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Nick Salter
Nick Salter
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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 working in characteristic zeroultimately doing, however, I only care about characteristic zero.)

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. I'm working in characteristic zero, however.)

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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Nick Salter
Nick Salter
  • 2.8k
  • 22
  • 32

Classification or bounds on "totally symmetric" arrangements of subspaces

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. I'm working in characteristic zero, however.)