Translation Planes Research Articles

Nearly flag-transitive affine planes

Abstract Spreads of orthogonal vector spaces are used to construct many translation planes of even order qm , for odd m > 1, having a collineation with a (qm – 1)-cycle on the line at infinity and on each of two affine lines.

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2r) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2r). As an application, we show that a translation plane of order qn, q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).

A note on generalized twisted fields and autotopisms of translation planes

The first part of this paper is a discussion of autotopism groups of certain translation planes. The second part shows that there is a certain class of semifields of order q4 having nuclei of order q2 , and their translation planes are not desarguesian and not isomorphic to generalized twisted field planes.

Enumeration of orthogonal Buekenhout unitals

In this paper we develop general techniques for enumerating orthogonal Buekenhout unitals embedded in two-dimensional translation planes. We then apply these techniques in the regular nearfield planes, the odd-order Hall planes, and the odd-order flag-transitive affine planes. Stabilizers of the resulting unitals also are computed.

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and πt, the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that πt is flag-transitive, πt is not isomorphic to π, and πt is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S2b and S2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and πt (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and πt in some order.

Parallel mechanisms with decoupled rotation of the moving platform in planar motion

The current article presents a new family of T2R1-type spatial parallel mechanisms (PMs) with decoupled and unlimited rotation of the moving platform in planar motion. The moving platform performs two independent translations (T2) and one independent unlimited rotation (R1) whose axis is perpendicular to the plane of translations. A method is proposed for structural synthesis of T2R1-type PMs based on the theory of linear transformations. The moving platform has unlimited rotational capabilities and is decoupled with respect to translational motion. To the best of the author's knowledge, this article presents for the first time solutions of T2R1-type spatial PMs with decoupled and unlimited rotation of the moving platform in planar motion.

The group fixing a completely regular line–oval

We prove that the action of the full collineation group of a symplectic translation plane of even order on the set of completely regular line---ovals is transitive. This provides us with a complete description of the group of collineations fixing a completely regular line---oval.

Polarities and unitals in the Coulter–Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter---Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter---Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.

On the planes of Suetake

The Suetake planes and the simple full non-square Andre planes are shown to be exactly those finite translation planes that are coupled with a semifield plane. Also, the Suetake planes are never generalized Andre planes.

Translation planes constructed by multiple hyper-regulus replacement

New classes of mutually disjoint hyper-reguli of order q n , for n > 2, are determined, which are not Andre hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q−1) = 1, there are (q−1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2(q-1)/2 translation planes constructed is not Andre or generalized Andre. There are also new constructions of mixed subgeometry partitions.

Polarities of shift planes

Abstract We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes of polarities. We show that a translation plane admits an orthogonal polarity if, and only if, it is a shift plane. The corresponding planes are exactly those that can be coordinatized by commutative semifields. The orthogonal polarities form a single conjugacy class. Finally, we construct examples of compact connected shift planes with more conjugacy classes of polarities than the corresponding classical planes.

A new semifield of order 2 10

In [G. Lunardon, Semifields and linear sets of P G ( 1 , q t ) , Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q 2 n , n > 1 and n odd, with left nucleus F q n , middle and right nuclei both F q 2 and center F q . When n = 3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1–34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87–104]. For n > 3 , no example of a semifield belonging to this family is known. In this paper we first prove that, when n > 3 , any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2 10 belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.

Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.

Quotients of projective spaces and spheres

Abstract In this paper we prove results on topological properties of quotients of manifolds, in particular of projective spaces and spheres. Specifically, under suitable conditions we exhibit results on homology and homotopy groups of these quotients. As an application we generalize the characterization of topological translation planes obtained by Löwen, Steinke, and Van Maldeghem in [Adv. Geom.: S59–S74, 2003] to all possible (finite) dimensions. Our approach also yields a more unified proof for this characterization result. Most of the results in this paper are obtained by means of standard results from algebraic topology.

Factorized spread sets and translation planes with large homology groups

Abstract In this note we construct some series of translation planes of order q 2 which possess two commuting homology groups of order q – 1 and (q + 1)(q – 1, 2) respectively. These translation planes can be considered as relatives of nearfield planes: for nearfield planes the nontrivial part of the associated spread set is a group whereas in our case this set is the product of two groups.

A new family of locally compact 4-dimensional translation planes admitting a 7-dimensional collineation group

Abstract The present paper contains the construction of a family of 4-dimensional translation planes with a 7-dimensional collineation group. The planes of this family do not appear in the comprehensive classification [Betten, J. Reine Angew. Math. 285: 126–148, 1976, Betten, Geometriae Dedicata 3: 405–440, 1974/75, Betten, Math. Z. 154: 125–141, 1977]. We furthermore determine the isomorphism types of the new planes.

A New Class of Translation Planes Constructed by Hyper-regulus Replacement

A new class of translation planes of order qn that admit very small affine homology groups is constructed. These planes are constructed by the replacement of a new hyper-regulus that cannot be an Andre hyper-regulus. Hence, the planes cannot be Andre or generalized Andre.

HMO-planes

Abstract Hiramine, Matsumoto and Oyama [Osaka J. Math. 24: 123–137, 1987] made the remarkable discovery that every translation plane of order q 2 that is 2-dimensional over its kernel produces translation planes of order q 4. This construction is studied, with emphasis on isomorphisms among the resulting planes.

Classification of Projective Translation Planes of Order q2 Admitting a Two-Transitive Orbit of Length q + 1

A classification is given of all projective translation planes of order q2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points.

On the isotopism classes of finite semifields

A projective plane is called a translation plane if there exists a line L such that the group of elations with axis L acts transitively on the points not on L. A translation plane whose dual plane is also a translation plane is called a semifield plane. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a semifield, and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. In [S. Ball, G. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (1) (2007) 117–129] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a BEL-configuration, and vice versa that each BEL-configuration gives rise to a semifield. In this manuscript we investigate the question when two BEL-configurations determine isotopic semifields. We show that there is a one-to-one correspondence between the isotopism classes of finite semifields and the orbits of the action a subgroup of index two of the automorphism group of a Segre variety on subspaces of maximum dimension skew to a determinantal hypersurface.