Polar space

In mathematics, in the field of combinatorics, a polar space of rank "n" ("n" ≥ 3), or "projective index" "n"−1, consists of a set "P", conventionally the set of points, together with certain subsets of "P", called "subspaces", that satisfy these axioms :

* Every subspace, together with its own subspaces, is isomorphic with a partial geometry "PG"("d","q") with −1 ≤ "d" ≤ ("n"−1) and "q" a prime power. By definition, for each subspace the corresponding "d" is its dimension.
* The intersection of two subspaces is always a subspace.
* For each point "p" not in a subspace "A" of dimension of "n"−1, there is a unique subspace "B" of dimension "n"−1 such that "A"∩"B" is ("n"−2)-dimensional. The points in "A"∩"B" are exactly the points of "A" that are in a common subspace of dimension 1 with "p".
* There are at least two disjoint subspaces of dimension "n"−1.

A polar space of rank two is a generalized quadrangle.

Examples

* In "PG"("d","q"), with "d" odd and "d" ≥ 3, the set of all points, with as subspaces the totally isotropic subspaces of a random symplectic polarity, forms a polar space of rank ("d"+1)/2.
* Let "Q" be a nonsingular quadric in "PG"("n","q") with character ω. Then the index of "Q" will be "g" = ("n"+"w"−3)/2. The set of all points on the quadric, together with the subspaces on the quadric, forms a polar space of rank "g"+1.
* Let "H" be a nonsingular Hermitian variety in "PG"("n","q"²). The index of "H" will be $leftlfloor\; frac\{n-1\}\{2\}\; ight\; floor\; ,!$. The points on "H", together with the subspaces on it, form a polar space of rank $leftlfloorfrac\{n+1\}\{2\}\; ight\; floor\; ,!$.

Classification

Jacques Tits proved that a finite polar space of rank at least three, is always isomorphic with one of the three structures given above. This leaves only the problem of classifying generalized quadrangles.