def test_get_symmetric_group_sgs(): assert get_symmetric_group_sgs(2) == ([0], [Permutation(3)(0, 1)]) assert get_symmetric_group_sgs(2, 1) == ([0], [Permutation(0, 1)(2, 3)]) assert get_symmetric_group_sgs(3) == ([0, 1], [Permutation(4)(0, 1), Permutation(4)(1, 2)]) assert get_symmetric_group_sgs(3, 1) == ([0, 1], [Permutation(0, 1)(3, 4), Permutation(1, 2)(3, 4)]) assert get_symmetric_group_sgs(4) == ([0, 1, 2], [Permutation(5)(0, 1), Permutation(5)(1, 2), Permutation(5)(2, 3)]) assert get_symmetric_group_sgs(4, 1) == ([0, 1, 2], [Permutation(0, 1)(4, 5), Permutation(1, 2)(4, 5), Permutation(2, 3)(4, 5)])
def test_canonicalize_no_dummies(): base1, gens1 = get_symmetric_group_sgs(1) base2, gens2 = get_symmetric_group_sgs(2) base2a, gens2a = get_symmetric_group_sgs(2, 1) # A commuting # A^c A^b A^a; ord = [a,b,c]; g = [2,1,0,3,4] # T_c = A^a A^b A^c; can = list(range(5)) g = Permutation([2, 1, 0, 3, 4]) can = canonicalize(g, [], 0, (base1, gens1, 3, 0)) assert can == list(range(5)) # A anticommuting # A^c A^b A^a; ord = [a,b,c]; g = [2,1,0,3,4] # T_c = -A^a A^b A^c; can = [0,1,2,4,3] g = Permutation([2, 1, 0, 3, 4]) can = canonicalize(g, [], 0, (base1, gens1, 3, 1)) assert can == [0, 1, 2, 4, 3] # A commuting and symmetric # A^{b,d}*A^{c,a}; ord = [a,b,c,d]; g = [1,3,2,0,4,5] # T_c = A^{a c}*A^{b d}; can = [0,2,1,3,4,5] g = Permutation([1, 3, 2, 0, 4, 5]) can = canonicalize(g, [], 0, (base2, gens2, 2, 0)) assert can == [0, 2, 1, 3, 4, 5] # A anticommuting and symmetric # A^{b,d}*A^{c,a}; ord = [a,b,c,d]; g = [1,3,2,0,4,5] # T_c = -A^{a c}*A^{b d}; can = [0,2,1,3,5,4] g = Permutation([1, 3, 2, 0, 4, 5]) can = canonicalize(g, [], 0, (base2, gens2, 2, 1)) assert can == [0, 2, 1, 3, 5, 4] # A^{c,a}*A^{b,d} ; g = [2,0,1,3,4,5] # T_c = A^{a c}*A^{b d}; can = [0,2,1,3,4,5] g = Permutation([2, 0, 1, 3, 4, 5]) can = canonicalize(g, [], 0, (base2, gens2, 2, 1)) assert can == [0, 2, 1, 3, 4, 5]
def test_canonicalize_no_slot_sym(): # cases in which there is no slot symmetry after fixing the # free indices; here and in the following if the symmetry of the # metric is not specified, it is assumed to be symmetric. # If it is not specified, tensors are commuting. # A_d0 * B^d0; g = [1,0, 2,3]; T_c = A^d0*B_d0; can = [0,1,2,3] base1, gens1 = get_symmetric_group_sgs(1) dummies = [0, 1] g = Permutation([1, 0, 2, 3]) can = canonicalize(g, dummies, 0, (base1, gens1, 1, 0), (base1, gens1, 1, 0)) assert can == [0, 1, 2, 3] # equivalently can = canonicalize(g, dummies, 0, (base1, gens1, 2, None)) assert can == [0, 1, 2, 3] # with antisymmetric metric; T_c = -A^d0*B_d0; can = [0,1,3,2] can = canonicalize(g, dummies, 1, (base1, gens1, 1, 0), (base1, gens1, 1, 0)) assert can == [0, 1, 3, 2] # A^a * B^b; ord = [a,b]; g = [0,1,2,3]; can = g g = Permutation([0, 1, 2, 3]) dummies = [] t0 = t1 = (base1, gens1, 1, 0) can = canonicalize(g, dummies, 0, t0, t1) assert can == [0, 1, 2, 3] # B^b * A^a g = Permutation([1, 0, 2, 3]) can = canonicalize(g, dummies, 0, t0, t1) assert can == [1, 0, 2, 3] # A symmetric # A^{b}_{d0}*A^{d0, a} order