def test_no_metric_symmetry():
# no metric symmetry
# A^d1_d0 * A^d0_d1; ord = [d0,-d0,d1,-d1]; g= [2,1,0,3,4,5]
# T_c = A^d0_d1 * A^d1_d0; can = [0,3,2,1,4,5]
g = Permutation([2,1,0,3,4,5])
can = canonicalize(g, list(range(4)), None, [[], [Permutation(list(range(4)))], 2, 0])
assert can == [0,3,2,1,4,5]
# A^d1_d2 * A^d0_d3 * A^d2_d1 * A^d3_d0
# ord = [d0,-d0,d1,-d1,d2,-d2,d3,-d3]
# 0 1 2 3 4 5 6 7
# g = [2,5,0,7,4,3,6,1,8,9]
# T_c = A^d0_d1 * A^d1_d0 * A^d2_d3 * A^d3_d2
# can = [0,3,2,1,4,7,6,5,8,9]
g = Permutation([2,5,0,7,4,3,6,1,8,9])
#can = canonicalize(g, list(range(8)), 0, [[], [list(range(4))], 4, 0])
#assert can == [0, 2, 3, 1, 4, 6, 7, 5, 8, 9]
can = canonicalize(g, list(range(8)), None, [[], [Permutation(list(range(4)))], 4, 0])
assert can == [0, 3, 2, 1, 4, 7, 6, 5, 8, 9]
# A^d0_d2 * A^d1_d3 * A^d3_d0 * A^d2_d1
# g = [0,5,2,7,6,1,4,3,8,9]
# T_c = A^d0_d1 * A^d1_d2 * A^d2_d3 * A^d3_d0
# can = [0,3,2,5,4,7,6,1,8,9]
g = Permutation([0,5,2,7,6,1,4,3,8,9])
can = canonicalize(g, list(range(8)), None, [[], [Permutation(list(range(4)))], 4, 0])
assert can == [0,3,2,5,4,7,6,1,8,9]
g = Permutation([12,7,10,3,14,13,4,11,6,1,2,9,0,15,8,5,16,17])
can = canonicalize(g, list(range(16)), None, [[], [Permutation(list(range(4)))], 8, 0])
assert can == [0,3,2,5,4,7,6,1,8,11,10,13,12,15,14,9,16,17]
def test_riemann_invariants1():
skip('takes too much time')
baser, gensr = riemann_bsgs
g = Permutation([
17, 44, 11, 3, 0, 19, 23, 15, 38, 4, 25, 27, 43, 36, 22, 14, 8, 30, 41,
20, 2, 10, 12, 28, 18, 1, 29, 13, 37, 42, 33, 7, 9, 31, 24, 26, 39, 5,
34, 47, 32, 6, 21, 40, 35, 46, 45, 16, 48, 49
])
can = canonicalize(g, list(range(48)), 0, (baser, gensr, 12, 0))
assert can == [
0, 2, 4, 6, 1, 3, 8, 10, 5, 7, 12, 14, 9, 11, 16, 18, 13, 15, 20, 22,
17, 19, 24, 26, 21, 23, 28, 30, 25, 27, 32, 34, 29, 31, 36, 38, 33, 35,
40, 42, 37, 39, 44, 46, 41, 43, 45, 47, 48, 49
]
g = Permutation([
0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 19, 22, 24, 26, 5, 21, 28,
30, 32, 34, 36, 38, 40, 42, 44, 46, 13, 48, 50, 52, 15, 49, 54, 56, 17,
33, 41, 58, 9, 23, 60, 62, 29, 35, 63, 64, 3, 45, 66, 68, 25, 37, 47,
57, 11, 31, 69, 70, 27, 39, 53, 72, 1, 59, 73, 74, 55, 61, 67, 76, 43,
65, 75, 78, 51, 71, 77, 79, 80, 81
])
can = canonicalize(g, list(range(80)), 0, (baser, gensr, 20, 0))
assert can == [
0, 2, 4, 6, 1, 8, 10, 12, 3, 14, 16, 18, 5, 20, 22, 24, 7, 26, 28, 30,
9, 15, 32, 34, 11, 36, 23, 38, 13, 40, 42, 44, 17, 39, 29, 46, 19, 48,
43, 50, 21, 45, 52, 54, 25, 56, 33, 58, 27, 60, 53, 62, 31, 51, 64, 66,
35, 65, 47, 68, 37, 70, 49, 72, 41, 74, 57, 76, 55, 67, 59, 78, 61, 69,
71, 75, 63, 79, 73, 77, 80, 81
]
def test_no_metric_symmetry():
