Number of linear independent equations Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a multi-linear function.
Here is a simple example:
Given equations
$$\partial_i b_{jk}+\partial_j b_{ki}+\partial_k b_{ij}=0$$
where $i,j,k\in\{1,2,...,n\}$ and $B=\{b_{ij}\}$ is anti-symmetric matrix.
How do I know how many independent equations there are? I don't think it is $C(n,3)$ since I don't use the property of anti-symmetric matrix.
 A: In this generality, it is not at all clear what you mean by 'any general rule'.  Of course, there is a general rule: Compute the rank of the appropriate matrix or linear map.  However, computing that rank can easily be nontrivial, and there is no universal way to simplify this problem.
In your specific case, of the system of equations on the $\partial_kb_{ij}$, the number of independent equations is $C(n,3)$ after all, because you are asking for the rank the exterior multiplication mapping
$$
V\otimes\Lambda^2(V)\to\Lambda^3(V).
$$
This mapping is, of course, onto (because the map isn't zero and the right hand side is an irreducible $\mathrm{GL}(V)$-module), so the rank is the dimension of $\Lambda^3(V)$, which is $C(\dim V, 3)$.
As an example, though, of a case in which you don't necessarily know the correct answer right away, consider the case of the Bianchi identities
$$
R_{ijkl}+R_{iklj}+R_{iljk} = 0
$$
where $R_{ijkl}=-R_{jikl}=-R_{ijlk}$.  It is less obvious how many independent equations this is because you may not know the range of the corresponding map
$$
\Lambda^2(V)\otimes\Lambda^2(V)\to V\otimes\Lambda^3(V)
$$
because the right hand side isn't irreducible as a $\mathrm{GL}(V)$-module.  (However, it turns out that it is indeed onto in this case.)  
As this discussion suggests, sometimes representation theory helps, but it's not a panacea.
