A COORDINATIZATION OF GENERALIZED

QUADRANLGES OF ORDER (s − 1, s + 1)

Marijke de Soete

A generalized quadrangle is an incidence structure S = (P, B, I) with P and B sets of objects called points and lines respectively, with a symmetric incidence relation I which satisfies:

(i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line;

(ii) each line is incident with 1+s points (s ≥ 1) and two distinct lines are incident

with at most one point; (iii) for each point x and each line L, x I̸L, there exists a unique pair (y, M) ∈ P ×B

such that x I M I y I L. We call (s, t) the order of S. Let us consider a generalized quadrangle S of order (s−1, s+1) containing a spread

R (i.e., a subset R of B such that each point is incident with a unique line of R). R is called a spread of symmetry for the generalized quadrangle S if the group G_{R} of automorphisms of S fixing R linewise acts transitively on each line of R. If S has a spread of symmetry, then from S there arises a generalized quadrangle S′ of order s having a center of symmetry. So in this case we are able to give a coordinatization of S, using a planar ternary ring, and G_{R}, which is derived from the coordinatization of S′ due to S. E. Payne [3].

We also investigate the converse problem. Given a planar ternary ring and a group G, which are the conditions to obtain a generazlied quadrangle of order (s − 1, s + 1). Examples are given for the known models AS(q), T_{2}^{∗}(O) and the dual of P (T_{2}(O), x) with x a point of the oval O.

References

1. M. De Soete, J. A. Thas, Acoordinatization of generalized quadrangles of order (s, s + 2), J. Combin. Theory Ser. A 48 (1988), 1–11.

2. Stanley E. Payne, Quadrangles of order (s − 1, s + 1), J. Algebra 22 (1972), 97–119.

3. Stanley E. Payne, Generalized quadrangles of even order, J. Algebra 31 (1974), 367–391.

Seminarie voor hogere Meetkunde, State University of Ghent, Krijgslaan 281, B-9000 Gent, Belguim

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