# no metric symmetry
# A^d1_d0 * A^d0_d1; ord = [d0,-d0,d1,-d1]; g= [2,1,0,3,4,5]
# T_c = A^d0_d1 * A^d1_d0; can = [0,3,2,1,4,5]
g = Permutation([2,1,0,3,4,5])
can = canonicalize(g, list(range(4)), None, [[], [Permutation(list(range(4)))], 2, 0])
assert can == [0,3,2,1,4,5]
# A^d1_d2 * A^d0_d3 * A^d2_d1 * A^d3_d0
# ord = [d0,-d0,d1,-d1,d2,-d2,d3,-d3]
# 0 1 2 3 4 5 6 7
# g = [2,5,0,7,4,3,6,1,8,9]
# T_c = A^d0_d1 * A^d1_d0 * A^d2_d3 * A^d3_d2
# can = [0,3,2,1,4,7,6,5,8,9]
g = Permutation([2,5,0,7,4,3,6,1,8,9])
#can = canonicalize(g, list(range(8)), 0, [[], [list(range(4))], 4, 0])
#assert can == [0, 2, 3, 1, 4, 6, 7, 5, 8, 9]
can = canonicalize(g, list(range(8)), None, [[], [Permutation(list(range(4)))], 4, 0])
assert can == [0, 3, 2, 1, 4, 7, 6, 5, 8, 9]
# A^d0_d2 * A^d1_d3 * A^d3_d0 * A^d2_d1
# g = [0,5,2,7,6,1,4,3,8,9]
# T_c = A^d0_d1 * A^d1_d2 * A^d2_d3 * A^d3_d0
# can = [0,3,2,5,4,7,6,1,8,9]
g = Permutation([0,5,2,7,6,1,4,3,8,9])
can = canonicalize(g, list(range(8)), None, [[], [Permutation(list(range(4)))], 4, 0])
assert can == [0,3,2,5,4,7,6,1,8,9]
g = Permutation([12,7,10,3,14,13,4,11,6,1,2,9,0,15,8,5,16,17])
can = canonicalize(g, list(range(16)), None, [[], [Permutation(list(range(4)))], 8, 0])
assert can == [0,3,2,5,4,7,6,1,8,11,10,13,12,15,14,9,16,17]
def test_riemann_invariants():
baser, gensr = riemann_bsgs
# R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1]; g = [0,2,3,1,4,5]
# T_c = -R^{d0 d1}_{d0 d1}; can = [0,2,1,3,5,4]
g = Permutation([0, 2, 3, 1, 4, 5])
can = canonicalize(g, list(range(2, 4)), 0, (baser, gensr, 1, 0))
assert can == [0, 2, 1, 3, 5, 4]
# use a non minimal BSGS
can = canonicalize(g, list(range(2, 4)), 0, ([2, 0], [
Permutation([1, 0, 2, 3, 5, 4]),
Permutation([2, 3, 0, 1, 4, 5])
], 1, 0))
assert can == [0, 2, 1, 3, 5, 4]
"""
The following tests in test_riemann_invariants and in
test_riemann_invariants1 have been checked using xperm.c from XPerm in
in [1] and with an older version contained in [2]
[1] xperm.c part of xPerm written by J. M. Martin-Garcia
http://www.xact.es/index.html
[2] test_xperm.cc in cadabra by Kasper Peeters, http://cadabra.phi-sci.com/
"""
# R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} *
# R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10}
# ord: contravariant d_k ->2*k, covariant d_k -> 2*k+1
# T_c = R^{d0 d1 d2 d3} * R_{d0 d1}^{d4 d5} * R_{d2 d3}^{d6 d7} *
# R_{d4 d5}^{d8 d9} * R_{d6 d7}^{d10 d11} * R_{d8 d9 d10 d11}
g = Permutation([
23, 2, 1, 10, 12, 8, 0, 11, 15, 5, 17, 19, 21, 7, 13, 9, 4, 14, 22, 3,
16, 18, 6, 20, 24, 25
])
can = canonicalize(g, list(range(24)), 0, (baser, gensr, 6, 0))
assert can == [
0, 2, 4, 6, 1, 3, 8, 10, 5, 7, 12, 14, 9, 11, 16, 18, 13, 15, 20, 22,
17, 19, 21, 23, 24, 25
