A tensor product action on \(q\)-clan generalized quadrangles with \(q=2^ e\)
type: Item
instance of: scholarly article
title: A tensor product action on \(q\)-clan generalized quadrangles with \(q=2^ e\)
zbMATH Open document ID: 0841.51003
DOI: 10.1016/0024-3795(94)00345-E
author: Stanley E. Payne
published in: Linear Algebra and its Applications
publication date: 1996-07-18T00:00:00Z
review text: 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).
reviewed by: Erhard Quaisser
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 803734
zbMATH Keywords: generalized quadrangles
zbMATH Keywords: \(q\)-clan
zbMATH Keywords: automorphisms
zbMATH Keywords: ovals
MaRDI profile type: Publication
cites work: On \(q\)-clan geometry, \(q=2^ e\)
cites work: Flocks and ovals
cites work: Some generalized quadrangles with parameters \(q^ 2,q\)
cites work: Q4028448
cites work: Collineations of the Subiaco generalized quadrangles
cites work: Isomorphisms between Subiaco \(q\)-clan geometries
cites work: Q3974202
cites work: Q3344217
cites work: Generalized quadrangles and flocks of cones
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description: scientific article; zbMATH DE number 803734