# Sum over cyclic permutation of indices

To define the Schouten bracket I need to be able to sum over a cyclic permutation of the indices: $$[\Phi,\Xi]_S=\mathfrak S_{i,j,k} \left(\Phi^{is}\partial_s\Xi^{jk}+ \Xi^{is}\partial_s\Phi^{jk}\right)\,,$$

where $$\mathfrak S_{i,j,k}$$ denotes the cyclic sum.

I am aware of the related posts however trying it myself I am not sure the implementation is correct:

ps = Permute[{i, j, k}, CyclicGroup[3]];

Sch[A_, B_] := Sum[Sum[A[[ps[[r]][[1]],s]]D[B[[ps[[r]][[2]],ps[[r]][[3]]]],X[[s]]]+B[[ps[[r]][[1]],s]] D[A[[ps[[r]][[2]],ps[[r]][[3]]]],X[[s]]],{s,4},{r,3}],{i,4},{j,4},{k,4}]


The indices $$i,j,k$$ range over 1 to 4, and $$X$$ are my local coordinates.

EDIT: Indeed I was to brief, I won't to check that for example the following matrix $$\Pi$$:

{{0, 0, (1 + z1 zb1), z1 zb2}, {0, 0, z2 zb1, (1 - z1 zb1)}, {-(1 + z1 zb1), -z2 zb1, 0, 0}, {-z1 zb2, -(1 - z1 zb1), 0, 0}}


defines a Poisson bivector for the 2-dimensional projective space i.e. $$[\Pi,\Pi]_S=0$$. The local coordinates are $$z_1,\bar z_1,z_2,\bar z_2$$.

The reference that I am using is M. V. Karasev and V. P. Maslov - Nonlinear Poisson Brackets. Geometry and Quantization, this is the pertinent excerpt

• This is hard to check without any context. Please give examples of the entities A and B that want to stuff into Sch. – Henrik Schumacher Mar 23 at 12:26
• It's a bit hard to understand as you have not given examples of expected inputs and outputs. That being said, I suppose you want a sum like Sum[f @@ idx, {i, 4}, {j, 4}, {k, 4}, {idx, NestList[RotateLeft, {i, j, k}, 2]}]. – J. M. will be back soon Mar 23 at 12:30
• can you clarify separately your letter meaning by code also! because it get me confused! so we try to help you – Alrubaie Mar 23 at 16:26
• @Sch, J. M. is slightly pensive and Alrubaie Thank you already taking a look. I added details :-) – Anne O'Nyme Mar 23 at 18:03