A tensor product action on \(q\)-clan generalized quadrangles with \(q=2^ e\)
A \(q\)-clan is a set \(C = \left\{A_t = {x_t \;y_t \choose\;0\;\;z_t} : t \in F\right\}\) of \(q\) \(2 \times 2\) matrices over \(F = \text{GF}(q)\), where \(q\) be any prime power and the matrix \(A_s - A_t\) is anisotropic for distinct \(s, t \in F\). With any \(q\)-clan \(C\) is associated a collection of special kinds of geometrical structures: generalized quadrangle \(\text{GQ}(C)\) of order \((q^2, q)\), flock of a quadratic cone, spread of \(\text{PG}(3,q)\), translation plane of dimension at most over their kernel [see \textit{W. M. Kantor}, Math. Z. 192, 45-50 (1986; Zbl 0592.51003), the author, Congr. Numerantium 49, 115-128 (1985; Zbl 0632.51011), and \textit{J. A. Thas}, Eur. J. Comb. 8, 441-452 (1987; Zbl 0646.51019)]. In consequence of the fundamental theorem of \(q\)-clan geometry all automorphisms of the \(\text{GQ}(C)\) of order \((q^2, q)\) which fix a special pair of points are automorphisms of the elation group of the quadrangle. When \(q = 2^e\), the author describes these automorphisms in terms of tensor products of pairs of matrices in \(\text{GL} (2,q)\). He develops this tensor product action so that collineations of \(\text{GQ} (C)\) can be studied very efficiently. \(\text{GQ} (C)\) has subquadrangles of order \(q\) associated with ovals in \(\text{PG} (2,q)\). The general theory is applied to give an improved description of the induced stabilizers of the ovals in \(\text{PG} (2,q)\) that are associated with special \(q\)-clans (Subiaco \(q\)-clans).
scientific article; zbMATH DE number 803734
scientific article