # A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0 # ord = [a0,a1,a2,a3,d0,-d0,d1,-d1,d2,-d2,d3,-d3] # 0 1 2 3 4 5 6 7 8 9 10 11 # g = [10,4,8, 0,7,9, 6,11,1, 2,3,5, 12,13] # T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3} # can = [0,4,6, 1,5,8, 2,3,10, 7,9,11, 12,13] g = Permutation([10, 4, 8, 0, 7, 9, 6, 11, 1, 2, 3, 5, 12, 13]) can = canonicalize(g, list(range(4, 12)), 0, (base3, gens3, 4, 0)) assert can == [0, 4, 6, 1, 5, 8, 2, 3, 10, 7, 9, 11, 12, 13] # A commuting symmetric, B antisymmetric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # ord = [d0,-d0,d1,-d1,d2,-d2,d3,-d3] # g = [0,2,4,5,7,3,1,6,8,9] # in this esxample and in the next three, # renaming dummy indices and using symmetry of A, # T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = 0 g = Permutation([0, 2, 4, 5, 7, 3, 1, 6, 8, 9]) can = canonicalize(g, list(range(8)), 0, (base3, gens3, 2, 0), (base2a, gens2a, 1, 0)) assert can == 0 # A anticommuting symmetric, B anticommuting # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} # can = [0,2,4, 1,3,6, 5,7, 8,9] can = canonicalize(g, list(range(8)), 0, (base3, gens3, 2, 1), (base2a, gens2a, 1, 0)) assert can == [0, 2, 4, 1, 3, 6, 5, 7, 8, 9] # A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} # can = [0,2,4, 1,3,6, 5,7, 9,8] can = canonicalize(g, list(range(8)), 1, (base3, gens3, 2, 1), (base2a, gens2a, 1, 0)) assert can == [0, 2, 4, 1, 3, 6, 5, 7, 9, 8] # A anticommuting symmetric, B anticommuting anticommuting, # no metric symmetry # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = [0,2,4, 1,3,7, 5,6, 8,9] can = canonicalize(g, list(range(8)), None, (base3, gens3, 2, 1), (base2a, gens2a, 1, 0)) assert can == [0, 2, 4, 1, 3, 7, 5, 6, 8, 9] # Gamma anticommuting # Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha} # ord = [alpha, rho, mu,-mu,nu,-nu] # g = [3,5,1,4,2,0,6,7] # T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu} # can = [2,4,1,0,3,5,7,6]] g = Permutation([3, 5, 1, 4, 2, 0, 6, 7]) t0 = (base2a, gens2a, 1, None) t1 = (base1, gens1, 1, None) t2 = (base3a, gens3a, 1, None) can = canonicalize(g, list(range(2, 6)), 0, t0, t1, t2) assert can == [2, 4, 1, 0, 3, 5, 7, 6] # Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha} # ord = [alpha, beta, gamma, -rho, mu,-mu,nu,-nu] # 0 1 2 3 4 5 6 7 # g = [5,7,2,1,3,6,4,0,8,9] # T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu} # can = [4,6,1,2,3,0,5,7,8,9] t0 = (base2a, gens2a, 2, None) g = Permutation([5, 7, 2, 1, 3, 6, 4, 0, 8, 9]) can = canonicalize(g, list(range(4, 8)), 0, t0, t1, t2) assert can == [4, 6, 1, 2, 3, 0, 5, 7, 8, 9] # f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry # f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b} # ord = [mu,-mu,nu,-nu,a,-a,b,-b,c,-c,d,-d, e, -e] # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 # g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15] # T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e} # can = [4,6,8, 5,10,12, 0,7, 1,11, 2,9, 3,13, 15,14] g = Permutation([8, 11, 5, 9, 13, 7, 1, 10, 3, 4, 2, 12, 0, 6, 14, 15]) base_f, gens_f = bsgs_direct_product(base1, gens1, base2a, gens2a) base_A, gens_A = bsgs_direct_product(base1, gens1, base1, gens1) t0 = (base_f, gens_f, 2, 0) t1 = (base_A, gens_A, 4, 0) can = canonicalize(g, [list(range(4)), list(range(4, 14))], [0, 0], t0, t1) assert can == [4, 6, 8, 5, 10, 12, 0, 7, 1, 11, 2, 9, 3, 13, 15, 14]
def test_riemann_products(): baser, gensr = riemann_bsgs base1, gens1 = get_symmetric_group_sgs(1) base2, gens2 = get_symmetric_group_sgs(2) base2a, gens2a = get_symmetric_group_sgs(2, 1) # R^{a b d0}_d0 = 0 g = Permutation([0,1,2,3,4,5]) can = canonicalize(g, list(range(2,4)), 0, (baser, gensr, 1, 0)) assert can == 0 # R^{d0 b a}_d0 ; ord = [a,b,d0,-d0}; g = [2,1,0,3,4,5] # T_c = -R^{a d0 b}_d0; can = [0,2,1,3,5,4] g = Permutation([2,1,0,3,4,5]) can = canonicalize(g, list(range(2, 4)), 0, (baser, gensr, 1, 0)) assert can == [0,2,1,3,5,4] # R^d1_d2^b_d0 * R^{d0 a}_d1^d2; ord=[a,b,d0,-d0,d1,-d1,d2,-d2] # g = [4,7,1,3,2,0,5,6,8,9] # T_c = -R^{a d0 d1 d2}* R^b_{d0 d1 d2} # can = [0,2,4,6,1,3,5,7,9,8] g = Permutation([4,7,1,3,2,0,5,6,8,9]) can = canonicalize(g, list(range(2,8)), 0, (baser, gensr, 2, 0)) assert can == [0,2,4,6,1,3,5,7,9,8] can1 = canonicalize_naive(g, list(range(2,8)), 0, (baser, gensr, 2, 0)) assert can == can1 # A symmetric commuting # R^{d6 d5}_d2^d1 * R^{d4 d0 d2 d3} * A_{d6 d0} A_{d3 d1} * A_{d4 d5} # g = [12,10,5,2, 8,0,4,6, 13,1, 7,3, 9,11,14,15] # T_c = -R^{d0 d1 d2 d3} * R_d0^{d4 d5 d6} * A_{d1 d4}*A_{d2 d5}*A_{d3 d6} g = Permutation([12,10,5,2,8,0,4,6,13,1,7,3,9,11,14,15]) can = canonicalize(g, list(range(14)), 0, ((baser,gensr,2,0)), (base2,gens2,3,0)) assert can == [0, 2, 4, 6, 1, 8, 10, 12, 3, 9, 5, 11, 7, 13, 15, 14] # R^{d2 a0 a2 d0} * R^d1_d2^{a1 a3} * R^{a4 a5}_{d0 d1} # ord = [a0,a1,a2,a3,a4,a5,d0,-d0,d1,-d1,d2,-d2] # 0 1 2 3 4 5 6 7 8 9 10 11 # can = [0, 6, 2, 8, 1, 3, 7, 10, 4, 5, 9, 11, 12, 13] # T_c = R^{a0 d0 a2 d1}*R^{a1 a3}_d0^d2*R^{a4 a5}_{d1 d2} g = Permutation([10,0,2,6,8,11,1,3,4,5,7,9,12,13]) can = canonicalize(g, list(range(6,12)), 0, (baser, gensr, 3, 0)) assert can == [0, 6, 2, 8, 1, 3, 7, 10, 4, 5, 9, 11, 12, 13] #can1 = canonicalize_naive(g, list(range(6,12)), 0, (baser, gensr, 3, 0)) #assert can == can1 # A^n_{i, j} antisymmetric in i,j # A_m0^d0_a1 * A_m1^a0_d0; ord = [m0,m1,a0,a1,d0,-d0] # g = [0,4,3,1,2,5,6,7] # T_c = -A_{m a1}^d0 * A_m1^a0_d0 # can = [0,3,4,1,2,5,7,6] base, gens = bsgs_direct_product(base1, gens1, base2a, gens2a) dummies = list(range(4, 6)) g = Permutation([0,4,3,1,2,5,6,7]) can = canonicalize(g, dummies, 0, (base, gens, 2, 0)) assert can == [0, 3, 4, 1, 2, 5, 7, 6] # A^n_{i, j} symmetric in i,j # A^m0_a0^d2 * A^n0_d2^d1 * A^n1_d1^d0 * A_{m0 d0}^a1 # ordering: first the free indices; then first n, then d # ord=[n0,n1,a0,a1, m0,-m0,d0,-d0,d1,-d1,d2,-d2] # 0 1 2 3 4 5 6 7 8 9 10 11] # g = [4,2,10, 0,11,8, 1,9,6, 5,7,3, 12,13] # if the dummy indices m_i and d_i were separated, # one gets # T_c = A^{n0 d0 d1} * A^n1_d0^d2 * A^m0^a0_d1 * A_m0^a1_d2 # can = [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 12, 13] # If they are not, so can is # T_c = A^{n0 m0 