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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).

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 ?


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 ?

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When $q = 2$ you can use the discrete Fourier transform, identifying $P(V)$ with the set of functions $V \to \mathbb{F}_2$. – Qiaochu Yuan Sep 23 '12 at 22:25

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