\h"6SSKJrJrJrJr SSKJrJrJr SSK J r J r Sr Sr SrSrSrS rS rS rS rS rSrSSjrSSjrSSjrSS/\"SS5"SS5\"SS5"SS5\"S5"SS5"SS5/4rSrSrSrSSjrSrg) ) Permutation_af_rmul _af_invert_af_new)PermutationGroup_orbit_orbit_transversal)_distribute_gens_by_base_orbits_transversals_from_bsgsc[U5U:a [S5e/nUb^USSS2HRn[[US-55nUS:XaUS-XR'X%US-'XTS-XTsXT'XTS-'UR U5 MT USSS2HGn[[US-55nXTS-XTS-XTXTS-4XTUS-&UR U5 MI U$)a Return the strong generators for dummy indices. Parameters ========== dummies : List of dummy indices. `dummies[2k], dummies[2k+1]` are paired indices. In base form, the dummy indices are always in consecutive positions. sym : symmetry under interchange of contracted dummies:: * None no symmetry * 0 commuting * 1 anticommuting n : number of indices Examples ======== >>> from sympy.combinatorics.tensor_can import dummy_sgs >>> dummy_sgs(list(range(2, 8)), 0, 8) [[0, 1, 3, 2, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 5, 4, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 7, 6, 8, 9], [0, 1, 4, 5, 2, 3, 6, 7, 8, 9], [0, 1, 2, 3, 6, 7, 4, 5, 8, 9]] zList too largeN)len ValueErrorlistrangeappend)dummiessymnresjas V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/tensor_can.py dummy_sgsrs6 7|a)** C 1AU1q5\"Aax1u!a%1uXqtNAD!E( JJqM UbU^ q1u 1uXqQxqQx7AE  1  Jc[U5nUVs/sHoD(a [U5OSPM nnUSSn[U5H:n[U5H(up[U5Hn XpU ;dM XYXh' M& M* M< U$s snf)ao 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))], [0], list(range(10))) [0, 1, 2, 2, 2, 2, 2, 2, 8, 9] N)rminr enumerate) rrindices num_typesdxmricrs r _min_dummiesr)JsCI+237RBRD 7A3 !*C 9 g&DA9% ?TCF&' J 4sA?c8X2HnXUU:XdMUs $ g)zh Return the representative h satisfying s[h[b]] == j If there is not such a representative return None N)srbS_cosetshs r_trace_Sr0cs' [ qT7a<H rc.UHnX0U:XdM Us $ g)zh Return the representative h satisfying h[gj] == p_i If there is not such a representative return None Nr+)gjp_iDxtravr/s r_trace_Dr5os!  5C<H rcF/nUHnX$;aURU5 MURU5nUS-S:XaXES-nOXES- nURU5 URU5 URU5 URU5 URU5 M g)z remove p0 from dumx r rrN)rindexremove)dumx dumx_flatp0rr%k p0_paireds r _dumx_remover>{s C < JJrN  HHRL q5A:q5 Iq5 I "  )# 2rc/nSn[U5H]nXQ;a1UR[X$R555 US- nM9UR[ [U55/5 M_ [ U5S- nX4[ [U55/:Xa"US-nX4[ [U55/:XaM"USUS-$Nrr)rrsortedvaluesrr)sizebase transversalrrr's rtransversal2cosetrFs A A 4[ 9 HHVKN1134 5 FA HHd5;'( )  A A $4d $% % Q $4d $% % Va!e9rc2 ^^/^0URnURnUS- n[[U55n[ SU55(+n [ U5n USSn /n U Hn U R U 5 M USSnUVs/sHoRPM nnU(a [XbU5n/n[U 5H%nUR [U UUUU55 M' [[U55nUUU4/n[US- 5GHnUnUU;=(a UnU(a)UVs/sHn[U5PM nn[UUU5m/OU1m/U (a [XU5m0OO[US- 5VVs/sH3n[[UUVs/sHn[U5PM snU55PM5 snnm0[U/U04SjU55nUVs/sHn[U5PM nnU(a [UUUSSS9OSnU(aUVs/sHn[U5PM nn[U 5H<nUU U;dMUUbU UnO"U UR!U5S-nU UUSS2n O U/n/n U(GaqUR#5un!n"m[UU04SjT/55U:waM8T/Vs/sHnT0TUU:XdMUPM n#n[%U"U5n$[U$5n%U#Vs/sHnU!UPM n&nUVs/sHnU%UPM n'nU&Vs/sH nUU';dM UPM