In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). 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. Every generalized n-gon with n even is also a near polygon.
Definition A generalized 2-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For n ≥ 3 {\displaystyle n\geq 3} a generalized n-gon is an incidence structure ( P , L , I {\displaystyle P,L,I} ), where P {\displaystyle P} is the set of points, L {\displaystyle L} is the set of lines and I ⊆ P × L {\displaystyle I\subseteq P\times L} is the incidence relation, such that:
It is a partial linear space. It has no ordinary m-gons as subgeometry for 2 ≤ m < n {\displaystyle 2\leq m<n} . It has an ordinary n-gon as a subgeometry. For any { A 1 , A 2 } ⊆ P ∪ L {\displaystyle \{A_{1},A_{2}\}\subseteq P\cup L} there exists a subgeometry ( P ′ , L ′ , I ′ {\displaystyle P',L',I'} ) isomorphic to an ordinary n-gon such that { A 1 , A 2 } ⊆ P ′ ∪ L ′ {\displaystyle \{A_{1},A_{2}\}\subseteq P'\cup L'} . An equivalent but sometimes simpler way to express these conditions is: consider the bipartite incidence graph with the vertex set P ∪ L {\displaystyle P\cup L} and the edges connecting the incident pairs of points and lines.
The girth of the incidence graph is twice the diameter n of the incidence graph. A generalized polygon is of order (s,t) if:
all vertices of the incidence graph corresponding to the elements of L {\displaystyle 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 {\displaystyle 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 {\displaystyle P,L,I} ), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the converse relation of I {\displaystyle I} . It can easily be shown that this is again a generalized n-gon.
Examples The incidence graph of a generalized digon is a complete bipartite graph Ks+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 set inclusion as the 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.
Restriction on parameters Walter Feit and Graham Higman proved that finite generalized n-gons of order (s, t) with s ≥ 2, t ≥ 2 can exist only for the following values of n:
2, 3, 4, 6 or 8. Another proof of the Feit-Higman result was given by Kilmoyer and Solomon. Generalized "n"-gons for these values are referred to as generalized digons, triangles, quadrangles, hexagons and octagons. When Feit-Higman theorem is combined with the Haemers-Roos inequalities, we get the following restrictions,
If n = 2, the incidence graph is a complete bipartite graph and thus "s", "t" can be arbitrary integers. If n = 3, the structure is a finite projective plane, and s = t. If n = 4, the structure is a finite generalized quadrangle, and t1/2 ≤ s ≤ t2. If n = 6, then st is a square, and t1/3 ≤ s ≤ t3. If n = 8, then 2st is a square, and t1/2 ≤ 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. Every known finite generalized hexagon of order (s, t) for s, t > 1 has order
(q, q): the split Cayley hexagons and their duals, (q3, q): the twisted triality hexagon, or (q, q3): the dual twisted triality hexagon, where q is a prime power. Every known finite generalized octagon of order (s, t) for s, t > 1 has order
(q, q2): the Ree-Tits octagon or (q2, q): the dual Ree-Tits octagon, where q is an odd power of 2.
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