a,b,d0,-d0; T_c = A^{a d0}*A{b}_{d0} # g = [1,3,2,0,4,5]; can = [0,2,1,3,4,5] base2, gens2 = get_symmetric_group_sgs(2) dummies = [2, 3] g = Permutation([1, 3, 2, 0, 4, 5]) can = canonicalize(g, dummies, 0, (base2, gens2, 2, 0)) assert can == [0, 2, 1, 3, 4, 5] # with antisymmetric metric can = canonicalize(g, dummies, 1, (base2, gens2, 2, 0)) assert can == [0, 2, 1, 3, 4, 5] # A^{a}_{d0}*A^{d0, b} g = Permutation([0, 3, 2, 1, 4, 5]) can = canonicalize(g, dummies, 1, (base2, gens2, 2, 0)) assert can == [0, 2, 1, 3, 5, 4] # A, B symmetric # A^b_d0*B^{d0,a}; g=[1,3,2,0,4,5] # T_c = A^{b,d0}*B_{a,d0}; can = [1,2,0,3,4,5] dummies = [2, 3] g = Permutation([1, 3, 2, 0, 4, 5]) can = canonicalize(g, dummies, 0, (base2, gens2, 1, 0), (base2, gens2, 1, 0)) assert can == [1, 2, 0, 3, 4, 5] # same with antisymmetric metric can = canonicalize(g, dummies, 1, (base2, gens2, 1, 0), (base2, gens2, 1, 0)) assert can == [1, 2, 0, 3, 5, 4] # A^{d1}_{d0}*B^d0*C_d1 ord=[d0,-d0,d1,-d1]; g = [2,1,0,3,4,5] # T_c = A^{d0 d1}*B_d0*C_d1; can = [0,2,1,3,4,5] base1, gens1 = get_symmetric_group_sgs(1) base2, gens2 = get_symmetric_group_sgs(2) g = Permutation([2, 1, 0, 3, 4, 5]) dummies = [0, 1, 2, 3] t0 = (base2, gens2, 1, 0) t1 = t2 = (base1, gens1, 1, 0) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [0, 2, 1, 3, 4, 5] # A without symmetry # A^{d1}_{d0}*B^d0*C_d1 ord=[d0,-d0,d1,-d1]; g = [2,1,0,3,4,5] # T_c = A^{d0 d1}*B_d1*C_d0; can = [0,2,3,1,4,5] g = Permutation([2, 1, 0, 3, 4, 5]) dummies = [0, 1, 2, 3] t0 = ([], [Permutation(list(range(4)))], 1, 0) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [0, 2, 3, 1, 4, 5] # A, B without symmetry # A^{d1}_{d0}*B_{d1}^{d0}; g = [2,1,3,0,4,5] # T_c = A^{d0 d1}*B_{d0 d1}; can = [0,2,1,3,4,5] t0 = t1 = ([], [Permutation(list(range(4)))], 1, 0) dummies = [0, 1, 2, 3] g = Permutation([2, 1, 3, 0, 4, 5]) can = canonicalize(g, dummies, 0, t0, t1) assert can == [0, 2, 1, 3, 4, 5] # A_{d0}^{d1}*B_{d1}^{d0}; g = [1,2,3,0,4,5] # T_c = A^{d0 d1}*B_{d1 d0}; can = [0,2,3,1,4,5] g = Permutation([1, 2, 3, 0, 4, 5]) can = canonicalize(g, dummies, 0, t0, t1) assert can == [0, 2, 3, 1, 4, 5] # A, B, C without symmetry # A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1] # g=[4,2,0,3,5,1,6,7] # T_c=A^{d0 d1}*B_{a d1}*C_{d0 b}; can = [2,4,0,5,3,1,6,7] t0 = t1 = t2 = ([], [Permutation(list(range(4)))], 1, 0) dummies = [2, 3, 4, 5] g = Permutation([4, 2, 0, 3, 5, 1, 6, 7]) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [2, 4, 0, 5, 3, 1, 6, 7] # A symmetric, B and C without symmetry # A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1] # g=[4,2,0,3,5,1,6,7] # T_c = A^{d0 d1}*B_{a d0}*C_{d1 b}; can = [2,4,0,3,5,1,6,7] t0 = (base2, gens2, 1, 0) t1 = t2 = ([], [Permutation(list(range(4)))], 1, 0) dummies = [2, 3, 4, 5] g = Permutation([4, 2, 0, 3, 5, 1, 6, 7]) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [2, 4, 0, 3, 5, 1, 6, 7] # A and C symmetric, B without symmetry # A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1] # g=[4,2,0,3,5,1,6,7] # T_c = A^{d0 d1}*B_{a