n(nU(Hn)U(a-['U!U)UU5n*U*(dM!U*V+s/sHn+U!U+PM n*n+OU!n*U(a[)U$U)UU5n,U,(dMYO UU$U):waMeUn,U,U$U)U:XdeU"V+s/sHn+U,U+PM n,n+U*V+s/sH n+U,UU+PM n-n+U R+U*U,U-45 M U(aGMqU R-S S 9 S /U-n.U (aNU R#5un!n"mTSS U.SS :XaTS U.S :wa g OUR+U!U"T45 Tn.U (aMNUVs/sHoUU:XdM UPM nnUU;aUR/U5 [1XU5 /n[U 5H%nUR [U UUUU55 M' GM US S $s snfs snfs snfs snnfs snfs snfs snfs snfs snfs snfs sn+fs sn+fs sn+fs snf)aR" Butler-Portugal algorithm for tensor canonicalization with dummy indices. Parameters ========== 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. Returns ======= Return 0 if the tensor is zero, else return the array form of the permutation representing the canonical form of the tensor. Notes ===== 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 [3] 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[0]], \dots, 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 [4]). A slot symmetry transformation `s` is a permutation acting on the slots `t \rightarrow T(ind[(g*s)[0]], \dots, ind[(g*s)[n-1]])` A dummy symmetry transformation acts on `ind` `t \rightarrow T(ind[(d*g)[0]], \dots, 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`, or in other words to the set of all permutations allowed by the slot and dummy symmetries. 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` and then contravariant 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 ordering 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[0] is the slot symmetry `-(0, 2)` `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}` sgens[1] 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, 2)(1, 3), (0, 4)(1, 5)]` where the first three interchange covariant and contravariant positions of the same index (d1 <-> -d1) and the last two interchange the dummy indices themselves (d1 <-> d2). The dummy symmetry acts from the left `d = [1, 0, 2, 3, 4, 5, 6, 7]` exchange `d1 \leftrightarrow -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^{d2}{}_{d1 d3}{}^{d1 d3}{}_{d2}` `= -T^{d3}{}_{d1 d3}{}^{d1 d2}{}_{d2}` under slot symmetry `-(0,4)`; `= 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 \rightarrow S_{b0} \rightarrow S_{b0,b1} \rightarrow \dots` and `D \rightarrow D_{p0} \rightarrow D_{p0,p1} \rightarrow \dots` 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 \times g \times s_i` satisfying `h_i[j] = p_j` for `0 \le j < i` starting from `s_0 = id, d_0 = id, h_0 = g`. The equations `h_0[0] = p_0, h_1[1] = p_1, \dots` 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, \dots, b_{i-1}}*b_j = D_{p_0, \dots, p_{i-1}}*p_j` so that for dx in `D_{p_0,\dots,p_{i-1}}` and sx in `S_{base[0], \dots, 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 [3]: 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.