# Source code for sympy.combinatorics.tensor_can

from __future__ import print_function, division

from sympy.core.compatibility import range
from sympy.combinatorics.permutations import Permutation, _af_rmul, \
_af_invert, _af_new
from sympy.combinatorics.perm_groups import PermutationGroup, _orbit, \
_orbit_transversal
from sympy.combinatorics.util import _distribute_gens_by_base, \
_orbits_transversals_from_bsgs

"""
References for tensor canonicalization:

 R. Portugal "Algorithmic simplification of tensor expressions",
J. Phys. A 32 (1999) 7779-7789

 R. Portugal, B.F. Svaiter "Group-theoretic Approach for Symbolic
Tensor Manipulation: I. Free Indices"
arXiv:math-ph/0107031v1

 L.R.U. Manssur, R. Portugal "Group-theoretic Approach for Symbolic
Tensor Manipulation: II. Dummy Indices"
arXiv:math-ph/0107032v1

 xperm.c part of XPerm written by J. M. Martin-Garcia
http://www.xact.es/index.html
"""

def dummy_sgs(dummies, sym, n):
"""
Return the strong generators for dummy indices

Parameters
==========

dummies : list of dummy indices
dummies[2k], dummies[2k+1] are paired indices
sym : symmetry under interchange of contracted dummies::
* None  no symmetry
* 0     commuting
* 1     anticommuting

n : number of indices

in base form the dummy indices are always in consecutive positions

Examples
========

>>> from sympy.combinatorics.tensor_can import dummy_sgs
>>> dummy_sgs(range(2, 8), 0, 8)
[[0, 1, 3, 2, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 5, 4, 6, 7, 8, 9],
[0, 1, 2, 3, 4, 5, 7, 6, 8, 9], [0, 1, 4, 5, 2, 3, 6, 7, 8, 9],
[0, 1, 2, 3, 6, 7, 4, 5, 8, 9]]
"""
if len(dummies) > n:
raise ValueError("List too large")
res = []
# exchange of contravariant and covariant indices
if sym is not None:
for j in dummies[::2]:
a = list(range(n + 2))
if sym == 1:
a[n] = n + 1
a[n + 1] = n
a[j], a[j + 1] = a[j + 1], a[j]
res.append(a)
# rename dummy indices
for j in dummies[:-3:2]:
a = list(range(n + 2))
a[j:j + 4] = a[j + 2], a[j + 3], a[j], a[j + 1]
res.append(a)
return res

def _min_dummies(dummies, sym, indices):
"""
Return list of minima of the orbits of indices in group of dummies
see double_coset_can_rep for the description of dummies and sym
indices is the initial list of dummy indices

Examples
========

>>> from sympy.combinatorics.tensor_can import _min_dummies
>>> _min_dummies([list(range(2, 8))], , list(range(10)))
[0, 1, 2, 2, 2, 2, 2, 2, 8, 9]
"""
num_types = len(sym)
m = []
for dx in dummies:
if dx:
m.append(min(dx))
else:
m.append(None)
res = indices[:]
for i in range(num_types):
for c, i in enumerate(indices):
for j in range(num_types):
if i in dummies[j]:
res[c] = m[j]
break
return res

def _trace_S(s, j, b, S_cosets):
"""
Return the representative h satisfying s[h[b]] == j

If there is not such a representative return None
"""
for h in S_cosets[b]:
if s[h[b]] == j:
return h
return None

def _trace_D(gj, p_i, Dxtrav):
"""
Return the representative h satisfying h[gj] == p_i

If there is not such a representative return None
"""
for h in Dxtrav:
if h[gj] == p_i:
return h
return None

def _dumx_remove(dumx, dumx_flat, p0):
"""
remove p0 from dumx
"""
res = []
for dx in dumx:
if p0 not in dx:
res.append(dx)
continue
k = dx.index(p0)
if k % 2 == 0:
p0_paired = dx[k + 1]
else:
p0_paired = dx[k - 1]
dx.remove(p0)
dx.remove(p0_paired)
dumx_flat.remove(p0)
dumx_flat.remove(p0_paired)
res.append(dx)

def transversal2coset(size, base, transversal):
a = []
j = 0
for i in range(size):
if i in base:
a.append(sorted(transversal[j].values()))
j += 1
else:
a.append([list(range(size))])
j = len(a) - 1
while a[j] == [list(range(size))]:
j -= 1
return a[:j + 1]

[docs]def double_coset_can_rep(dummies, sym, b_S, sgens, S_transversals, g):
"""
Butler-Portugal algorithm for tensor canonicalization with dummy indices

dummies
list of lists of dummy indices,
one list for each type of index;
the dummy indices are put in order contravariant, covariant
[d0, -d0, d1, -d1, ...].

sym
list of the symmetries of the index metric for each type.

possible symmetries of the metrics
* 0     symmetric
* 1     antisymmetric
* None  no symmetry

b_S
base of a minimal slot symmetry BSGS.

sgens
generators of the slot symmetry BSGS.

