Finite Generalized Quadrangles (Ems Series of Lectures in by Stanley E. Payne and Joseph A. Thas PDF

Geometry Topology

By Stanley E. Payne and Joseph A. Thas

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T t 0 /. s C s 0 2 t 0 /. As d i ti2 . i ti /2 > 0, we 2 02 t / > 0. st C s 0P 0 0 0 be that Ps D s or Ps > s t . Further, we note that ti D . i ti /=d for all i 2 f1; : : : ; d g iff d i ti2 . , iff s D s 0 or s D s 0 t 0 . If s D s 0 , then ti D 1 C st 0 for all i . 3. Recognizing subquadrangles 23 ovoid of S 0 . If s D s 0 t 0 , then ti D 1 C s 0 for all i . (b) ([126]). 1, and let 1 be the indices of the points of S 0 , 2 the set of remaining indices. 1 C s 0 t 0 /, and each point indexed by an element of 1 is collinear with exactly 1 C s 0 C s 0 t 0 such points.

D is an incidence matrix of the structure S). t C 1/I , where A is an adjacency matrix of the point graph of S (cf. 2). s C t /. , rij D 0) otherwise. Then Q and R are permutation matrices for which DR D QD. Q 1 /T D DD T Q D MQ. Hence QM D MQ. 1 (C. T. Benson [10], cf. also [142]). mod s C t /: Proof. QM /n D Qn M n D M n . It follows that the eigenvalues of QM are the eigenvalues of M multiplied by the appropriate roots of 20 Chapter 1. Combinatorics of finite generalized quadrangles unity.

Fx; yg?? /, then u is on just one line joining a point of fx; yg? to a point of fx; yg?? ; if u 2 fx; yg? [ fx; yg?? , then u is on s C 1 lines joining a point of fx; yg? to a point of fx; yg?? L; u/, with L a line joining a point of fx; yg? to a point of fx; yg?? and with u a point of O which is incident with L. 1 C s/. Hence r D 2. Since no two points of O are collinear, there follows jO \ fx; yg?? j; jO \ fx; yg? j 2 f0; 2g. Let O be an ovoid of the GQ S of order s and let z be a regular point not on O.

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