Ed+\ddlmZddlmZejZdZd dZdZd dZd d Z d Z d Z dS) Permutation)_distribute_gens_by_basec6d|Dd|DkS)ao Compare two lists of permutations as sets. Explanation =========== This is used for testing purposes. Since the array form of a permutation is currently a list, Permutation is not hashable and cannot be put into a set. Examples ======== >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.testutil import _cmp_perm_lists >>> a = Permutation([0, 2, 3, 4, 1]) >>> b = Permutation([1, 2, 0, 4, 3]) >>> c = Permutation([3, 4, 0, 1, 2]) >>> ls1 = [a, b, c] >>> ls2 = [b, c, a] >>> _cmp_perm_lists(ls1, ls2) True c,h|]}t|Stuple.0as z"_cmp_perm_lists.. s $ $ $E!HH $ $ $c,h|]}t|Srr r s rrz"_cmp_perm_lists..!s % % %E!HH % % %rr)firstseconds r_cmp_perm_listsrs02 % $e $ $ $ % %f % % % &&rFc( ddlm} ddlmt |drt |d}d|jD  fd}g}|s8|D]4}||r'|tj |5n%|D]"}||r||#|St |d rt||||St |d rt|||g|SdS) NrPermutationGroup)_af_commutes_with generatorsTafcg|] }|j Sr) _array_formr xs r z+_naive_list_centralizer..Bs888! 888rc>tfdDS)Nc30K|]}|VdSNr)r genrrs r z<_naive_list_centralizer....Cs1*U*U+<+z)_naive_list_centralizer..Cs)s*U*U*U*U*UPT*U*U*U'U'Urgetitem array_form) sympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrhasattrlistgenerate_diminorappendr_af_new_naive_list_centralizer) selfotherrrelementscommutes_with_genscentralizer_listelementrr's @@rr2r2$s@@@@@@2CBBBBBul##L,,,556688u'7888UUUUU 5# J J%%g..J$++K,?,H,HIII J$ 5 5%%g..5$++G444  " "L&t-=-=e-D-DbIII  % %L&t-=-=ug-F-FKKKLLrcZddlm}t||}|}tt |D][}|||}||krdS|||}\|dkrdSdS)a Verify the correctness of a base and strong generating set. Explanation =========== This is a naive implementation using the definition of a base and a strong generating set relative to it. There are other procedures for verifying a base and strong generating set, but this one will serve for more robust testing. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> A = AlternatingGroup(4) >>> A.schreier_sims() >>> _verify_bsgs(A, A.base, A.strong_gens) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims rrFT)r+rrrangelenorder stabilizer)groupbaser'rstrong_gens_distrcurrent_stabilizeri candidates r _verify_bsgsrETs8A@@@@@0t<< 3t99  DD$$%6q%9::  # # % %):): : 55/::47CC!!Q&u 4rNc|||}t|d}t||d}t ||S)a3 Verify the centralizer of a group/set/element inside another group. This is used for testing ``.centralizer()`` from ``sympy.combinatorics.perm_groups`` Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup) >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.testutil import _verify_centralizer >>> S = SymmetricGroup(5) >>> A = AlternatingGroup(5) >>> centr = PermutationGroup([Permutation([0, 1, 2, 3, 4])]) >>> _verify_centralizer(S, A, centr) True See Also ======== _naive_list_centralizer, sympy.combinatorics.perm_groups.PermutationGroup.centralizer, _cmp_perm_lists NTr) centralizerr.r/r2r)r?argcentr centr_listcentr_list_naives r_verify_centralizerrL}sc: '!!