S_transversals
transversals for the slot BSGS.

g
permutation representing the tensor.

Return 0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.

A tensor with dummy indices can be represented in a number
of equivalent ways which typically grows exponentially with
the number of indices. To be able to establish if two tensors
with many indices are equal becomes computationally very slow
in absence of an efficient algorithm.

The Butler-Portugal algorithm  is an efficient algorithm to
put tensors in canonical form, solving the above problem.

Portugal observed that a tensor can be represented by a permutation,
and that the class of tensors equivalent to it under slot and dummy
symmetries is equivalent to the double coset D*g*S
(Note: in this documentation we use the conventions for multiplication
of permutations p, q with (p*q)(i) = p[q[i]] which is opposite
to the one used in the Permutation class)

Using the algorithm by Butler to find a representative of the
double coset one can find a canonical form for the tensor.

To see this correspondence,
let g be a permutation in array form; a tensor with indices ind
(the indices including both the contravariant and the covariant ones)
can be written as

t = T(ind[g,..., ind[g[n-1]]),

where n= len(ind);
g has size n + 2, the last two indices for the sign of the tensor
(trick introduced in ).

A slot symmetry transformation s is a permutation acting on the slots
t -> T(ind[(g*s)],..., ind[(g*s)[n-1]])

A dummy symmetry transformation acts on ind
t -> T(ind[(d*g)],..., ind[(d*g)[n-1]])

Being interested only in the transformations of the tensor under
these symmetries, one can represent the tensor by g, which transforms
as

g -> d*g*s, so it belongs to the coset D*g*S.

Let us explain the conventions by an example.

Given a tensor T^{d3 d2 d1}{}_{d1 d2 d3} with the slot symmetries
T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}

T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}

and symmetric metric, find the tensor equivalent to it which
is the lowest under the ordering of indices:
lexicographic ordering d1, d2, d3 then and contravariant index
before covariant index; that is the canonical form of the tensor.

The canonical form is -T^{d1 d2 d3}{}_{d1 d2 d3}
obtained using T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}.

To convert this problem in the input for this function,
use the following labelling of the index names
(- for covariant for short) d1, -d1, d2, -d2, d3, -d3

T^{d3 d2 d1}{}_{d1 d2 d3} corresponds to g = [4, 2, 0, 1, 3, 5, 6, 7]
where the last two indices are for the sign

sgens = [Permutation(0, 2)(6, 7), Permutation(0, 4)(6, 7)]

sgens is the slot symmetry -(0, 2)
T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}

sgens is the slot symmetry -(0, 4)
T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}

The dummy symmetry group D is generated by the strong base generators
[(0, 1), (2, 3), (4, 5), (0, 1)(2, 3),(2, 3)(4, 5)]

The dummy symmetry acts from the left
d = [1, 0, 2, 3, 4, 5, 6, 7]  exchange d1 -> -d1
T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}

g=[4, 2, 0, 1, 3, 5, 6, 7]  -> [4, 2, 1, 0, 3, 5, 6, 7] = _af_rmul(d, g)
which differs from _af_rmul(g, d).

The slot symmetry acts from the right
s = [2, 1, 0, 3, 4, 5, 7, 6]  exchanges slots 0 and 2 and changes sign
T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}

g=[4,2,0,1,3,5,6,7]  -> [0, 2, 4, 1, 3, 5, 7, 6] = _af_rmul(g, s)

Example in which the tensor is zero, same slot symmetries as above:
T^{d3}{}_{d1,d2}{}^{d1}{}_{d3}{}^{d2}

= -T^{d3}{}_{d1,d3}{}^{d1}{}_{d2}{}^{d2}   under slot symmetry -(2,4);

= T_{d3 d1}{}^{d3}{}^{d1}{}_{d2}{}^{d2}    under slot symmetry -(0,2);

= T^{d3}{}_{d1 d3}{}^{d1}{}_{d2}{}^{d2}    symmetric metric;

= 0  since two of these lines have tensors differ only for the sign.

The double coset D*g*S consists of permutations h = d*g*s corresponding
to equivalent tensors; if there are two h which are the same apart
from the sign, return zero; otherwise
choose as representative the tensor with indices
ordered lexicographically according to [d1, -d1, d2, -d2, d3, -d3]
that is rep = min(D*g*S) = min([d*g*s for d in D for s in S])

The indices are fixed one by one; first choose the lowest index
for slot 0, then the lowest remaining index for slot 1, etc.
Doing this one obtains a chain of stabilizers

S -> S_{b0} -> S_{b0,b1} -> ... and
D -> D_{p0} -> D_{p0,p1} -> ...

where [b0, b1, ...] = range(b) is a base of the symmetric group;
the strong base b_S of S is an ordered sublist of it;
therefore it is sufficient to compute once the
strong base generators of S using the Schreier-Sims algorithm;
the stabilizers of the strong base generators are the
strong base generators of the stabilizer subgroup.

dbase = [p0, p1, ...] is not in general in lexicographic order,
so that one must recompute the strong base generators each time;
however this is trivial, there is no need to use the Schreier-Sims
algorithm for D.

