# Solving antisymmetric tensorial equation

Assume we have the following Tensor objects: \begin{equation} F_{i}{}^{j}\;and\;S_{ij}{}^{k}, \end{equation} where the components of $F$ are known, and we would like to solve for the components of $S$ if they satisfy the following equation \begin{equation} F^{l}{}_{i}S_{jl}{}^{k}-F^{l}{}_{j}S_{il}{}^{k}=0. \end{equation} $l$ 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 \begin{equation} F= \begin{matrix} a & b & c & d\\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{matrix} \end{equation}

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.

• Hello, welcome to Mathematica.SE! Please consider registering your account so that any upvotes you get on this question are added to those you might get on future questions and answers. That way, over time you will be able to do more on the site (post graphics, edit things, etc). Another tip: after posting a question stay around for a little while, to answer questions raised by commenters. This will streamline the Q&A process considerably. – Dr. belisarius Sep 20 '12 at 12:14
• BTW ... Have you tried something? Do you have some approximate code? – Dr. belisarius 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... – Imagine 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. – Dr. belisarius 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 ).