d0}*C_{b d1}; can = [2,4,0,3,1,5,6,7] t0 = t2 = (base2, gens2, 1, 0) t1 = ([], [Permutation(list(range(4)))], 1, 0) dummies = [2, 3, 4, 5] g = Permutation([4, 2, 0, 3, 5, 1, 6, 7]) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [2, 4, 0, 3, 1, 5, 6, 7] # A symmetric, B without symmetry, C antisymmetric # A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1] # g=[4,2,0,3,5,1,6,7] # T_c = -A^{d0 d1}*B_{a d0}*C_{b d1}; can = [2,4,0,3,1,5,7,6] t0 = (base2, gens2, 1, 0) t1 = ([], [Permutation(list(range(4)))], 1, 0) base2a, gens2a = get_symmetric_group_sgs(2, 1) t2 = (base2a, gens2a, 1, 0) dummies = [2, 3, 4, 5] g = Permutation([4, 2, 0, 3, 5, 1, 6, 7]) can = canonicalize(g, dummies, 0, t0, t1, t2) assert can == [2, 4, 0, 3, 1, 5, 7, 6]
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) 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_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 graph_certificate(gr): """ Return a certificate for the graph gr adjacency list The graph is assumed to be unoriented and without external lines. Associate to each vertex of the graph a symmetric tensor with number of indices equal to the degree of the vertex; indices are contracted when they correspond to the same line of the graph. The canonical form of the tensor gives a certificate for the graph. This is not an efficient algorithm to get the certificate of a graph. Examples ======== >>> from diofant.combinatorics.testutil import graph_certificate >>> gr1 = {0:[1, 2, 3, 5], 1:[0, 2, 4], 2:[0, 1, 3, 4], 3:[0, 2, 4], 4:[1, 2, 3, 5], 5:[0, 4]} >>> gr2 = {0:[1, 5], 1:[0, 2, 3, 4], 2:[1, 3, 5], 3:[1, 2, 4, 5], 4:[1, 3, 5], 5:[0, 2, 3, 4]} >>> c1 = graph_certificate(gr1) >>> c2 = graph_certificate(gr2) >>> c1 [0, 2, 4, 6, 1, 8, 10, 12, 3, 14, 16, 18, 5, 9, 15, 7, 11, 17, 13, 19, 20, 21] >>> c1 == c2 True """ from diofant.combinatorics.permutations import _af_invert from diofant.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize items = list(gr.items()) items.sort(key=lambda x: len(x[1]), reverse=True) pvert = [x[0] for x in items] pvert = _af_invert(pvert) # the indices of the tensor are twice the number of lines of the graph num_indices = 0 for v, neigh in items: num_indices += len(neigh) # associate to each vertex its indices; for each line # between two vertices assign the # even index to the vertex which comes first in items, # the odd index to the other vertex vertices = [[] for i in items] i = 0 for v, neigh in items: for v2 in neigh: if pvert[v] < pvert[v2]: vertices[pvert[v]].append(i) vertices[pvert[v2]].append(i+1) i += 2 g = [] for v in vertices: g.extend(v) assert len(g) == num_indices g += [num_indices, num_indices + 1] size = num_indices + 2 assert sorted(g) == list(range(size)) g = Permutation(g) vlen = [0]*(len(vertices[0])+1) for neigh in vertices: vlen[len(neigh)] += 1 v = [] for i in range(len(vlen)): n = vlen[i] if n: base, gens = get_symmetric_group_sgs(i) v.append((base, gens, n, 0)) v.reverse() dummies = list(range(num_indices)) can = canonicalize(g, dummies, 0, *v) return can