The algorithm keeps a TAB of elements (s_i, d_i, h_i)
where h_i = d_i*g*s_i satisfying h_i[j] = p_j for 0 <= j < i
starting from s_0 = id, d_0 = id, h_0 = g.

The equations h_0 = p_0, h_1 = p_1,... are solved in this order,
choosing each time the lowest possible value of p_i

For j < i
d_i*g*s_i*S_{b_0,...,b_{i-1}}*b_j = D_{p_0,...,p_{i-1}}*p_j
so that for dx in D_{p_0,...,p_{i-1}} and sx in
S_{base,...,base[i-1]} one has dx*d_i*g*s_i*sx*b_j = p_j

Search for dx, sx such that this equation holds for j = i;
it can be written as s_i*sx*b_j = J, dx*d_i*g*J = p_j
sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})
dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})

s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})
d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i
h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i

h_n*b_j = p_j for all j, so that h_n is the solution.

Add the found (s, d, h) to TAB1.

At the end of the iteration sort TAB1 with respect to the h;
if there are two consecutive h in TAB1 which differ only for the
sign, the tensor is zero, so return 0;
if there are two consecutive h which are equal, keep only one.

Then stabilize the slot generators under i and the dummy generators
under p_i.

Assign TAB = TAB1 at the end of the iteration step.

At the end TAB contains a unique (s, d, h), since all the slots
of the tensor h have been fixed to have the minimum value according
to the symmetries. The algorithm returns h.

It is important that the slot BSGS has lexicographic minimal base,
otherwise there is an i which does not belong to the slot base
for which p_i is fixed by the dummy symmetry only, while i
is not invariant from the slot stabilizer, so p_i is not in
general the minimal value.

This algorithm differs slightly from the original algorithm :
the canonical form is minimal lexicographically, and
the BSGS has minimal base under lexicographic order.
Equal tensors h are eliminated from TAB.

Examples
========

>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.tensor_can import double_coset_can_rep, get_transversals
>>> gens = [Permutation(x) for x in [[2, 1, 0, 3, 4, 5, 7, 6], [4, 1, 2, 3, 0, 5, 7, 6]]]
>>> base = [0, 2]
>>> g = Permutation([4, 2, 0, 1, 3, 5, 6, 7])
>>> transversals = get_transversals(base, gens)
>>> double_coset_can_rep([list(range(6))], , base, gens, transversals, g)
[0, 1, 2, 3, 4, 5, 7, 6]

>>> g = Permutation([4, 1, 3, 0, 5, 2, 6, 7])
>>> double_coset_can_rep([list(range(6))], , base, gens, transversals, g)
0
"""
size = g.size
g = g.array_form
num_dummies = size - 2
indices = list(range(num_dummies))
all_metrics_with_sym = all([_ is not None for _ in sym])
num_types = len(sym)
dumx = dummies[:]
dumx_flat = []
for dx in dumx:
dumx_flat.extend(dx)
b_S = b_S[:]
sgensx = [h._array_form for h in sgens]
if b_S:
S_transversals = transversal2coset(size, b_S, S_transversals)
# strong generating set for D
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
ginv = _af_invert(g)
idn = list(range(size))
# TAB = list of entries (s, d, h) where h = _af_rmuln(d,g,s)
# for short, in the following d*g*s means _af_rmuln(d,g,s)
TAB = [(idn, idn, g)]
for i in range(size - 2):
b = i
testb = b in b_S and sgensx
if testb:
sgensx1 = [_af_new(_) for _ in sgensx]
deltab = _orbit(size, sgensx1, b)
else:
deltab = {b}
# p1 = min(IMAGES) = min(Union D_p*h*deltab for h in TAB)
if all_metrics_with_sym:
md = _min_dummies(dumx, sym, indices)
else:
md = [min(_orbit(size, [_af_new(
ddx) for ddx in dsgsx], ii)) for ii in range(size - 2)]

