Generalized n-gon

In combinatorial theory, a generalized "n"-gon is an incidence structure introduced by Jacques Tits. Generalized polygons encompass as special cases projective planes (generalized triangles, "n" = 3) and generalized quadrangles ("n" = 4), which form the most complex kinds of axiomatic projective and polar spaces. Many generalized polygons arise from finite groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Finite generalized polygons satisfying a technical condition known as the "Moufang property" have been completely classified by Tits and Weiss.

Definition

A generalized "n"-gon ("n" is a natural number greater than one) is an incidence structure ("P","L","I"), where "P" is the set of points, "L" is the set of lines and $Isubseteq\; P\; imes\; L$ is the incidence relation, satisfying certain regularity conditions. In order to express them, consider the bipartite "incidence graph" with the vertex set "P" ∪ "L" and the edges connecting the incident pairs of points and lines.

* The diameter of the incidence graph is "n".
* The girth of the incidence graph is 2"n".These two conditions are frequently stated as follows: any pair consisting of a point and a line is contained in an ordinary "n"-gon and there are no ordinary "k"-gons for "k" < "n".
* All vertices of the incidence graph corresponding to the elements of "L" have the same degree "s" + 1 for some natural number "s"; in other words, every line contains exactly "s" + 1 points.
* All vertices of the incidence graph corresponding to the elements of "P" have the same degree "t" + 1 for some natural number "t"; in other words, every point lies on exactly "t" + 1 lines.

Examples

* A generalized digon ("n" = 2) is a complete bipartite graph $K\_\{s+1,t+1\}.$

* For any natural "n" &ge; 3, consider the boundary of the ordinary polygon with "n" sides. Declare the vertices of the polygon to be the points and the sides to be the lines, with the usual incidence relation. This results in a generalized "n"-gon with "s" = "t" = 1.

* For each group of Lie type "G" of rank 2 there is an associated generalized polygon "X" with "n" equal to 3, 4, or 6 such that "G" acts transitively on the set of flags of "X".

Feit – Higman theorem

Walter Feit and Graham Higman proved that "finite" generalized "n"-gons with "s" &ge; 2, "t" &ge; 2 can exist only for the following values of "n":

:2, 3, 4, 6 or 8.

Moreover,

* If "n" = 2, the structure is a complete bipartite graph.
* If "n" = 3, the structure is a finite projective plane, and "s" = "t".
* If "n" = 4, the structure is a finite generalized quadrangle, and "t"^1/2 ≤ "s" ≤ "t"².
* If "n" = 6, then "st" is a square, and "t"^1/3 ≤ "s" ≤ "t"³.
* If "n" = 8, then "2st" is a square, and "t"^1/2 ≤ "s" ≤ "t"².
* If "s" or "t" is allowed to be 1 and the structure is not the ordinary "n"-gon then besides the values of "n" already listed, only "n" = 12 may be possible.

If "s" and "t" are both infinite then generalized "n"-gons exist for each "n" greater or equal to 2. It is unknown whether or not there exist generalized "n"-gons with one of the parameters finite and the other infinite (these cases are called "semi-finite").

See also

* Building (mathematics)
* (B, N) pair
* Ree group
* Moufang plane

References

* W. Feit and G. Higman, "The nonexistence of certain generalized polygons", J. Algebra, 1 (1964), 114–131 MathSciNet|id=0170955

* Hendrik van Maldeghem, "Generalized polygons", Monographs in Mathematics, 93, Birkhauser Verlag, Basel, 1998 ISBN 3-7643-5864-5 MathSciNet|id=1725957

* Dennis Stanton, "Generalized n-gons and Chebychev polynomials", J. Combin. Theory Ser. A, 34:1, 1983, 15–27 MathSciNet|id=685208

* Jacques Tits and Richard Weiss, "Moufang polygons", Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. x+535 pp. ISBN 3-540-43714-2 MathSciNet|id=1938841