Let $F$ be a field and let $W_1, \dots, W_k \subset F^n$ be a collection of $d$-dimensional subspaces of $F^n$ such that $W_i \cap W_j = \{0\}$ for all indices $i,j$. We say that such an arrangement is <i>totally symmetric</i> if for any permutation $\sigma \in S_k$, there is an automorphism $A_\sigma \in GL_n(F)$ such that 
$$
A_\sigma W_i = W_{\sigma(i)}
$$
holds for all $1 \le i \le k$. 

I am wondering if anyone has classified or bounded (in terms of $k,n$ and the characteristic of $F$) the size of such totally symmetric arrangements. For what I am doing, I can find a sufficiently good bound, so this is <i>not</i> a question about how to study such objects, merely a reference request - I'd like to avoid re-inventing the wheel if possible. Does this appear anywhere, e.g. in the combinatorics literature?

(N.B. the characteristic of $F$ matters! It's a fun exercise to classify totally-symmetric line arrangements, and see that things behave differently in characteristics 2 and 3. I'm working in characteristic zero, however.)