Orthogonal Tits Quadrangles

We show that every 4-plump razor-sharp normal Tits quadrangle X is uniquely determined by a non-degenerate quadratic space whose Witt index m is at least 2. If this Witt index is finite, then X is the Tits quadrangle arising from the corresponding building of type B_m or D_m by a standard construction.

The generating rank of a polar Grassmannian

In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

APPLICATIONS OF SYSTEMS OF QUADRATIC FORMS TO GENERALISED QUADRATIC FORMS

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

Orthogonal Buildings

This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of positive dimension, (K, V, q) is a regular quadratic space of positive Witt index, S is the vertex set of the Coxeter diagram, (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n greater than or equal to 3, S is the vertex set of the Coxeter diagram for some n greater than or equal to 3, and Dn.l,script small l is the Tits index of absolute type Dn for n greater than or equal to 3. The chapter also considers propositions dealing with regular quadratic spaces and hyperbolic quadratic spaces.

ON THE WITT INDEX OF THE BILINEAR FORM DETERMINED BY A LEONARD PAIR

Let a,b,d be nonnegative integers such that a + b = d + 1. Let ℝ be the field of real numbers. We prove that there is always a Leonard pair A, A⋆ of d + 1 by d + 1 matrices over ℝ such that the associated bilinear form of P. Terwilliger on ℝd + 1 has the Witt index a - b.