The Structure Of Matroids

Let S be a set of matroids. Excluded minors for S are the matroids that are minor minimal with respect to not in S.Let q be a prime power, andC(q) the class of matroids that can be representable over GF(q).Rota conjectured that the number of excluded minors for L(q) is finite, which has only been solved for q=2,3,4. The presence of inequivalent representations of matroids over fields is the major barrier to Rota's Conjecture. Kahn conjectured that for any finite field, the number of inequivalent representations of a 3-connected matroid is finite.Unfortunately, Kahn's conjecture has been refuted(Oxley et al.,in J.Combin Theory Ser. B 67:325-343,1996).The counterexamples given by Oxley et al.to show Kahn's Conjecture is false for all finite fields with at least 7 elements have many mutually interacting 3-separations.Thus, it is not possible to decompose a 3-connected matroid across 3-separations in a reasonable way like 1-connected matroids or 2-connected ones having done(Cunningham and Edmonds,in Canada. J.Math 32:734-765,1980);however,it encourages the belief that a version of Kahn's Conjecture could be recovered for matroids with a higher connectivity than 3-connectivity. Recently, Geelen et al. proved that for each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F.(The paper is in preparation.)However, in the process of trying to extend the result to the non-prime case, they glumly found that for every finite field F of non-prime order≥9, there are a class of 4-connected matroids having arbitrarily many inequivalent representations over F.In Chapter 2,we prove that via an operation "reducing",every 3-connected representable matroid M with at least 9 elements can be decomposed into a set of sequentially 4-connected matroids and 3 special matroids which we call freely-placed-line matroids, spike-like matroids and swirl-like matroids; more concretely, there is a labeled tree that gives a precise description of the way that M built from its pieces.Strictly 4-connectivity is a notion too strong to be really useful, which ex-cludes highly structured objected such as matroids of complete graphs and projec- tive geometry. Moreover, it does not appear possible to find a reasonable analogue of Tutte's Wheels and Whirls Theorem, which holds for 3-connected matroids and sequentially 4-connected matroids.Given this and the results recently obtained by Geelen et at.,it is natural to hope that for any finite field of prime order, the number of inequivalent representations of a sequentially 4-connected matroid is finite.On the other hand, to better understand the behavior of inequivalent rep-resentation of 3-connected matroids, the notion, totally free matroids, is defined (Geelen et al.,in J.Combin Theory Ser.B 84:130-179,2002).It turns out that the number of inequivalent representation of a 3-connected matroid is bounded above by the number of inequivalent representation of a totally free minor.In Chapter 3, we prove matroids obtained from any totally free matroid by a se-quence of segment-cosegment and cosegment-segment exchanges are also totally free.Let U(q) be the class of matroids with no U2,q+2-minor.Obviously, U2,q+2 is an excluded minor for L(q),so all other excluded minors of L(q) except for U2,q+2 must be in U(q).Hence, it will be of help if we can know the structure of matroids in U(q) well.And it is natural to consider the extremal matroids in U(q),where a matroid in U(q)is extremal if it is simple and has no simple rank-preserving single-element extension in U(q).In Chapter 4,we focus on the following problem:for any integer l≥2, given some extremal matroids in U(l), how can we construct more? We study amalgams of extremal matroids in U(l): we determine which amalgams are in U(l) and which are extremal in U(l),where an amalgam of two matroids M1 on E1 and M2 on E2 is a matroid on E1∪E2 whose restrictions to E1 and E2 are M1 and M2,respectively.