# Solving antisymmetric tensorial equation

Assume we have the following Tensor objects: $$F_{i}{}^{j}\;and\;S_{ij}{}^{k},$$ where the components of $F$ are known, and we would like to solve for the components of $S$ if they satisfy the following equation $$F^{l}{}_{i}S_{jl}{}^{k}-F^{l}{}_{j}S_{il}{}^{k}=0.$$ $l$ is summed over, all the indices run from 1 to 4, and $S$ is symmetric in the lower indices.

My attempt:

First, suppose we know all the components of $F$, and they are given by $$F= \begin{matrix} a & b & c & d\\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{matrix}$$

Then I defined the components of $S$ by:

S[i_, j_, k_] := S[i, j, k]


The first term of the equation I defined it as:

SF[i_, j_, k_] := SF[i, j, k] = S[1, i, j].F[k, 1] +
V[2, i, j].F[k, 2] +
V[3, i, j].F[k, 3] +
V[4, i, j].F[k, 4];


As for the second term in the equation, I think it can be found using Transpose

FS[i, j, k] = Transpose[SF, {i, k}]


Then for example:

Solve[SF==FS,{S[i,j,k]},{i,4},{j,4},{k,4}]


is not working. I'm sure there is something wrong in my commands, but I can't figure out what it is. The functions $a$,$b$,$c$,... in the expression of $F$ are some complicated scalar functions of space coordinates.

• Have you tried something? Do you have some approximate code? Sep 20 '12 at 12:15
• @belisarius Thanks for your comments, I have a code that gives me $F$ which is an endomorphisms acting on the tangent space of some moduli space. I will update the questions with what I tried to do... Sep 20 '12 at 13:01

Perhaps

f = RandomInteger[{-1, 1}, {4, 4}];
Solve[
And @@ Join[

• @Imagine I would recommend the following: 1) Check if my answer works as you expect with a simple case. 2) Report that here 3) The optimization problem depends on your F entries, so if the above is working you should post another question with details about your F's and tag it as optimization. Sep 21 '12 at 15:04
I've posted a detailed write-up on this method in the "asked-and-answered" question Conveniently solving tensor equations in which the contained tensors have various symmetries, where I use the above posted question as the example (solving the equation $F^{l}{}_{i}S_{jl}{}^{k}-F^{l}{}_{j}S_{il}{}^{k}=0$ where $S_{ij}{}^{k}$ is symmetric in i and j ).