d0} A^n1_m0^d1 A^{d2 a0}_d0 A_d2^a1_d1 # can = [0, 4, 6, 1, 5, 8, 10, 2, 7, 11, 3, 9, 12, 13] # case with single type of indices base, gens = bsgs_direct_product(base1, gens1, base2, gens2) dummies = list(range(4, 12)) g = Permutation([4,2,10, 0,11,8, 1,9,6, 5,7,3, 12,13]) can = canonicalize(g, dummies, 0, (base, gens, 4, 0)) assert can == [0, 4, 6, 1, 5, 8, 10, 2, 7, 11, 3, 9, 12, 13] # case with separated indices dummies = [list(range(4, 6)), list(range(6,12))] sym = [0, 0] can = canonicalize(g, dummies, sym, (base, gens, 4, 0)) assert can == [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 12, 13] # case with separated indices with the second type of index # with antisymmetric metric: there is a sign change sym = [0, 1] can = canonicalize(g, dummies, sym, (base, gens, 4, 0)) assert can == [0, 6, 8, 1, 7, 10, 4, 2, 9, 5, 3, 11, 13, 12]
def test_canonicalize1(): base1, gens1 = get_symmetric_group_sgs(1) base1a, gens1a = get_symmetric_group_sgs(1, 1) base2, gens2 = get_symmetric_group_sgs(2) base3, gens3 = get_symmetric_group_sgs(3) base2a, gens2a = get_symmetric_group_sgs(2, 1) base3a, gens3a = get_symmetric_group_sgs(3, 1) # A_d0*A^d0; ord = [d0,-d0]; g = [1,0,2,3] # T_c = A^d0*A_d0; can = [0,1,2,3] g = Permutation([1,0,2,3]) can = canonicalize(g, [0, 1], 0, (base1, gens1, 2, 0)) assert can == list(range(4)) # A commuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0; ord=[d0,-d0,d1,-d1,d2,-d2] # g = [1,3,5,4,2,0,6,7] # T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2; can = list(range(8)) g = Permutation([1,3,5,4,2,0,6,7]) can = canonicalize(g, list(range(6)), 0, (base1, gens1, 6, 0)) assert can == list(range(8)) # A anticommuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0; ord=[d0,-d0,d1,-d1,d2,-d2] # g = [1,3,5,4,2,0,6,7] # T_c 0; can = 0 g = Permutation([1,3,5,4,2,0,6,7]) can = canonicalize(g, list(range(6)), 0, (base1, gens1, 6, 1)) assert can == 0 can1 = canonicalize_naive(g, list(range(6)), 0, (base1, gens1, 6, 1)) assert can1 == 0 # A commuting symmetric # A^{d0 b}*A^a_d1*A^d1_d0; ord=[a,b,d0,-d0,d1,-d1] # g = [2,1,0,5,4,3,6,7] # T_c = A^{a d0}*A^{b d1}*A_{d0 d1}; can = [0,2,1,4,3,5,6,7] g = Permutation([2,1,0,5,4,3,6,7]) can = canonicalize(g, list(range(2,6)), 0, (base2, gens2, 3, 0)) assert can == [0,2,1,4,3,5,6,7] # A, B commuting symmetric # A^{d0 b}*A^d1_d0*B^a_d1; ord=[a,b,d0,-d0,d1,-d1] # g = [2,1,4,3,0,5,6,7] # T_c = A^{b d0}*A_d0^d1*B^a_d1; can = [1,2,3,4,0,5,6,7] g = Permutation([2,1,4,3,0,5,6,7]) can = canonicalize(g, list(range(2,6)), 0, (base2,gens2,2,0), (base2,gens2,1,0)) assert can == [1,2,3,4,0,5,6,7] # A commuting symmetric # A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1] # g = [4,2,1,0,5,3,6,7] # T_c = A^{a d0 d1}*A^{b}_{d0 d1}; can = [0,2,4,1,3,5,6,7] g = Permutation([4,2,1,0,5,3,6,7]) can = canonicalize(g, list(range(2,6)), 0, (base3, gens3, 2, 0)) assert can == [0,2,4,1,3,5,6,7] # A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0 # ord = [a0,a1,a2,a3,d0,-d0,d1,-d1,d2,-d2,d3,-d3] # 0 1 2 3 4 5 6 7 8 9 10 11 # g = [10,4,8, 0,7,9, 6,11,1, 2,3,5, 