Consider vector space V over finite field $F_q$ and 
$V^ * $ its dual space. Denote $P(V), P(V^ * )$ the sets of ALL subsets in $V$ and $V^*$. 

**Question** How to construct GL_n(F_q) equivariant bijection between P(V) and P(V^*) ?(Which exists if I understand correctly Andreas Blass MO-reply  [here][1]).


Remark: In comment Andreas Blass mentioned that the proof is not entirely constructive.

Remark: Trivial example V=F_2 - obvious. V=F_2xF_2 - Klein group - here is SURPRISE: V =canonically = V^* . So F_2xF_2xF_2 seems to be first non-trivial case.

Remark: My guess is that if subset "L"  of V is linear we should correspond to it 
orthogonal linear subspace $L^{ort}$ in $V^ *$. So it is a kind of projective duality. But what to do with 
non-linear subsets ? Especially with points ? 

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PS

**Question** what are the natural automorphisms of P(X) ? (Except obvious coming from 
automorhisms of X itself) ?  Do they correspond to some "correspondences"  or whatever ? 

**Question** How to make the theorem constructive ? 




  [1]: https://mathoverflow.net/questions/106945/sets-m-n-with-g-action-such-that-cm-cn-as-g-modules-how-are-they-related/106958#106958