Ernest E. Shult

Line graphs, root systems, and elliptic geometry

Publisher Summary The chapter discusses star-closed sets of lines at 60 ° and 90 ° , leading to a theorem that leaves only a restricted number of possibilities, of a specific structure. These possibilities are realized by the root systems A n , D n , E 8 , E 7 , E 6 defined in terms of lines. The chapter discusses the relations to the official Root Systems, as they occur in geometry and algebra. The chapter also discusses the graphs represented by subsets of the root systems: line graphs of complete bipartite graphs for A n , Hoffmans generalized line graphs for D n , and various exceptional graphs for E 8 . The chapter also presents the application to Hadamard matrices.

Affine polar spaces

Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of ‘locally polar spaces’. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.

m -systems of polar spaces

Abstract Let P be a finite classical polar space of rank r , with r ⩾ 2. A partial m -system M of P , with 0 ≤ m ≦ r − 1, is any set { π 1 , π 2 , …, π k } of k (≠ 0) totally singular m -spaces of P such that no maximal totally singular space containing π i has a point in common with ( ν 1 ∪ π 2 ∪ ⋯ ∪ π k ) − π i , i = 1, 2, …, k . In each of the respective cases an upper bound δ for | M | is obtained. If | M | = δ, then M is called an m -system of P . For m = 0 the m -systems are the ovoids of P ; for m = r − 1 the m -systems are the spreads of P . Surprisingly δ is independent of m , giving the explanation why an ovoid and a spread of a polar space P have the same size. In the paper many properties of m -systems are proved. We show that with m -systems of three types of polar spaces there correspond strongly regular graphs and two-weight codes. Also, we describe several ways to construct an m ′-system from a given m -system. Finally, examples of m -systems are given.

Geometric hyperplanes of embeddable Grassmannians

Abstract It is proved that every geometric hyperplane of an embeddable Grassmann space of finite rank, arises from an embedding. An equivalent formulation of this result amounts to a geometric characterization of alternating k-linear forms.

Steiner triple systems with doubly transitive automorphism groups: A corollary to the classification theorem for finite simple groups

Assuming that the classification theorem for finite simple groups is complete, a conjecture of M. Hall (Proc. Sympos. Pure Math. 6 (1962), 47–66; and in “Proceedings of the International Conference on Theory of Groups”, pp. 115–144, Australian National University, Canberra, Australia, 1965) that a Steiner triple system with a doubly transitive automorphism group is a projective or affine geometry, is verified.

Nonexistence of Ovoides in &Ω + (10,3)

On etablit la non-existence des ovoides dans Ω + (10,3), ce qui elimine du meme coup les ovoides pour tous les Ω + (2n,3), n≥5

Theory of Fields

If F is a subfield of a field K, then K is said to be an extension of the field F. For \(\alpha \in K\), \(F(\alpha )\) denotes the subfield generated by \(F\cup \{\alpha \}\), and the extension \(F\subseteq F(\alpha )\) is called a simple extension of F. The element \(\alpha \) is algebraic over F if \(\dim _FF(\alpha )\) is finite. Field theory is largely a study of field extensions. A central theme of this chapter is the exposition of Galois theory, which concerns a correspondence between the poset of intermediate fields of a finite normal separable extension \(F\subseteq K\) and the poset of subgroups of \(\textit{Gal}_F(K)\), the group of automorphis ms of K which leave the subfield F fixed element-wise. A pinnacle of this theory is the famous Galois criterion for the solvability of a polynomial equation by radicals. Important side issues include the existence of normal and separable closures, the fact that trace maps for separable extensions are non-zero (needed to show that rings of integral elements are Noetherian in Chap. 9), the structure of finite fields, the Chevalley-Warning theorem, as well as Luroth’s theorem and transcendence degree. Attached are two appendices that may be of interest. One gives an account of fields with valuations, while the other gives several proofs that finite division rings are fields. There are abundant exercises.

Combinatorial Construction of Some Near Polygons

We give a construction which takes a rank two incidence geometry with three points on a line and returns a geometry of the same type, i.e., with three points on a line. It is also demonstrated that embeddings of the original geometry can be extended to the new geometry. It is shown that the family of dual polar spaces of type Sp(2n, 2) arise recursively from the construction starting with the geometry consisting of one point and no lines. Making use of this construction we inductively construct projective embeddings for these geometries, in particular the embedding in the spin module for the group Sp(2n, 2). We also show that if we apply the construction to a classical near polygon which is isometrically embedded in the near 2n-gon of type Sp(2n, 2) the resulting space is a near polygon. Examples of such classical, isometrically embedding spaces are near 2n-gon of Hamming type on a three letter alphabet and the product of dual polar spaces of types Sp(2k, 2) and Sp(2l, 2) withk+l=n.

On the classification of generalized Fischer spaces

In this paper we show that a generalized Fischer space is either a Fischer space or is locally a polar space. As a corollary we obtain the classification of the finite irreducible generalized Fischer spaces.

Construction of hjelmslev planes from (t, k)-nets

A (t, k)-net is an abstract generalization of the incidence structures which occur as the point and line neighborhoods of a finite Hjelmslev plane. A (t, k)-net contains ‘substructures’ which are nets of ordert and degreek. Every (t, r) Hjelmslev plane (brieflyH-plane) can be constructed from a suitable collection of (t, r+1)- and (t, r)-nets. A (t, r)H-plane or (t, k)-net is called extremal provided: each two points which are joined by more than two lines are joined by preciselyt lines and dually. IfB is a ‘properly’ extremal (t, r)H-plane (means both 2 andt≠2 occur among the joining numbers), thent is even; andr=2 orr=1+(t/2). All the 3-uniform [J. Combinat. Theory9, 267–288 (1970, this Zbl.204, 210)] (4, 2)H-planes are examples. Two further examples are constructed in the paper: an (8, 2) translationH-planeC and an (8, 2) projectiveH-planeD, none of whose affineH-planes are translationH-planes. All point neighborhoods fromC andD and all line neighborhoods fromD are isomorphic to a give (8, 3)-netE;E is constructed by considering the subspaces of a 64-point symplectic geometry overZ2.E is also used to answer (affirmatively) the question of the existence of proper fairly near affineH-planes [J. Combinat. Theory16A, 34–50 (1974)].