p_i = min([min([md[h[x]] for x in deltab]) for s, d, h in TAB])
dsgsx1 = [_af_new(_) for _ in dsgsx]
Dxtrav = _orbit_transversal(size, dsgsx1, p_i, False, af=True) \
if dsgsx else None
if Dxtrav:
Dxtrav = [_af_invert(x) for x in Dxtrav]
# compute the orbit of p_i
for ii in range(num_types):
if p_i in dumx[ii]:
# the orbit is made by all the indices in dum[ii]
if sym[ii] is not None:
deltap = dumx[ii]
else:
# the orbit is made by all the even indices if p_i
# is even, by all the odd indices if p_i is odd
p_i_index = dumx[ii].index(p_i) % 2
deltap = dumx[ii][p_i_index::2]
break
else:
deltap = [p_i]
TAB1 = []
nTAB = len(TAB)
while TAB:
s, d, h = TAB.pop()
if min([md[h[x]] for x in deltab]) != p_i:
continue
deltab1 = [x for x in deltab if md[h[x]] == p_i]
# NEXT = s*deltab1 intersection (d*g)**-1*deltap
dg = _af_rmul(d, g)
dginv = _af_invert(dg)
sdeltab = [s[x] for x in deltab1]
gdeltap = [dginv[x] for x in deltap]
NEXT = [x for x in sdeltab if x in gdeltap]
# d, s satisfy
# d*g*s*base[i-1] = p_{i-1}; using the stabilizers
# d*g*s*S_{base,...,base[i-1]}*base[i-1] =
# D_{p_0,...,p_{i-1}}*p_{i-1}
# so that to find d1, s1 satisfying d1*g*s1*b = p_i
# one can look for dx in D_{p_0,...,p_{i-1}} and
# sx in S_{base,...,base[i-1]}
# d1 = dx*d; s1 = s*sx
# d1*g*s1*b = dx*d*g*s*sx*b = p_i
for j in NEXT:
if testb:
# solve s1*b = j with s1 = s*sx for some element sx
# of the stabilizer of ..., base[i-1]
# sx*b = s**-1*j; sx = _trace_S(s, j,...)
# s1 = s*trace_S(s**-1*j,...)
s1 = _trace_S(s, j, b, S_transversals)
if not s1:
continue
else:
s1 = [s[ix] for ix in s1]
else:
s1 = s
# assert s1[b] == j  # invariant
# solve d1*g*j = p_i with d1 = dx*d for some element dg
# of the stabilizer of ..., p_{i-1}
# dx**-1*p_i = d*g*j; dx**-1 = trace_D(d*g*j,...)
# d1 = trace_D(d*g*j,...)**-1*d
# to save an inversion in the inner loop; notice we did
# Dxtrav = [perm_af_invert(x) for x in Dxtrav] out of the loop
if Dxtrav:
d1 = _trace_D(dg[j], p_i, Dxtrav)
if not d1:
continue
else:
if p_i != dg[j]:
continue
d1 = idn
assert d1[dg[j]] == p_i  # invariant
d1 = [d1[ix] for ix in d]
h1 = [d1[g[ix]] for ix in s1]
# assert h1[b] == p_i  # invariant
TAB1.append((s1, d1, h1))

# if TAB contains equal permutations, keep only one of them;
# if TAB contains equal permutations up to the sign, return 0
TAB1.sort(key=lambda x: x[-1])
nTAB1 = len(TAB1)
prev =  * size
while TAB1:
s, d, h = TAB1.pop()
if h[:-2] == prev[:-2]:
if h[-1] != prev[-1]:
return 0
else:
TAB.append((s, d, h))
prev = h

# stabilize the SGS
sgensx = [h for h in sgensx if h[b] == b]
if b in b_S:
b_S.remove(b)
_dumx_remove(dumx, dumx_flat, p_i)
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
return TAB[-1]

def canonical_free(base, gens, g, num_free):
"""
canonicalization of a tensor with respect to free indices
choosing the minimum with respect to lexicographical ordering
in the free indices

base, gens  BSGS for slot permutation group
g               permutation representing the tensor
num_free        number of free indices
The indices must be ordered with first the free indices

see explanation in double_coset_can_rep
The algorithm is a variation of the one given in .

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import canonical_free
>>> gens = [[1, 0, 2, 3, 5, 4], [2, 3, 0, 1, 4, 5],[0, 1, 3, 2, 5, 4]]
>>> gens = [Permutation(h) for h in gens]
>>> base = [0, 2]
>>> g = Permutation([2, 1, 0, 3, 4, 5])
>>> canonical_free(base, gens, g, 4)
[0, 3, 1, 2, 5, 4]

Consider the product of Riemann tensors
T = R^{a}_{d0}^{d1,d2}*R_{d2,d1}^{d0,b}
The order of the indices is [a, b, d0, -d0, d1, -d1, d2, -d2]
The permutation corresponding to the tensor is
g = [0, 3, 4, 6, 7, 5, 2, 1, 8, 9]

In particular a is position 0, b is in position 9.
Use the slot symmetries to get T is a form which is the minimal
in lexicographic order in the free indices a and b, e.g.
-R^{a}_{d0}^{d1,d2}*R^{b,d0}_{d2,d1} corresponding to
[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]