12,13] # T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3} # can = [0,4,6, 1,5,8, 2,3,10, 7,9,11, 12,13] g = Permutation([10,4,8, 0,7,9, 6,11,1, 2,3,5, 12,13]) can = canonicalize(g, list(range(4,12)), 0, (base3, gens3, 4, 0)) assert can == [0,4,6, 1,5,8, 2,3,10, 7,9,11, 12,13] # A commuting symmetric, B antisymmetric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # ord = [d0,-d0,d1,-d1,d2,-d2,d3,-d3] # g = [0,2,4,5,7,3,1,6,8,9] # in this esxample and in the next three, # renaming dummy indices and using symmetry of A, # T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = 0 g = Permutation([0,2,4,5,7,3,1,6,8,9]) can = canonicalize(g, list(range(8)), 0, (base3, gens3,2,0), (base2a,gens2a,1,0)) assert can == 0 # A anticommuting symmetric, B anticommuting # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} # can = [0,2,4, 1,3,6, 5,7, 8,9] can = canonicalize(g, list(range(8)), 0, (base3, gens3,2,1), (base2a,gens2a,1,0)) assert can == [0,2,4, 1,3,6, 5,7, 8,9] # A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} # can = [0,2,4, 1,3,6, 5,7, 9,8] can = canonicalize(g, list(range(8)), 1, (base3, gens3,2,1), (base2a,gens2a,1,0)) assert can == [0,2,4, 1,3,6, 5,7, 9,8] # A anticommuting symmetric, B anticommuting anticommuting, # no metric symmetry # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = [0,2,4, 1,3,7, 5,6, 8,9] can = canonicalize(g, list(range(8)), None, (base3, gens3,2,1), (base2a,gens2a,1,0)) assert can == [0,2,4,1,3,7,5,6,8,9] # Gamma anticommuting # Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha} # ord = [alpha, rho, mu,-mu,nu,-nu] # g = [3,5,1,4,2,0,6,7] # T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu} # can = [2,4,1,0,3,5,7,6]] g = Permutation([3,5,1,4,2,0,6,7]) t0 = (base2a, gens2a, 1, None) t1 = (base1, gens1, 1, None) t2 = (base3a, gens3a, 1, None) can = canonicalize(g, list(range(2, 6)), 0, t0, t1, t2) assert can == [2,4,1,0,3,5,7,6] # Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha} # ord = [alpha, beta, gamma, -rho, mu,-mu,nu,-nu] # 0 1 2 3 4 5 6 7 # g = [5,7,2,1,3,6,4,0,8,9] # T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu} # can = [4,6,1,2,3,0,5,7,8,9] t0 = (base2a, gens2a, 2, None) g = Permutation([5,7,2,1,3,6,4,0,8,9]) can = canonicalize(g, list(range(4, 8)), 0, t0, t1, t2) assert can == [4,6,1,2,3,0,5,7,8,9] # f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry # f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b} # ord = [mu,-mu,nu,-nu,a,-a,b,-b,c,-c,d,-d, e, -e] # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 # g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15] # T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e} # can = [4,6,8, 5,10,12, 0,7, 1,11, 2,9, 3,13, 15,14] g = Permutation([8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15]) base_f, gens_f = bsgs_direct_product(base1, gens1, base2a, gens2a) base_A, gens_A = bsgs_direct_product(base1, gens1, base1, gens1) t0 = (base_f, gens_f, 2, 0) t1 = (base_A, gens_A, 4, 0) can = canonicalize(g, [list(range(4)), list(range(4, 14))], [0, 0], t0, t1) assert can == [4,6,8, 5,10,12, 0,7, 1,11, 2,9, 3,13, 15,14]