The geometries PG(n-1,q)/PG(k-1,q)

We show that the simple matroid PG(n − 1, q)PG(k − 1, q), for n ≥ 4 and 1 ≤ k ≤ n − 2, is characterized by a variety of numerical and polynomial invariants.; In the first chapter, we introduce matroids by first refreshing our knowledge of finite projective and affine geometries. The projective geometry, PG(n − 1, q), plays an important role in extremal matroid theory. In particular, Bose and Burton [13] showed that, for a fixed m with m < n, among subgeometries of PG(n − 1, q) that contain no restriction isomorphic to PG(m − 1, q), the unique maximal example is PG(n − 1, q)PG(n − m, q), the deletion of PG(n − m, q) from PG( n − 1, q).; In the second chapter, we describe the structure of these Bose-Burton geometries (that is, the geometries of the form PG( n − 1,q)PG(n − m, q)); in particular, we explore their flats. We first ascertain the form and size of flats of different ranks and show that among GF(q)-representable geometries of rank n, the Bose-Burton geometries maximize not only the number of points for geometries with no restriction isomorphic to PG( m − 1, q), but also the number of ( q + 1)-point lines, the number of (q2 + q + 1)-point planes, in general, the number of flats of the form PG(f − 1, q) for 1 ≤ f ≤ m − 1. In addition, we show that the Bose-Burton geometries have no modular flats apart from the flats of this form and the largest flat. Thus, these geometries are not supersolvable. After finding the number, sizes, and forms of the flats of PG(n − 1, q) PG(n − m, q), we show that these geometries are characterized by information about flats of only four ranks, namely, 1, m, n − 2, and n − 1.; In the next two chapters, we move from flats to exploring matroid polynomials that record some of the flat information of a matroid. First, we discuss the characteristic polynomial and show that, subject to additional hypotheses, the Bose-Burton geometries are determined by their characteristic polynomials. In Chapter 3, we also explore the critical exponents of these geometries.; The characteristic polynomial is an evaluation of a very powerful matroid invariant, the Tutte polynomial. In Chapter 4, we study this polynomial and show several examples of non-isomorphic matroids with the same Tutte polynomial. We show that, although it is unusual for a matroid to be determined by its Tutte polynomial, any matroid that has the same Tutte polynomial as the geometry PG(n − 1, q)PG( n − m, q) is isomorphic to this geometry, with no additional hypotheses. The proof of this surprising property opens up additional possibilities for other matroids that may be determined by their Tutte polynomials, and we discuss some of these questions in the final chapter.