>>> from sympy.combinatorics.tensor_can import riemann_bsgs, tensor_gens
>>> base, gens = riemann_bsgs
>>> size, sbase, sgens = tensor_gens(base, gens, [[], []], 0)
>>> g = Permutation([0, 3, 4, 6, 7, 5, 2, 1, 8, 9])
>>> canonical_free(sbase, [Permutation(h) for h in sgens], g, 2)
[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]
"""
g = g.array_form
size = len(g)
if not base:
return g[:]

transversals = get_transversals(base, gens)
m = len(base)
for x in sorted(g[:-2]):
if x not in base:
base.append(x)
h = g
for i, transv in enumerate(transversals):
b = base[i]
h_i = [size]*num_free
# find the element s in transversals[i] such that
# _af_rmul(h, s) has its free elements with the lowest position in h
s = None
for sk in transv.values():
h1 = _af_rmul(h, sk)
hi = [h1.index(ix) for ix in range(num_free)]
if hi < h_i:
h_i = hi
s = sk
if s:
h = _af_rmul(h, s)
return h

def _get_map_slots(size, fixed_slots):
res = list(range(size))
pos = 0
for i in range(size):
if i in fixed_slots:
continue
res[i] = pos
pos += 1
return res

def _lift_sgens(size, fixed_slots, free, s):
a = []
j = k = 0
fd = list(zip(fixed_slots, free))
fd = [y for x, y in sorted(fd)]
num_free = len(free)
for i in range(size):
if i in fixed_slots:
a.append(fd[k])
k += 1
else:
a.append(s[j] + num_free)
j += 1
return a

[docs]def canonicalize(g, dummies, msym, *v):
"""
canonicalize tensor formed by tensors

Parameters
==========

g : permutation representing the tensor

dummies : list representing the dummy indices
it can be a list of dummy indices of the same type
or a list of lists of dummy indices, one list for each
type of index;
the dummy indices must come after the free indices,
and put in order contravariant, covariant
[d0, -d0, d1,-d1,...]
msym :  symmetry of the metric(s)
it can be an integer or a list;
in the first case it is the symmetry of the dummy index metric;
in the second case it is the list of the symmetries of the
index metric for each type
v : list, (base_i, gens_i, n_i, sym_i) for tensors of type i

base_i, gens_i : BSGS for tensors of this type.
The BSGS should have minimal base under lexicographic ordering;
if not, an attempt is made do get the minimal BSGS;
in case of failure,
canonicalize_naive is used, which is much slower.

n_i :    number of tensors of type i.

sym_i :  symmetry under exchange of component tensors of type i.

Both for msym and sym_i the cases are
* None  no symmetry
* 0     commuting
* 1     anticommuting

Returns
=======

0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.

Algorithm
=========

First one uses canonical_free to get the minimum tensor under
lexicographic order, using only the slot symmetries.
If the component tensors have not minimal BSGS, it is attempted
to find it; if the attempt fails canonicalize_naive

Compute the residual slot symmetry keeping fixed the free indices
using tensor_gens(base, gens, list_free_indices, sym).

Reduce the problem eliminating the free indices.

Then use double_coset_can_rep and lift back the result reintroducing
the free indices.

Examples
========

one type of index with commuting metric;

A_{a b} and B_{a b} antisymmetric and commuting

T = A_{d0 d1} * B^{d0}{}_{d2} * B^{d2 d1}

ord = [d0,-d0,d1,-d1,d2,-d2] order of the indices

g = [1, 3, 0, 5, 4, 2, 6, 7]

T_c = 0

>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize, bsgs_direct_product
>>> from sympy.combinatorics import Permutation
>>> base2a, gens2a = get_symmetric_group_sgs(2, 1)
>>> t0 = (base2a, gens2a, 1, 0)
>>> t1 = (base2a, gens2a, 2, 0)
>>> g = Permutation([1, 3, 0, 5, 4, 2, 6, 7])
>>> canonicalize(g, range(6), 0, t0, t1)
0

same as above, but with B_{a b} anticommuting

T_c = -A^{d0 d1} * B_{d0}{}^{d2} * B_{d1 d2}

can = [0,2,1,4,3,5,7,6]

>>> t1 = (base2a, gens2a, 2, 1)
>>> canonicalize(g, range(6), 0, t0, t1)
[0, 2, 1, 4, 3, 5, 7, 6]

two types of indices [a,b,c,d,e,f] and [m,n], in this order,
both with commuting metric

f^{a b c} antisymmetric, commuting

A_{m a} no symmetry, commuting

T = f^c{}_{d a} * f^f{}_{e b} * A_m{}^d * A^{m b} * A_n{}^a * A^{n e}

ord = [c,f,a,-a,b,-b,d,-d,e,-e,m,-m,n,-n]

g = [0,7,3, 1,9,5, 11,6, 10,4, 13,2, 12,8, 14,15]

The canonical tensor is
T_c = -f^{c a b} * f^{f d e} * A^m{}_a * A_{m d} * A^n{}_b * A_{n e}

can = [0,2,4, 1,6,8, 10,3, 11,7, 12,5, 13,9, 15,14]

