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A q-clan with q a power of 2 is equivalent to a certain generalized quadrangle with a family of subquadrangles each associated with an oval in the Desarguesian plane of order 2. It is also equivalent to a flock of a quadratic cone, and hence to a line-spread of 3-dimensional projective space and thus to a translation plane, and more. These geometric objects are tied together by the so-called Fundamental Theorem of q-Clan Geometry. The book gives a complete proof of this theorem, followed by a detailed study of the known examples. The collineation groups of the associated generalized quadrangles and the stabilizers of their associated ovals are worked out completely.



Chapter 1. q-Clans and Their Geometries

Let q = 2e, F = GF(q). Let \( A = \left( \begin{gathered} x y \hfill \\ w z \hfill \\ \end{gathered} \right) \) and \( A' = \left( \begin{gathered} x' y' \hfill \\ w' z' \hfill \\ \end{gathered} \right) \) be arbitrary 2 × 2 matrices over F.

Chapter 2. The Fundamental Theorem

Since in this section we depend so strongly on the computational setup, we review the notation one more time.

Chapter 3. Aut(GQ(C))

Recall that \( \mathcal{G} \) denotes the full group of collineations of GQ(C), and that \( \mathcal{G}_0 \) denotes the subgroup of \( \mathcal{G} \) fixing the points ((0, 0), (0, 0), 0) and (∞).

Chapter 4. The Cyclic q-Clans

By a cyclic q-clan we mean one for which there is some m modulo q + 1 for which the automorphism θ(id, M ⊗ Mm) of G⊗ given explicitly by Eq. (3.21) is a collineation of GQ(C). (See Theorem 3.8.1.) In [COP03] the authors gave a unified construction that included three previously known cyclic families plus a new one. We have modified their presentation to obtain what we call the canonical version. (See [Pa02a] for the connection between the original construction, which we do not need, and the one given here.) Moreover, we go on to show that the unified construction really does give cyclic GQ (see [CP03]).

Chapter 5. Applications to the Known Cyclic q-Clans

To obtain the canonical form of the classical q-clan put k = m = 1 in Eqs. (4.7) and (4.8). A simple computation shows that if \( w = \frac{1} {{\delta \frac{1} {2}}} \) , so tr(w) = 1, then the classical q-clan C in 1/2-normalized form is given by
$$ C = \left\{ {A_t = \left( {\begin{array}{*{20}c} {wt^{\tfrac{1} {2}} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {t^{\tfrac{1} {2}} } \\ {wt^{\tfrac{1} {2}} } \\ \end{array} } \right):t \in F} \right\}. $$

Chapter 6. The Subiaco Oval Stabilizers

An algebraic plane curve of degree n (n ≥1) in PG(2, q) is a set of points C = V(f) = {(x, y, z) ∈ PG(2, q): f(x, y, z) = 0}, where f is a homogeneous nonzero polynomial of degree n in the variables x, y, z.

Chapter 7. The Adelaide Oval Stabilizers

We now pick up right where we left off at the end of Section 4.8, except that from now on we assume that q = 2e with e even, and \( m = \frac{{q - 1}} {3} \equiv \frac{{ - 2}} {3} \) (mod q + 1), so we are in the Adelaide case. The unique linear map known that stabilizes the oval \( \mathcal{O}'_\alpha \) is the involution given by
$$ \begin{gathered} ([j],[j + 1],[jm] + 1) \mapsto \hfill \\ ([j],[j + 1],[jm] + 1)\left( {\begin{array}{*{20}c} 1 \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} {[1]} \\ 1 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ \end{array} } \right) = ([j],[j - 1],[jm] + 1). \hfill \\ \end{gathered} $$
The fixed points of this involution are the points of the line x = 0, i.e., the points (0, y, z). But clearly the unique oval point on this line is the point (0, δ, 1), hence this line is a tangent line. The generator of the known stabilizer is \( \hat \theta ' \) , which acts on the points of this line as (0, y, z) ↦ (0, y2/δ, z2, from which it follows that exactly three points on this line are fixed: the oval point (0, δ, 1) and two others: (0, 1, 0) and (0, 0, 1). But the secant line through pj′ and p-j′ passes through the point (0, 1, 0), implying that the nucleus must be (0, 0, 1).

Chapter 8. The Payne q-Clans

Suppose that \( C = \{ A_t \equiv \left( {\begin{array}{*{20}c} {x_t } \\ 0 \\ \end{array} \begin{array}{*{20}c} {y_t } \\ {z_t } \\ \end{array} } \right):t \in F\} \) is a q-clan for which each of the functions xt, yt, zt is a monomial function. In a rather remarkable paper, T. Penttila and L. Storme [PS98] show that up to the usual equivalence of q-clans, the three known examples are the only ones. Since the two non-classical families exist only for e odd, we assume throughout this chapter that e is odd. Then the three known families have the following appearance. There is some positive integer i for which
$$ C = \{ A_t = \left( {\begin{array}{*{20}c} {t^{\frac{1} {{2i}}} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {t^{\frac{1} {2}} } \\ {t^{\frac{{2i - 1}} {{2i}}} } \\ \end{array} } \right):t \in F\} . $$

Chapter 9. Other Good Stuff

Let S be a GQ with parameters (s, t), s ≥ 1, t ≥ 1. A spread of S is a set M of lines that partition the points of S. Dually, an ovoid of S is a set \( \mathcal{M} \) of points of S such that each line of S is incident with a unique point of \( \mathcal{O} \) . It is easy to see that a spread must have 1 + st lines and an ovoid must have 1 + st points. For example, if a GQ S′ of order q is contained as a subquadrangle in a GQ S with order (q2, q), then each point X of a line l exterior to S′ is on a unique line of S′. Hence the q2 +1 lines of S′ that meet l form a spread of S′ said to be subtended by l. Spreads and ovoids of GQ have been studied a great deal and have a wide variety of connections with other geometric objects. For a general reference see J. A. Thas and S. E. Payne [TP94]. For q = 2e see especially [BOPPR1] and [BOPPR2]. In this section we give a very brief introduction to the material contained in these latter two papers.


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