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))], [0], 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))], [0], base, gens, transversals, g) 0 r c3(# UHoSLv M g7fNr+).0_s r 'double_coset_can_rep..s":c9csNc3X>^# UHupm[UU4SjT55v M g7f)c34># UH nTTUv M g7frIr+rJxr/mds rrL1double_coset_can_rep...s/1b1hN)r!)rJr,dr/deltabrRs @rrLrMs#CsGA!#////ss&*FT)afc34># UH nTTUv M g7frIr+rPs rrLrMs,V2ad8VrTc US$)Nr+)rQs r&double_coset_can_rep..s"r)keyrrZ)rC array_formrranyrextend _array_formrFrrrr)r!r rr7poprr0r5rsortr8r>)1rrb_SsgensS_transversalsgrC num_dummiesr#all_metrics_with_symr$r9r:r%r/sgensxdsgsxr'idnTABr-testbrKsgensx1iiddxr3dsgsx1r4rQdeltap p_i_indexTAB1r,rUdeltab1dgdginvsdeltabgdeltapNEXTrs1ixd1h1prevrVrRs1 ` @@rdouble_coset_can_reprsC\ 66D A(K5%&G"":c":::CI 1:DI a&C%* +UmmUF + *4nE E 9  YtAwA <= uT{ C a=/C 4!8_ S#V +126awqz6G2D'1-FSF d1B8=TAXH7FfT %$' %%,% %$'(*,-7FHBCsCC&+,e'!*e,$D&#uF  -34VjmVF4 "Bd2hr7&!"XF!%Rs 3a 7I!"Xill3F#UFggiGAq!,V,,3"(<&QBqtHOq&G<!QBrNE%,-WqtWG-)/0AuQxG0&7w!!w,AwD7 "!Q>:B .01baeb1B!"Q%f5B be| B"Q%yC''''()qbfq)*,-"Bb2i"- RRL)A)cp o &sTzhhjGAq!"vcr"R5DH$% Aq!9%Dd$1Vtqy!V1 8 JJqMTc*y!A LL47CFK@ A"[^ q6":u,3$'H-5(=.07(2&*-$2srS(S5S! SS! S'S,(S1<S1S64S; TTTT /T T$TS!cURn[U5nU(dUSS$[X5n[USS5HnX`;dM UR U5 M UnUHunU/U-n Sn UR 5HBn [ X{5n [U5V s/sHoRU 5PM nn X:dM>Un U n MD U (dMj[ Xz5nMw U$s sn f)a Canonicalization of a tensor with respect to free indices choosing the minimum with respect to lexicographical ordering in the free indices. Explanation =========== ``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 [2]. 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] Nr^) r_rget_transversalsrArrBrrr7)rDgensrhnum_freerC transversalsrQr/transvh_ir,skrr~his rcanonical_freers` A q6D t #D/L AcrF^ = KKN AfXo --/B!B).x92((2,B9x " 1A H :sCcv[[U55nSn[U5HnXA;aM X2U'US- nM U$r@)rr)rC fixed_slotsrposr's r_get_map_slotsr_sC uT{ C C 4[  A q  Jrc/nS=pV[[X55VVs/sHupxUPM n nn[U5n [U5H=n X;aUR X5 US- nM"UR X5U -5 US- nM? U$s snnfr@)rAziprrr) rCrfreer,rrr<rKyfdrr's r _lift_sgensrjs A IAs;56 76!6B 74yH 4[  HHRUO FA HHQTH_ % FA  H 8sBc SSKJn [U[5(dUS;a [ S5eSnOG[ U5n[ SU55(d [ S5e[ U5U:wa [ S5eURnSn/nUHVupp[X5(d$[X5n U (dU"XU/UQ76nUs $U upURX//U -U 45 X{- nMX US:Xa[U[5(dU/nU/n/nUHnURU5 M U(a-U[[USUS S-55:wa [ S 5e[U6unnnUU:wa[ S UU4-5e[US - 5Vs/sH nUU;dM UPM nn[ U5n[UUUU5nU(dU$US US- :XaSOSnSn[U5Hunupp/nU SRS - n[U 5HTnUUUU-n/nUH+nUU;dM URUR!U55 M- URU5 UU- nMV XUU 4UU'M [U6unnn[U5V s/sHn UR!U 5PM n!n UU- n"UV s/sHn U U;dM U U- PM n#n U(aU#RU"S- U"S - /5 OU#RU"S - U"S- /5 [#UU!5n$UVs/sHnUU!;dM U$UPM n%nUV&Vs/sH4n&[%U&R&Vs/sHnUU!;dM U$UPM sn5PM6 n'n&nUV&V s/sHn&U&V s/sHn U U- PM sn PM n(n&n [)U%U'5n)[%U#5n#[+U(UU%U'U)U#5n*U*S:Xag[-UU!UU*5n+U+$s snfs sn fs sn fs snfs snfs snn&fs sn fs sn n&f) a 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 is used instead. 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] r)canonicalize_naiverrNzmsym must be 0, 1 or Nonerc3*# UH oS;v M g7f)rNr+)rJmsymxs rrLcanonicalize..s;dUL(dsz!msym entries must be 0, 1 or Nonez6dummies and msym must have the same number of elementsrZzdummies is not validz&g has size %d, generators have size %dr )sympy.combinatorics.testutilr isinstancerrrallrC_is_minimal_bsgsget_minimal_bsgsrrar gens_productsrr"r7rrrbrrr),rhrmsymvrr$rC num_tensorsv1base_igens_in_isym_imbsgscan flat_dummiesr9size1sbaserfr'rrg1signstartfree_ilen_tensrr/frr<rQpos_freesize_redg1_red map_slots sbase_redr sgens_red dummies_red transv_redg2g3s, r canonicalizerys-z@ dD ! ! | #89 9 I ;d;;;@A A w<9 $HJ J 66DK B&'" //$V4E(T>A> "NF 6B4#:u56 '(A~jt44)vLD! U<?LQSDTWXDX-Y(ZZ/00(,E5% u} 4e} DF FTAX @!!<*?AD @4yH ua 2B  2$("1D E+4Q<' 'FC!9>>A%sA5%(*,AB6IIaggaj) MM"  X E/1,8"'+D%&+8_5_ _H5hH$& g!,!QAL!,gK>!)Y7J V_F T9iV EB Qw T8T2 .B Ik AD6 < CSd,>s` O O O& O4 O O O'O$ O O O$* O/4O* O/O$*O/c UVs/sHn[U5PM nnUVs/sHn[U5PM nnU(aSOSn[US5U- n[US5U- n[[U55n[[XUU-55nU(aSUV s/sHn U SSU-U SU-U SU-/-PM nn UV Vs/sHoU Vs/sHo3U-PM sn-PM nn nO=UV s/sHoU-PM nn UV Vs/sHoU Vs/sHo3U-PM sn-PM nn nX-n U $s snfs snfs sn fs snfs snn fs sn fs snfs snn f)a Direct products of the generators gens1 and gens2. Examples ======== >>> from sympy.combinatorics.tensor_can import perm_af_direct_product >>> gens1 = [[1, 0, 2, 3], [0, 1, 3, 2]] >>> gens2 = [[1, 0]] >>> perm_af_direct_product(gens1, gens2, False) [[1, 0, 2, 3, 4, 5], [0, 1, 3, 2, 4, 5], [0, 1, 2, 3, 5, 4]] >>> gens1 = [[1, 0, 2, 3, 5, 4], [0, 1, 3, 2, 4, 5]] >>> gens2 = [[1, 0, 2, 3]] >>> perm_af_direct_product(gens1, gens2, True) [[1, 0, 2, 3, 4, 5, 7, 6], [0, 1, 3, 2, 4, 5, 6, 7], [0, 1, 2, 3, 5, 4, 6, 7]] r rNr^rZ)rrr) gens1gens2signedrQr,n1n2rendgenrs rperm_af_direct_productrYsX"$ $eT!WeE $# $eT!WeE $ A U1X B U1X B rOE uRb! "C !#!SSbC3r7R<R2">>! #:?@%3#.#Qb&#..%@&+,esse,:?@%3#.#Qb&#..%@ -C J# % $#.@,.@sFD,D1#D6 ED; E-E E E  E;E EcU(aSOSnUSRU- n[U5nXrVs/sHoU-PM sn- nUV s/sHoRPM nn UV s/sHoRPM nn [XU5n [ U S5n [[ U 55n U V s/sH oU :wdM U PM n n U (dU /n XzV s/sHn [ U 5PM sn 4$s snfs sn fs sn fs sn fs sn f)a Direct product of two BSGS. Parameters ========== 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.tensor_can import (get_symmetric_group_sgs, bsgs_direct_product) >>> base1, gens1 = get_symmetric_group_sgs(1) >>> base2, gens2 = get_symmetric_group_sgs(2) >>> bsgs_direct_product(base1, gens1, base2, gens2) ([1], [(4)(1 2)]) r r)rCrrbrrrr) base1rbase2rrr,rrDrQr/rrCid_afs rbsgs_direct_productr~s0A q B ;DU #UVU ##D$) *Eq]]EE *$) *Eq]]EE * !% 7D tAw>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs >>> get_symmetric_group_sgs(3) ([0, 1], [(4)(0 1), (4)(1 2)]) rrr)rrrrrb)rantisymr'rrQrDr/s rget_symmetric_group_sgsrs& AvGDqN+,,,>CAEl KlKA qa% ( 4 4lD K!|(,-1QAJ-(,-1QUAJ- a!e D d+d'!