Intersection pattern of the classical ovoids in symplectic 3-space of even order
The authors prove several theorems on classical ovoids in symplectic 3- space \(W(s)\) for \(s=2^ e\), \(e>1\) odd. They determine how the copies of the Suzuki group \(Sz(s)\) in the symplectic group \(Sp(4,s)\) intersect and how the classical ovoids in symplectic 3-space \(W(s)\) meet and obtain a complete set of double coset representatives of \(Sz(s)\) in \(Sp(4,s)\). Towards the study of the permutation representation they prove that the complex Hecke algebra of permutation representation is isomorphic to the center of the complex group algebra of \(Sz(s)\). They present a construction of a semi-biplane from any pair ofovoids of \(W(s)\) where in other words a semi-biplane is a Buekenhout diagram geometry of type \(\circ\overset{{}^\subset} {\mathrel{\mkern-8mu}\relbar\joinrel\relbar\mathrel{\mkern-8mu}} \circ\overset{{}^\supset} {\mathrel{\mkern-8mu}\relbar\joinrel\relbar\mathrel{\mkern-8mu}} \circ\). So they get the construction of several new series of that diagram geometries which are embedded as subgeometries of miquelian and Suzuki- Tits inversive planes.
scientific article; zbMATH DE number 4122727
scientific article