Generalized polygon

In combinatorial theory, a generalized polygon 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 groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss.

1 Definition
2 Examples
3 Feit-Higman theorem
4 See also
5 References

Definition

A generalized polygon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and $I\subseteq P\times L$ 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 girth of the incidence graph is twice the diameter of the incidence graph, which is usually denoted by n.

This condition is 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. When it is important to specify the diameter, a generalized polygon of diameter n is called a generalized n-gon, and the normal names for small polygons are used.

A generalized polygon is of order (s,t) if:

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.

We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.

The dual of a generalized n-gon (P,L,I) is the incidence structure (P,L,I^-1), which is again a generalized n-gon.

Examples

A generalized digon (n = 2) is a complete bipartite graph K_s+1,t+1.

For any natural n ≥ 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 n-gon X with n equal to 3, 4, 6 or 8 such that G acts transitively on the set of flags of X. In the finite case, for n=6, one obtains the Split Cayley hexagon of order (q,q) for G₂(q) and the twisted triality hexagon of order (q³,q) for ³D₄(q³), and for n=8, one obtains the Ree-Tits octagon of order (q,q²) for ²F₄(q) with q=2²ⁿ⁺¹. Up to duality, these are the only known thick finite generalized hexagons or octagons.

Feit-Higman theorem

Walter Feit and Graham Higman proved that finite generalized n-gons with s ≥ 2, t ≥ 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 ≤ t².
If n = 6, then st is a square, and t^1/3 ≤ s ≤ t³.
If n = 8, then 2st is a square, and t^1/2 ≤ s ≤ t².
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 polygons exist for each n greater or equal to 2. It is unknown whether or not there exist generalized polygons with one of the parameters finite and the other infinite (these cases are called semi-finite).

References

Godsil, Chris; Gordon Royle (2001). Algebraic Graph Theory. New York: Springer-Verlag. ISBN 0-387-95220-9.

W. Feit and G. Higman, The nonexistence of certain generalized polygons, J. Algebra, 1 (1964), 114–131 MR 0170955

Hendrik van Maldeghem, Generalized polygons, Monographs in Mathematics, 93, Birkhauser Verlag, Basel, 1998 ISBN 3-7643-5864-5 MR 1725957

Dennis Stanton, Generalized n-gons and Chebychev polynomials, J. Combin. Theory Ser. A, 34:1, 1983, 15–27 MR 685208

Jacques Tits and Richard Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. x+535 pp. ISBN 3-540-43714-2 MR 1938841

Generalized polygon

Contents

Definition

Examples

Feit-Higman theorem

See also

References