The Fundamental Theorem of q-Clan Geometry

Abstract

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.