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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 totally symmetric 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 not 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. For what I'm ultimately doing, however, I only care about characteristic zero.)

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  • $\begingroup$ Do you want $\sigma \mapsto A_\sigma$ to be a group homomorphism? $\endgroup$
    – ArB
    Commented Dec 23, 2021 at 18:17
  • $\begingroup$ @ArB - not necessarily. I've noticed in practice, however, that it is often possible to choose the $A_\sigma$ so that it is. $\endgroup$
    – Nick Salter
    Commented Dec 23, 2021 at 18:25
  • $\begingroup$ Not a reference or a solution, just an observation: since there is probably some matroid or matroid-like structure floating around that has to be uniform. For line arrangements this would be precisely the matroid represented by vectors spanning the lines; for $d>1$ maybe you can do something with exterior products. This could be a way to produce some bounds (not necessarily sharp). Interesting problem. $\endgroup$
    – Jeremy Martin
    Commented Jan 4, 2022 at 2:14

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