#&&e++t+4455J.ucdCCC :'7 8 88rcddlm} |||}t}t |dr|j}n&t |dr|}nt |dr|g}|D]}|D]}|||z  |t|}| |S)Nrrr __getitem__r*) r+rnormal_closuresetr-rr/addr. is_subgroup) r?rHclosurer conjugates subgr_genselr$ naive_closures r_verify_normal_closurerXs@@@@@@.,&&s++JsL!!^ m $ $ l # #U ##%%%% % %C NN38 $ $ $ $ %$$T*%5%566M   } - --rc ddlm}ddlm}m}ddlm}g}tt|D],} || \} } } } | | | gg| z| f-||\}}}||||dz }t|tr d}|g}|g}nt|}g}t|D]1} | ||| || |dz 2||}|d|D}t|d }|j}t!}|d D]A}|||}|D]0}t#|||}||1Bt|}|d |z}|D]/}|d d |d d kr|d |d krdS|}0t|dS)au Canonicalize tensor formed by tensors of the different types. Explanation =========== sym_i symmetry under exchange of two component tensors of type `i` None no symmetry 0 commuting 1 anticommuting Parameters ========== g : Permutation representing the tensor. dummies : List of dummy indices. msym : Symmetry of the metric. v : A list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`. base_i, gens_i BSGS for tensors of this type n_i number ot tensors of type `i` Returns ======= Returns 0 if the tensor is zero, else returns the array form of the permutation representing the canonical form of the tensor. Examples ======== >>> from sympy.combinatorics.testutil import canonicalize_naive >>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs >>> from sympy.combinatorics import Permutation >>> g = Permutation([1, 3, 2, 0, 4, 5]) >>> base2, gens2 = get_symmetric_group_sgs(2) >>> canonicalize_naive(g, [2, 3], 0, (base2, gens2, 2, 0)) [0, 2, 1, 3, 4, 5] rr) gens_products dummy_sgs)_af_rmulr:c,g|]}t|Srrrs rr z&canonicalize_naive..s888Q+a..888rTrrN)r+rsympy.combinatorics.tensor_canrZr[r,r\r;r<r0 isinstanceintextendr.generater*rPr rQsort)gdummiessymvrrZr[r\v1rCbase_igens_in_isym_isizesbasesgensdgens num_typesSDdliststshdqr prevs rcanonicalize_naiversNA@@@@@GGGGGGGG999999 B 3q66]]55%&qT"U 66B48U34444&+D% IgsDF + +E#s )eHH E 9  >> YYwqz3q64!8<<====A88%88899A t$$ % %E A B ZZ4Z  HQNN  Ahhq!nn%%A FF1IIII  RAFFHHH 9D  SbS6T#2#Y  uR  qq !::rczddlm}ddlm}m}t |}|ddd|D}||}d}|D]\}}|t|z }d|D} d} |D]f\}}|D]^} |||| krJ| || | | ||  | d z| d z } _gg} | D]}| |t| |ksJ| ||d zgz } |d z} t| t t| ksJt| } dgt| dd zz}| D]}|t|xxd z cc< g}tt|D]3} || }|r'|| \}}| |||df4|t t|}|| |dg|R}|S) a 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 r) _af_invert)get_symmetric_group_sgs canonicalizec,t|dS)Nr:)r<)rs rr(z#graph_certificate..>sS1YYrT)keyreversecg|] }|d Sr_rrs rr z%graph_certificate..?s ! ! !aQqT ! ! !rcg|]}gSrr)r rCs rr z%graph_certificate..Js"""q"""rr:r])r,rrbrrr.itemsrgr<r0resortedr;rr)grrrrrpvert num_indicesrkneighverticesrCv2rhrqvlennr@r'ricans rgraph_certificatersF<;;;;;TTTTTTTT   E JJ&&J555 ! !5 ! ! !E Ju  EK""5s5zz! #"E"""H A5  BQx%)# q"))!,,,r#**1Q3///Q   A   q66[    +{Q ''A ?D !99U4[[)) ))))AA 3HQK  " #D SZZA A 3t99  )) G  )0033JD$ HHdD!Q' ( ( (IIKKK5%%&&G ,q'1 )q ) ) )C Jr)Fr#) sympy.combinatoricsrsympy.combinatorics.utilrrmulrr2rErLrXrrrrrrs++++++======&&&:-L-L-L-L`&&&R!9!9!9!9H&.&.&.&.RKKK\NNNNNr