>>> base_f, gens_f = get_symmetric_group_sgs(3, 1)
>>> base1, gens1 = get_symmetric_group_sgs(1)
>>> 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)
>>> dummies = [range(2, 10), range(10, 14)]
>>> g = Permutation([0, 7, 3, 1, 9, 5, 11, 6, 10, 4, 13, 2, 12, 8, 14, 15])
>>> canonicalize(g, dummies, [0, 0], t0, t1)
[0, 2, 4, 1, 6, 8, 10, 3, 11, 7, 12, 5, 13, 9, 15, 14]
"""
from sympy.combinatorics.testutil import canonicalize_naive
if not isinstance(msym, list):
if not msym in [0, 1, None]:
raise ValueError('msym must be 0, 1 or None')
num_types = 1
else:
num_types = len(msym)
if not all(msymx in [0, 1, None] for msymx in msym):
raise ValueError('msym entries must be 0, 1 or None')
if len(dummies) != num_types:
raise ValueError(
'dummies and msym must have the same number of elements')
size = g.size
num_tensors = 0
v1 = []
for i in range(len(v)):
base_i, gens_i, n_i, sym_i = v[i]
# check that the BSGS is minimal;
# this property is used in double_coset_can_rep;
# if it is not minimal use canonicalize_naive
if not _is_minimal_bsgs(base_i, gens_i):
mbsgs = get_minimal_bsgs(base_i, gens_i)
if not mbsgs:
can = canonicalize_naive(g, dummies, msym, *v)
return can
base_i, gens_i = mbsgs
v1.append((base_i, gens_i, [[]] * n_i, sym_i))
num_tensors += n_i

if num_types == 1 and not isinstance(msym, list):
dummies = [dummies]
msym = [msym]
flat_dummies = []
for dumx in dummies:
flat_dummies.extend(dumx)

if flat_dummies and flat_dummies != list(range(flat_dummies, flat_dummies[-1] + 1)):
raise ValueError('dummies is not valid')

# slot symmetry of the tensor
size1, sbase, sgens = gens_products(*v1)
if size != size1:
raise ValueError(
'g has size %d, generators have size %d' % (size, size1))
free = [i for i in range(size - 2) if i not in flat_dummies]
num_free = len(free)

# g1 minimal tensor under slot symmetry
g1 = canonical_free(sbase, sgens, g, num_free)
if not flat_dummies:
return g1
# save the sign of g1
sign = 0 if g1[-1] == size - 1 else 1

# the free indices are kept fixed.
# Determine free_i, the list of slots of tensors which are fixed
# since they are occupied by free indices, which are fixed.
start = 0
for i in range(len(v)):
free_i = []
base_i, gens_i, n_i, sym_i = v[i]
len_tens = gens_i.size - 2
# for each component tensor get a list od fixed islots
for j in range(n_i):
# get the elements corresponding to the component tensor
h = g1[start:(start + len_tens)]
fr = []
# get the positions of the fixed elements in h
for k in free:
if k in h:
fr.append(h.index(k))
free_i.append(fr)
start += len_tens
v1[i] = (base_i, gens_i, free_i, sym_i)
# BSGS of the tensor with fixed free indices
# if tensor_gens fails in gens_product, use canonicalize_naive
size, sbase, sgens = gens_products(*v1)

# reduce the permutations getting rid of the free indices
pos_dummies = [g1.index(x) for x in flat_dummies]
pos_free = [g1.index(x) for x in range(num_free)]
size_red = size - num_free
g1_red = [x - num_free for x in g1 if x in flat_dummies]
if sign:
g1_red.extend([size_red - 1, size_red - 2])
else:
g1_red.extend([size_red - 2, size_red - 1])
map_slots = _get_map_slots(size, pos_free)
sbase_red = [map_slots[i] for i in sbase if i not in pos_free]
sgens_red = [_af_new([map_slots[i] for i in y._array_form if i not in pos_free]) for y in sgens]
dummies_red = [[x - num_free for x in y] for y in dummies]
transv_red = get_transversals(sbase_red, sgens_red)
g1_red = _af_new(g1_red)
g2 = double_coset_can_rep(
dummies_red, msym, sbase_red, sgens_red, transv_red, g1_red)
if g2 == 0:
return 0
# lift to the case with the free indices
g3 = _lift_sgens(size, pos_free, free, g2)
return g3

def perm_af_direct_product(gens1, gens2, signed=True):
"""
direct products of the generators gens1 and gens2

