Generalized quadrangles, flocks, and BLT sets

Approximately five years ago Thas observed a remarkable coincidence relating certain generalized quadrangles constructed in [Ka3] to flocks of cones and translation planes [Th] (cf. [Ka2]). While there is a rapidly growing literature concerning this connection, there as yet has been no explanation for it. This note contains an observation providing at least some kind of explanation. It also contains a seemingly useless method for “embedding” translation planes associated with the flocks “into” the generalized quadrangle. Just as in [BLT] we will only be able to deal with the case of odd parameters. It is not the purpose of this note to introduce all of the notation required. Instead, we refer to [Ka2; Ka3; Pal; Pa2; Th]. Let F= GF(q) and Q = F2 x Fx F2, with multiplication in Q given by (u, c, u)(u’, c’, 0’) = (U + u’, c + c’ + v . u’, v + v’), where v . U’ denotes the dot product. The construction in [Ka3; Pal; Pa21 of generalized quadrangles with parameters s=q2, t = q, assumes that 2 x 2 matrices B, and M, = B, + B:, Y E F, are given, and the construction uses the following family 9 of q + 1 subgroups ofQ: