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  "body": "\n## Definition\n\nLet $V$ be a [[vector space|000AYPA9AJ]] of finite dimension $n + 1 \\ge 3$ over a division ring.\nLet $\\mathcal{L}(V)$ be the set of its [[linear subspaces|00YC1D652K]], which are the subspaces of the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}(V)$.\nA *correlation* of $\\mathbb{P}(V)$ is a bijection $\\delta \\colon \\mathcal{L}(V) \\to \\mathcal{L}(V)$ such that $U \\subseteq W$ if and only if $\\delta(W) \\subseteq \\delta(U)$, for all $U, W \\in \\mathcal{L}(V)$.\nA *polarity* is a correlation $\\delta$ with $\\delta(\\delta(U)) = U$ for all $U \\in \\mathcal{L}(V)$.\n\n## Remarks\n\nA correlation sends subspaces of dimension $k$ to subspaces of dimension $n + 1 - k$, so it exchanges points and hyperplanes.\nFor a polarity $\\delta$, the hyperplane $\\delta(P)$ is the polar of the point $P$, and $P$ is the pole of that hyperplane.\n\nExample: let $F$ be a field, $V = F^{n+1}$ and $\\delta(U) = U^\\perp = \\set{y : x \\cdot y = 0 \\text{ for all } x \\in U}$, with the dot product $x \\cdot y = \\sum_i x_i y_i$.\nIf $U \\subseteq W$ then $W^\\perp \\subseteq U^\\perp$; since the dot product is a non-degenerate symmetric [[bilinear form|0SQZMXB6GP]], $\\dim U^\\perp = n + 1 - \\dim U$ and $(U^\\perp)^\\perp = U$.\nSo $\\delta$ is a bijection that reverses inclusion and has $\\delta^2 = \\id$: it is a polarity.\n\nNon-example: on $F^3$ let $M$ be the matrix with rows $(1, 1, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, and let $\\delta(U) = \\set{y : x^T M y = 0 \\text{ for all } x \\in U}$.\nThis is a correlation, but $\\delta(\\delta(U)) = M^{-1} M^T U$.\nFor $U = \\langle e_1 \\rangle$ one gets $M^T e_1 = (1, 1, 0)$ and $M^{-1} (1, 1, 0) = (0, 1, 0)$, so $\\delta^2(\\langle e_1 \\rangle) = \\langle e_2 \\rangle \\ne \\langle e_1 \\rangle$, and $\\delta$ is not a polarity.\n\nEvery correlation of $\\mathbb{P}(V)$ comes from a non-degenerate [[sesquilinear form|054YEQ0Y8M]] $f$ by $\\delta(U) = \\set{y : f(x, y) = 0 \\text{ for all } x \\in U}$.\nIt is a polarity exactly when $f$ is reflexive, that is $f(x, y) = 0$ implies $f(y, x) = 0$; up to a scalar, such forms are symmetric, alternating or Hermitian.\n",
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