Examples
========

>>> from sympy.combinatorics.tensor_can import perm_af_direct_product
>>> gens1 = [[1, 0, 2, 3], [0, 1, 3, 2]]
>>> gens2 = [[1, 0]]
>>> perm_af_direct_product(gens1, gens2, False)
[[1, 0, 2, 3, 4, 5], [0, 1, 3, 2, 4, 5], [0, 1, 2, 3, 5, 4]]
>>> gens1 = [[1, 0, 2, 3, 5, 4], [0, 1, 3, 2, 4, 5]]
>>> gens2 = [[1, 0, 2, 3]]
>>> perm_af_direct_product(gens1, gens2, True)
[[1, 0, 2, 3, 4, 5, 7, 6], [0, 1, 3, 2, 4, 5, 6, 7], [0, 1, 2, 3, 5, 4, 6, 7]]
"""
gens1 = [list(x) for x in gens1]
gens2 = [list(x) for x in gens2]
s = 2 if signed else 0
n1 = len(gens1) - s
n2 = len(gens2) - s
start = list(range(n1))
end = list(range(n1, n1 + n2))
if signed:
gens1 = [gen[:-2] + end + [gen[-2] + n2, gen[-1] + n2]
for gen in gens1]
gens2 = [start + [x + n1 for x in gen] for gen in gens2]
else:
gens1 = [gen + end for gen in gens1]
gens2 = [start + [x + n1 for x in gen] for gen in gens2]

res = gens1 + gens2

return res

[docs]def bsgs_direct_product(base1, gens1, base2, gens2, signed=True):
"""
direct product of two BSGS

base1    base of the first BSGS.

gens1    strong generating sequence of the first BSGS.

base2, gens2   similarly for the second BSGS.

signed   flag for signed permutations.

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import (get_symmetric_group_sgs, bsgs_direct_product)
>>> Permutation.print_cyclic = True
>>> base1, gens1 = get_symmetric_group_sgs(1)
>>> base2, gens2 = get_symmetric_group_sgs(2)
>>> bsgs_direct_product(base1, gens1, base2, gens2)
(, [(4)(1 2)])
"""
s = 2 if signed else 0
n1 = gens1.size - s
base = list(base1)
base += [x + n1 for x in base2]
gens1 = [h._array_form for h in gens1]
gens2 = [h._array_form for h in gens2]
gens = perm_af_direct_product(gens1, gens2, signed)
size = len(gens)
id_af = list(range(size))
gens = [h for h in gens if h != id_af]
if not gens:
gens = [id_af]
return base, [_af_new(h) for h in gens]

[docs]def get_symmetric_group_sgs(n, antisym=False):
"""
Return base, gens of the minimal BSGS for (anti)symmetric tensor

n  rank of the tensor

antisym = False symmetric tensor
antisym = True  antisymmetric tensor

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs
>>> Permutation.print_cyclic = True
>>> get_symmetric_group_sgs(3)
([0, 1], [(4)(0 1), (4)(1 2)])
"""
if n == 1:
return [], [_af_new(list(range(3)))]
gens = [Permutation(n - 1)(i, i + 1)._array_form for i in range(n - 1)]
if antisym == 0:
gens = [x + [n, n + 1] for x in gens]
else:
gens = [x + [n + 1, n] for x in gens]
base = list(range(n - 1))
return base, [_af_new(h) for h in gens]

riemann_bsgs = [0, 2], [Permutation(0, 1)(4, 5), Permutation(2, 3)(4, 5),
Permutation(5)(0, 2)(1, 3)]

def get_transversals(base, gens):
"""
Return transversals for the group with BSGS base, gens
"""
if not base:
return []
stabs = _distribute_gens_by_base(base, gens)
orbits, transversals = _orbits_transversals_from_bsgs(base, stabs)
transversals = [{x: h._array_form for x, h in y.items()} for y in
transversals]
return transversals

def _is_minimal_bsgs(base, gens):
"""
Check if the BSGS has minimal base under lexigographic order.

base, gens BSGS

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import riemann_bsgs, _is_minimal_bsgs
>>> _is_minimal_bsgs(*riemann_bsgs)
True
>>> riemann_bsgs1 = ([2, 0], ([Permutation(5)(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3)]))
>>> _is_minimal_bsgs(*riemann_bsgs1)
False
"""
base1 = []
sgs1 = gens[:]
size = gens.size
for i in range(size):
if not all(h._array_form[i] == i for h in sgs1):
base1.append(i)
sgs1 = [h for h in sgs1 if h._array_form[i] == i]
return base1 == base

def get_minimal_bsgs(base, gens):
"""
Compute a minimal GSGS

base, gens BSGS

If base, gens is a minimal BSGS return it; else return a minimal BSGS
if it fails in finding one, it returns None

TODO: use baseswap in the case in which if it fails in finding a
minimal BSGS

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import get_minimal_bsgs
>>> Permutation.print_cyclic = True
>>> riemann_bsgs1 = ([2, 0], ([Permutation(5)(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3)]))
>>> get_minimal_bsgs(*riemann_bsgs1)
([0, 2], [(0 1)(4 5), (5)(0 2)(1 3), (2 3)(4 5)])
"""
G = PermutationGroup(gens)
base, gens = G.schreier_sims_incremental()
if not _is_minimal_bsgs(base, gens):
return None
return base, gens

def tensor_gens(base, gens, list_free_indices, sym=0):
"""
Returns size, res_base, res_gens BSGS for n tensors of the
same type

base, gens BSGS for tensors of this type
list_free_indices  list of the slots occupied by fixed indices
for each of the tensors

sym symmetry under commutation of two tensors
sym   None  no symmetry
sym   0     commuting
sym   1     anticommuting

