Matroid designs of prime power index

A class of BIBD's is constructed with parameters generalizing those of the finite projective geometries. These designs are used to construct matroids in which the hyperplanes are equicardinal and the complement of every hyperplane has prime power cardinality. These so-called matroid designs of prime power index “almost” have the property that the flats of any given rank are equicardinal.     

On the simple connectedness of hyperplane complements in dual polar spaces

Let Δ be a dual polar space of rank n≥4, H be a hyperplane of Δ and Γ?Δ?H be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.     

On the embeddability of polar spaces

We show that every nondegenerate polar space of rank at least 4 with at least three points on each line can be embedded in a projective space. Together with some results from [9] and [12], this provides a particularly elementary proof that any such polar space is of classical type. Our methods involve the use of geometric hyperplanes and work equally well for spaces of finite or infinite rank.     

Embedding the planar structure of 4-dimensional linear spaces into projective spaces

It is known that a linear spaces of dimensiond has at least as many hyperplanes as points with equality if it is a (possibly degenerate) projective space. If there are only a few more hyperplanes than points, then the linear space can still be embedded in a projective space of the same dimension. But even if the difference between the number of hyperplanes and points is too big to ensure an embedding, it seems likely that the linear space is closely related to a projective space. We shall demonstrate this in the cased=4.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Classification of irreducible weight modules over higher rank Virasoro algebras

Let G be a rank n additive subgroup of C and Vir[G] the corresponding Virasoro algebra of rank n. In the present paper, irreducible weight modules with finite dimensional weight spaces over Vir[G] are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank n−1. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.     

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K ₅\ e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F ₇ with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in P ₉. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F ₇ and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R ₁₀, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).     

Minimal non-orientable matroids in a projective plane

We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field.     

Hyperplane partitions and difference systems of sets

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper gives new constructions of DSS obtained from partitions of hyperplanes in a finite projective space, as well as DSS obtained from balanced generalized weighing matrices and partitions of the complement of a hyperplane in a finite projective space.     

On a class of finite linear spaces with few lines

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421–423) states that a finite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing finite linear spaces for which the difference between the number b of lines and the number v of points is assigned.

In this paper finite linear spaces with b−vm, m being the minimum number of lines on a point, are characterized.     

On a theorem of de Bruijn and Erdös

A theorem of de Bruijn and Erdös [2] asserts that every finite geometry (see section 1 for definition) has at least as many lines as points. The present paper uses linear algebra as a technique to establish the de Bruijn-Erdös result and a particular higher dimensional generalization. These results are special cases of theorems due to Basterfield and Kelly [1] and Green [3].     

Classification of line-transitive point-imprimitive linear spaces with line size at most 12

In this paper we complete a classification of finite linear spaces with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders 4, 7, 9 and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.      

Projective Equivalence of Δ-Matroids with Coefficients and Symplectic Geometries

The projective equivalence of matroid representations over fields and of oriented matroids is well studied. This paper is devoted to the study of projective equivalence of Δ-matroids with coefficients, which covers the concept of projective equivalence of matroids with coefficients and thus in particular the projective equivalence of represented and oriented matroids. A necessary and sufficient condition for the projective equivalence of Δ-matroids with coefficients is established in terms of the inner Tutte group T _M ⁽⁰⁾ of the underlying combinatorial geometry M. The structure of T_M ⁽⁰⁾ of symplectic projective spaces M of odd dimensions d≥3 over fields is computed.     

A characterization of certain 3-dimensional affino-projective spaces

The finite planar spaces containing at least one pair of planes intersecting in exactly one point and in which for every such pair of planes Π and Π′, any line intersecting Π intersects Π′ (or is contained in Π) are completely classified. These spaces are essentially obtained from projective spaces PG(3, k) by deleting either k collinear points or an affino-projective (but not projective) plane of order k.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

A polynomial Hutton theorem with applications

Extending a classical linear result due to Hutton to a nonlinear setting, we prove that a continuous homogeneous polynomial between Banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators. Among other consequences, we apply this result to generalize a classical result due to Aron and Schottenloher about the approximation property on spaces of polynomials and a recent result due to Çaliskan and Rueda about the quasi-approximation property on projective symmetric tensor products.     

A characterization of quadrics by intersection numbers

This work is inspired by a paper of Hertel and Pott on maximum non-linear functions (Hertel and Pott, A characterization of a class of maximum non-linear functions. Preprint, 2006). Geometrically, these functions correspond with quasi-quadrics; objects introduced in De Clerck et al. (Australas J Combin 22:151–166, 2000). Hertel and Pott obtain a characterization of some binary quasi-quadrics in affine spaces by their intersection numbers with hyperplanes and spaces of codimension 2. We obtain a similar characterization for quadrics in projective spaces by intersection numbers with low-dimensional spaces. Ferri and Tallini (Rend Mat Appl 11(1): 15–21, 1991) characterized the non-singular quadric Q(4,q) by its intersection numbers with planes and solids. We prove a corollary of this theorem for Q(4,q) and then extend this corollary to all quadrics in PG(n,q),n ≥ 4. The only exceptions occur for q even, where we can have an oval or an ovoid as intersection with our point set in the non-singular part.      

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).