Non-classical polar unitals in finite Figueroa planes

The finite Figueroa planes are non-Desarguesian projective planes of order q ³ for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundh?fer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O??Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.     

M-P invertible matrices and unitary groups over Fq 2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Unitary block designs

Given a projective plane $\text{E}$ over the field of q² elements and a unitary polarity π of $\text{E}$ it is possible to construct the well-known unitary design $\text{U}$ whose points are the absolute points of π and whose blocks are the non-absolute lines of π. A relation of perpendicularity is defined between blocks and it is shown that this relation can be described in terms of the incidence structure of $\text{U}$. The projective plane $\text{E}$ together with the polarity π can then be reconstructed from the design $\text{U}$ in such a way that any automorphism of $\text{U}$ extends to a collineation of $\text{U}$ which commutes with π.     

Polyhedral 2-manifolds inE 3 with unusually large genus

An equivelar polyhedral 2-manifold in the class ?_p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. In this paper, examples are constructed, to show that each of the classes ?_3,q (q≧7), ?_4,q (q≧5) and ?_p,4 (p≧5) contains infinitely many distinct combinatorial types. As particular examples, there are polyhedral 2-manifolds with 576 vertices and genus 577, and with 4096 faces and genus 4097. A modification of one construction shows that there is a constantk, such that for eachg≧2, there exists a closed polyhedral 2-manifold inE ³ of genusg with at mostkg/logg vertices.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

On the elgenvectors of Schur's matrix

A basis of eigenvectors is given for the matrix $\text{U}$ = (e^2πimn/q), (1 ≤ m, n ≤ q). The eigenvectors arise from the characters on the reduced residue class group (mod q).     

Slices of the unitary spread

We prove that slices of the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q≡2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of PΓO ⁺(8,q) fixing the unitary spread. When q is even, there is a connection between spreads of Q⁺(7,q)\mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5,q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173–194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q=2^2h+1. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151–165, 1982. When q=3 ^h , we classify, up to the action of the stabilizer in PΓO(7,q) of the unitary spread of Q(6,q), those among its slices producing spreads of the elliptic quadric Q^-(5,q)\mathcal{Q}^{-}(5,q).     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Note on primitive permutation groups and a diophantine equation

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(m,q) ? H ? PFL(m,q), for any prime power q and integer m ? 3. In the course of the proof the diophantine equation 5^m + 11 = x^p2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.     

Configurations in projective planes and quadrilateral-star Ramsey numbers

Some classes of configurations in projective planes with polarity are constructed. As the main result, lower bounds for the Ramsey numbers r(n)=r(C₄;K_1,n) are derived from these geometric structures, which improve some bounds due to Parsons about 30 years ago, and also yield a new class of optimal values: r(q²-2q+1)=q²-q+1 whenever q is a power of 2. Moreover, the constructions also imply a known result on C₄-K_1,n bipartite Ramsey numbers.     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

On 1-Systems of Q(6, q), q Even

In this paper, a method is developed to study locally hermitian 1-systems of Q(6, q), q even, by associating a kind of flock in PG(4, q) to them. This method is applied to a known locally hermitian 1-system of Q(6, 2^2e ), which was discovered by Offer as a spread of the hexagon H(2^2e ). The results concerning this spread appear to be suitable for generalization and enable us to find new classes of 1-systems of Q(6, q), q even. We also prove that a locally hermitian 1-system of Q(6, q), q even, which is not contained in a 5-dimensional subspace, is semi-classical if and only if it belongs to the new classes we describe. Finally, from the new classes of 1-systems arise new classes of semipartial geometries.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

The Topological Tverberg Theorem and winding numbers

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d?1, and if q is a prime power, but not yet in general.)The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a “Winding Number Conjecture” that concerns only maps of the (d-1)-skeleton of a (q-1)(d+1)-simplex to R^d. “Many Tverberg partitions” arise if and only if there are “many q-winding partitions.”The d=2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K_3q-2 in the plane. We investigate graphs that are minimal with respect to the winding number condition.     

Multiple blocking sets and multisets in Desarguesian planes

In AG(2, q ²), the minimum size of a minimal (q ? 1)-fold blocking set is known to be q ³ ? 1. Here, we construct minimal (q ? 1)-fold blocking sets of size q ³ in AG(2, q ²). As a byproduct, we also obtain new two-character multisets in PG(2, q ²). The essential idea in this paper is to investigate q ³-sets satisfying the opposite of Ebert’s discriminant condition.     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

Punctured plane partitions and the q-deformed Knizhnik-Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik-Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley-Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ²-weighted punctured cyclically symmetric transpose complement plane partitions where τ=−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ²-enumerations of vertically symmetric alternating sign matrices and modifications thereof.     

Intersections of t-reguli,rational curves,and orthogonal Latin squares

The celebrated net-embedding theorem of R.H. Bruck asserts that a net with a large number of parallel classes, i.e., a net with small deficiency, can be embedded in an affine plane. The only known class of examples of unembeddable nets of small deficiency has been constructed by geometry, using partial spreads of PG(3, q). In this note we attempt to generalize that discussion to PG(n, q), with particular emphasis on the case n=5.