*d+ ++ L--+s+C.CC7Cr rrrc U(d/$[X5n[X5up4UVVVs/sH1oUR5VVs0sHupgXgR_M snnPM3 nnnnU$s snnfs snnnf)z8 Return transversals for the group with BSGS base, gens )r r itemsrb)rDrstabsorbitsrrrQr/s rrrsq  $T 0E9$FF " BC'')<)$!Q %)< " ="sA,A&A,&A,c ^/nUSSnUSRn[U5HVm[U4SjU55(aMURT5 UVs/sHoURTT:XdMUPM nnMX X :H$s snf)a 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 Nrc3F># UHoRTT:Hv M g7frI)rb)rJr/r's rrL#_is_minimal_bsgs..s7$Q==#q($s!)rCrrrrb)rDrrsgs1rCr/r's @rrrs}" E 7D 7<s B1Bcd[U5nUR5up[X5(dgX4$)a Compute a minimal GSGS base, gens BSGS If base, gens is a minimal BSGS return it; else return a minimal BSGS if it fails in finding one, it returns None TODO: use baseswap in the case in which if it fails in finding a minimal BSGS Examples ======== >>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.tensor_can import get_minimal_bsgs >>> riemann_bsgs1 = ([2, 0], ([Permutation(5)(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3)])) >>> get_minimal_bsgs(*riemann_bsgs1) ([0, 2], [(0 1)(4 5), (5)(0 2)(1 3), (2 3)(4 5)]) N)rschreier_sims_incrementalr)rDrGs rrrs4* A,,.JD D ' ' :rc SnU(dZUR/5S:aE[U5nUSRnXVS- -S-nU/[[ [ U555/4$[ U5(a [U5nOSn/nUSRn[ [ U55n US- n US(d#UR[ [ U 555 U"XpXS5up[ S[U55HZn U"XpXU 5up[XXS5upX-(d-UR[ [ US- US- U -555 Xj- nM\ US- nU Vs/sHnURU :wdMUPM n nUbU(dU (d [U 5/n XkU 4$/n[ [U5S- 5Hn XnXS-n[ [ US55nURU5 URU5 URUS5 UR[ [ USS-U555 US:XaURUUS-/5 OURUS-U/5 U R[U55 M [ U 5n UHn X;dM U RU 5 M U R5 U (d [U 5/n XkU 4$s snf)a Returns size, res_base, res_gens BSGS for n tensors of the same type. Explanation =========== 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.tensor_can import tensor_gens, get_symmetric_group_sgs 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, [[1], [0]]) (8, [0, 4], [(7)(0 2), (7)(4 5)]) four symmetric tensors with 3 indices, two of which with free indices cpU(d USSUSS4$URU5nUR5upX4$)z: return the BSGS for G.pointwise_stabilizer(free_indices) N)pointwise_stabilizerr)rrDr free_indicesHsgss r _get_bsgstensor_gens.._get_bsgs.sB7DG# #&&|4A335ID9 rr rNrrZ) countrrCrrrr`rrrrbrard)rDrlist_free_indicesrrrrCrno_freer num_indicesres_baseres_gensr'rrnrr/ base_commind1ind2rs r tensor_gensr sH  %++B/!3 ! "Aw||1H~!R'$uT{"34555  T " G 7<8aq}}'=8H> {''Hx''I 3w>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, gens_products >>> base, gens = get_symmetric_group_sgs(2) >>> gens_products((base, gens, [[], []], 0)) (6, [0, 2], [(5)(0 1), (5)(2 3), (5)(0 2)(1 3)]) >>> gens_products((base, gens, [[1], []], 0)) (6, [2], [(5)(2 3)]) rr)rrrrrCr) rres_sizerrr'rCrDrrr/s rrrs2$/!#5 H 1c!f &-D0T15q:({H x !E#28aEz8H2 7 x ''3s / B<BN)T)F)r) sympy.combinatorics.permutationsrrrrsympy.combinatorics.perm_groupsrrr sympy.combinatorics.utilr r rr)r0r5r>rFrrrrrrrr riemann_bsgsrrrrrr+rrrs#&,^2  * sl G T  ]@"J$,N,:1v Aq)!Q/Q1B1a1H#Aq!,Q244  68u$p#(r