Intersection pattern of the classical ovoids in symplectic 3-space of even order

Abstract

For s = 2^e, e > 1 odd, we determine how the copies of the Suzuki group Sz(s) in the symplectic group Sp(4, s) intersect. Using this information we determine how the classical ovoids in symplectic 3-space W(s) meet and obtain a complete set of double coset representatives of Sz(s) in Sp(4, s). We also note that the permutation representation of Sp(4, s) on the cosets of Sz(s) is multiplicity free, and its irreducible constituents are explicitly determined. Indeed, we show that the complex Hecke algebra of this permutation representation is isomorphic to the center of the complex group algebra of Sz(s). A combinatorial offshoot of this study is the construction of several new series of Buekenhout diagram geometries of type which are embedded as subgeometries of miquelian and Suzuki-Tits inversive planes.