]
# use a non minimal BSGS
can = canonicalize(g, list(range(24)), 0, ([2, 0], [
Permutation([1, 0, 2, 3, 5, 4]),
Permutation([2, 3, 0, 1, 4, 5])
], 6, 0))
assert can == [
0, 2, 4, 6, 1, 3, 8, 10, 5, 7, 12, 14, 9, 11, 16, 18, 13, 15, 20, 22,
17, 19, 21, 23, 24, 25
]
g = Permutation([
0, 2, 5, 7, 4, 6, 9, 11, 8, 10, 13, 15, 12, 14, 17, 19, 16, 18, 21, 23,
20, 22, 25, 27, 24, 26, 29, 31, 28, 30, 33, 35, 32, 34, 37, 39, 36, 38,
1, 3, 40, 41
])
can = canonicalize(g, list(range(40)), 0, (baser, gensr, 10, 0))
assert can == [
0, 2, 4, 6, 1, 3, 8, 10, 5, 7, 12, 14, 9, 11, 16, 18, 13, 15, 20, 22,
17, 19, 24, 26, 21, 23, 28, 30, 25, 27, 32, 34, 29, 31, 36, 38, 33, 35,
37, 39, 40, 41
]
def test_riemann_invariants1():
skip('takes too much time')
baser, gensr = riemann_bsgs
g = Permutation([17, 44, 11, 3, 0, 19, 23, 15, 38, 4, 25, 27, 43, 36, 22, 14, 8, 30, 41, 20, 2, 10, 12, 28, 18, 1, 29, 13, 37, 42, 33, 7, 9, 31, 24, 26, 39, 5, 34, 47, 32, 6, 21, 40, 35, 46, 45, 16, 48, 49])
can = canonicalize(g, list(range(48)), 0, (baser, gensr, 12, 0))
assert can == [0, 2, 4, 6, 1, 3, 8, 10, 5, 7, 12, 14, 9, 11, 16, 18, 13, 15, 20, 22, 17, 19, 24, 26, 21, 23, 28, 30, 25, 27, 32, 34, 29, 31, 36, 38, 33, 35, 40, 42, 37, 39, 44, 46, 41, 43, 45, 47, 48, 49]
g = Permutation([0,2,4,6, 7,8,10,12, 14,16,18,20, 19,22,24,26, 5,21,28,30, 32,34,36,38, 40,42,44,46, 13,48,50,52, 15,49,54,56, 17,33,41,58, 9,23,60,62, 29,35,63,64, 3,45,66,68, 25,37,47,57, 11,31,69,70, 27,39,53,72, 1,59,73,74, 55,61,67,76, 43,65,75,78, 51,71,77,79, 80,81])
can = canonicalize(g, list(range(80)), 0, (baser, gensr, 20, 0))
assert can == [0,2,4,6, 1,8,10,12, 3,14,16,18, 5,20,22,24, 7,26,28,30, 9,15,32,34, 11,36,23,38, 13,40,42,44, 17,39,29,46, 19,48,43,50, 21,45,52,54, 25,56,33,58, 27,60,53,62, 31,51,64,66, 35,65,47,68, 37,70,49,72, 41,74,57,76, 55,67,59,78, 61,69,71,75, 63,79,73,77, 80,81]
def test_canonical_free():
# t = A^{d0 a1}*A_d0^a0
# ord = [a0,a1,d0,-d0]; g = [2,1,3,0,4,5]; dummies = [[2,3]]
# t_c = A_d0^a0*A^{d0 a1}
# can = [3,0, 2,1, 4,5]
g = Permutation([2, 1, 3, 0, 4, 5])
dummies = [[2, 3]]
can = canonicalize(g, dummies, [None], ([], [Permutation(3)], 2, 0))
assert can == [3, 0, 2, 1, 4, 5]
def test_canonical_free():
# t = A^{d0 a1}*A_d0^a0
# ord = [a0,a1,d0,-d0]; g = [2,1,3,0,4,5]; dummies = [[2,3]]
# t_c = A_d0^a0*A^{d0 a1}
# can = [3,0, 2,1, 4,5]
base = [0]
gens = [Permutation(5)(0,2)(1,3)]
g = Permutation([2,1,3,0,4,5])
num_free = 2
dummies = [[2,3]]
can = canonicalize(g, dummies, [None], ([], [Permutation(3)], 2, 0))
assert can == [3,0, 2,1, 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_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_riemann_invariants():