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import tensor_gens, get_symmetric_group_sgs
>>> Permutation.print_cyclic = True

two symmetric tensors with 3 indices without free indices

>>> base, gens = get_symmetric_group_sgs(3)
>>> tensor_gens(base, gens, [[], []])
(8, [0, 1, 3, 4], [(7)(0 1), (7)(1 2), (7)(3 4), (7)(4 5), (7)(0 3)(1 4)(2 5)])

two symmetric tensors with 3 indices with free indices in slot 1 and 0

>>> tensor_gens(base, gens, [, ])
(8, [0, 4], [(7)(0 2), (7)(4 5)])

four symmetric tensors with 3 indices, two of which with free indices

"""
def _get_bsgs(G, base, gens, free_indices):
"""
return the BSGS for G.pointwise_stabilizer(free_indices)
"""
if not free_indices:
return base[:], gens[:]
else:
H = G.pointwise_stabilizer(free_indices)
base, sgs = H.schreier_sims_incremental()
return base, sgs

# if not base there is no slot symmetry for the component tensors
# if list_free_indices.count([]) < 2 there is no commutation symmetry
# so there is no resulting slot symmetry
if not base and list_free_indices.count([]) < 2:
n = len(list_free_indices)
size = gens.size
size = n * (gens.size - 2) + 2
return size, [], [_af_new(list(range(size)))]

# if any(list_free_indices) one needs to compute the pointwise
# stabilizer, so G is needed
if any(list_free_indices):
G = PermutationGroup(gens)
else:
G = None

# no_free list of lists of indices for component tensors without fixed
# indices
no_free = []
size = gens.size
id_af = list(range(size))
num_indices = size - 2
if not list_free_indices:
no_free.append(list(range(num_indices)))
res_base, res_gens = _get_bsgs(G, base, gens, list_free_indices)
for i in range(1, len(list_free_indices)):
base1, gens1 = _get_bsgs(G, base, gens, list_free_indices[i])
res_base, res_gens = bsgs_direct_product(res_base, res_gens,
base1, gens1, 1)
if not list_free_indices[i]:
no_free.append(list(range(size - 2, size - 2 + num_indices)))
size += num_indices
nr = size - 2
res_gens = [h for h in res_gens if h._array_form != id_af]
# if sym there are no commuting tensors stop here
if sym is None or not no_free:
if not res_gens:
res_gens = [_af_new(id_af)]
return size, res_base, res_gens

# if the component tensors have moinimal BSGS, so is their direct
# product P; the slot symmetry group is S = P*C, where C is the group
# to (anti)commute the component tensors with no free indices
# a stabilizer has the property S_i = P_i*C_i;
# the BSGS of P*C has SGS_P + SGS_C and the base is
# the ordered union of the bases of P and C.
# If P has minimal BSGS, so has S with this base.
base_comm = []
for i in range(len(no_free) - 1):
ind1 = no_free[i]
ind2 = no_free[i + 1]
a = list(range(ind1))
a.extend(ind2)
a.extend(ind1)
base_comm.append(ind1)
a.extend(list(range(ind2[-1] + 1, nr)))
if sym == 0:
a.extend([nr, nr + 1])
else:
a.extend([nr + 1, nr])
res_gens.append(_af_new(a))
res_base = list(res_base)
# each base is ordered; order the union of the two bases
for i in base_comm:
if i not in res_base:
res_base.append(i)
res_base.sort()
if not res_gens:
res_gens = [_af_new(id_af)]

return size, res_base, res_gens

def gens_products(*v):
"""
Returns size, res_base, res_gens BSGS for n tensors of different types

v is a sequence of (base_i, gens_i, free_i, sym_i)
where
base_i, gens_i  BSGS of tensor of type i
free_i          list of the fixed slots for each of the tensors
of type i; if there are n_i tensors of type i
and none of them have fixed slots, free = [[]]*n_i
sym   0 (1) if the tensors of type i (anti)commute among themselves

Examples
========

>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, gens_products
>>> Permutation.print_cyclic = True
>>> base, gens = get_symmetric_group_sgs(2)
>>> gens_products((base, gens, [[], []], 0))
(6, [0, 2], [(5)(0 1), (5)(2 3), (5)(0 2)(1 3)])
>>> gens_products((base, gens, [, []], 0))
(6, , [(5)(2 3)])
"""
res_size, res_base, res_gens = tensor_gens(*v)
for i in range(1, len(v)):
size, base, gens = tensor_gens(*v[i])
res_base, res_gens = bsgs_direct_product(res_base, res_gens, base,
gens, 1)
res_size = res_gens.size
id_af = list(range(res_size))
res_gens = [h for h in res_gens if h != id_af]
if not res_gens:
res_gens = [id_af]
return res_size, res_base, res_gens