A Note on Finite Self-Polar Generalized Hexagons and Partial Quadrangles

In 1976, Cameron et al. [2] proved that, if a finite generalized hexagon of order s admits a polarity, then either s or 3s is a perfect square. Their proof used standard eigenvalue arguments. Later on, Ott [3] showed that for a self polar finite thick generalized hexagon of order s, 3s always has to be a perfect square. His proof used Hecke algebras. It was surprising that one had to use this more complicated method to achieve this result. The goal of the present note is to prove it with the original elementary eigenvalue technique. Suppose that 1 is a finite generalized hexagon of order s admitting a polarity %. We number the points of 1 as p1 , p2 , ..., pv , v= (1+s)(1+s+s). Let A=(aij) be the (v_v)-matrix with aij=1 if pi I pj , and pij=0 otherwise. It is shown in Cameron et al. [2] that A has eigenvalues 1+s, 0, \s, and \3s. It is also shown there that tr(A)=1+s and that the multiplicity of the eigenvalue 1+s is 1. If we denote the multiplicity of = s by k= and the multiplicity of = 3s by l= , = # [+, &], and if we put k=k+&k& , l=l+&l& , then one has the equation