baser, gensr = riemann_bsgs
# R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1]; g = [0,2,3,1,4,5]
# T_c = -R^{d0 d1}_{d0 d1}; can = [0,2,1,3,5,4]
g = Permutation([0,2,3,1,4,5])
can = canonicalize(g, list(range(2, 4)), 0, (baser, gensr, 1, 0))
assert can == [0,2,1,3,5,4]
# use a non minimal BSGS
can = canonicalize(g, list(range(2, 4)), 0, ([2, 0], [Permutation([1,0,2,3,5,4]), Permutation([2,3,0,1,4,5])], 1, 0))
assert can == [0,2,1,3,5,4]
"""
The following tests in test_riemann_invariants and in
test_riemann_invariants1 have been checked using xperm.c from XPerm in
in [1] and with an older version contained in [2]
[1] xperm.c part of xPerm written by J. M. Martin-Garcia
http://www.xact.es/index.html
[2] test_xperm.cc in cadabra by Kasper Peeters, http://cadabra.phi-sci.com/
"""
# R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} *
# R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10}
# ord: contravariant d_k ->2*k, covariant d_k -> 2*k+1
# T_c = R^{d0 d1 d2 d3} * R_{d0 d1}^{d4 d5} * R_{d2 d3}^{d6 d7} *
# R_{d4 d5}^{d8 d9} * R_{d6 d7}^{d10 d11} * R_{d8 d9 d10 d11}
g = Permutation([23,2,1,10,12,8,0,11,15,5,17,19,21,7,13,9,4,14,22,3,16,18,6,20,24,25])
can = canonicalize(g, list(range(24)), 0, (baser, gensr, 6, 0))
assert can == [0,2,4,6,1,3,8,10,5,7,12,14,9,11,16,18,13,15,20,22,17,19,21,23,24,25]
# use a non minimal BSGS
can = canonicalize(g, list(range(24)), 0, ([2, 0], [Permutation([1,0,2,3,5,4]), Permutation([2,3,0,1,4,5])], 6, 0))
assert can == [0,2,4,6,1,3,8,10,5,7,12,14,9,11,16,18,13,15,20,22,17,19,21,23,24,25]
g = Permutation([0,2,5,7,4,6,9,11,8,10,13,15,12,14,17,19,16,18,21,23,20,22,25,27,24,26,29,31,28,30,33,35,32,34,37,39,36,38,1,3,40,41])
can = canonicalize(g, list(range(40)), 0, (baser, gensr, 10, 0))
assert can == [0,2,4,6,1,3,8,10,5,7,12,14,9,11,16,18,13,15,20,22,17,19,24,26,21,23,28,30,25,27,32,34,29,31,36,38,33,35,37,39,40,41]
def graph_certificate(gr):
"""
Return a certificate for the graph
Parameters
==========
gr : adjacency list
Explanation
===========
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 sympy.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 sympy.combinatorics.permutations import _af_invert
from sympy.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
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)
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 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 sympy.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 sympy.combinatorics.permutations import _af_invert
from sympy.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
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_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]
