most recent 30 from http://mathoverflow.net 2013-06-19T13:42:26Z http://mathoverflow.net/feeds/question/107912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107912/what-are-natural-automorphisms-of-set-of-subsets-how-to-constructify-andreas Alexander Chervov 2012-09-23T17:25:39Z 2012-09-23T17:25:39Z

<p>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 $V^*$. </p> <p><strong>Question</strong> 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 <a href="http://mathoverflow.net/questions/106945/sets-m-n-with-g-action-such-that-cm-cn-as-g-modules-how-are-they-related/106958#106958" rel="nofollow">here</a>).</p> <p>Remark: In comment Andreas Blass mentioned that the proof is not entirely constructive.</p> <p>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.</p> <p>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 ? </p> <hr> <p>PS</p> <p><strong>Question</strong> what are the natural automorphisms of P(X) ? (Except obvious coming from automorhisms of X itself) ? Do they correspond to some "correspondences" or whatever ? </p> <p><strong>Question</strong> How to make the theorem constructive ? </p>