Stanley E. Payne

An essay on skew translation generalized quadrangles

AstractWe study a class of generalized quadrangles satisfying a local Moufang condition that is more restrictive than the usual concept of skew translation GQ. A new infinite family is constructed whose point-line duals are translation generalized quadrangles with very small kernel. They have parameters (q2, q), q=3r, for r>2.

Conical flocks, partial flocks, derivation, and generalized quadrangles

L. Bader, G. Lunardon and J. A. Thas have shown that a flock ℱ0 of a quadratic cone in PG(3, q), q odd, determines a set ℱ={ℱ0,ℱ1,...,ℱq} of q+1 flocks. Each ℱj, 1≦j≦q, is said to be derived from ℱ0. We show that, by derivation, the flocks with q=3e arising from the Ganley planes yield an inequivalent flock for q≧27. Further, we prove that the Fisher flocks (q odd, q≧5) are the unique nonlinear flocks for which (q−1)/2 planes of the flock contain a common line. This result is used to show that each of the flocks derived from a Fisher flock is again a Fisher flock. Finally, we prove that any set of q−1 pairwise disjoint nonsingular conics of a cone can be extended to a flock. All these results have implications for the theory of translation planes.

Nonisomorphic generalized quadrangles

Abstract For each integer e > 2 a class of somewhat more than φ ( e ) pairwise non-isomorphic quadrangles is exhibited and shown to yield nonisomorphic ( v , k , λ)-designs. The collineation groups of these quadrangles and designs are determined. Also a class of quadrangles with s = q − 1, t = q + l , q any prime power, is constructed.

Affine representations of generalized quadrangles

Abstract It is shown that a general construction due to Tits of finite generalized quadrangles (4-gons) yields the “classical” examples and only these except when the characteristic of the underlying field is 2. In that case an affine representation of the quadrangles is used to obtain results concerning the self-duality and self-polarity of a “nice” class of quadrangles.

On maximizing det (ATA)

Let A be an n x p matrix of +/- 1s, n >= p. The problem considered is the destination of the maximal value of det(A^TA). The complete solution is given for p = 5, the maximum value is determined for n sufficiently large compared to p and provided certain Hadamard matrices exist.

Half Moufang implies Moufang for finite generalized quadrangles

SummaryA finite generalized quadrangle has two types of panels. If each panel of one type is Moufang, then every panel is Moufang. Hence by a theorem of Fong and Seitz [1] the quadrangle is classical or dual classical.

The equivalence of certain generalized quadrangles

Abstract A rather more general construction is shown to yield the generalized quadrangles of order ( s , t ) ( s = q − 1, t = q +1, q any prime power) constructed by Ahrens and Szekeres and by Hall.

The Fundamental Theorem of q -Clan Geometry

Let q be any prime power, F = GF(q). A q-clan is a set C = {A t : t ∈ F}of q, 2 × 2 matrices over F such that their pairwise differences are all anisotropic, i.e., for distinct \( s,t \in F,\left( {a,b} \right)\left( {{A_s} - {A_t}} \right)\left( {\begin{array}{*{20}{c}} a \\ b \\ \end{array} } \right) = 0 \) has only the trivial solution a = b = 0. Starting with a q-clan C, there are at least the following geometries associated with C in a canonical way (cf. [18]): a generalized quadrangle GQ(C) with parameters (q 2 , q); a flock F(C) of a quadratic cone in PG(3, q); a line spread S(C) of PG(3, q); a translation plane T(C) of dimension at most 2 over its kernel. Starting with a natural definition of equivalence for q-clans, the Fundamental Theorem of q-clan geometry (F.T.) interprets the equivalence of q-clans C 1 and C 2 as an isomorphism between G(C 1 ) and G(C 2 ), where G(C i ) is any of the geometries (mentioned above) associated with C i . The F.T. was first recognized in its present form in [1], but it was stated there in detail only for q = 2 e , and the proof was claimed to be only a slightly revised version of the proof given in [14] of an important special case of the F.T. However, the proof in [14] starts off by assuming a fairly technical result from [11] where it is embedded in a more general theory.

Collineations of the generalized quadrangles associated with q-clans

Abstract For each known finite generalized quadrangle associated with a q -clan (and hence with a flock of a quadratic cone), a description of its collineation group is given.

Spreads, flocks, and generalized quadrangles

A spread of PG(3,q), q an odd prime, recently constructed by R. Baker and G. Ebert, when generalized for q an odd prime power is isomorphic to a spread derived by J. A. Thas from a flock of a quadratic cone discovered by J. C. Fisher. The associated generalized quadrangle has an unusual colllneation.