Quotients of dilworth truncations

Sufficient conditions are given for an elementary quotient of the kth Dilworth truncation of a matroid M to be the kth Dilworth truncation of a quotient of M. As special cases, contractions of Dilworth truncations and principal truncations by connected flats of Dilworth truncations are characterised as Dilworth truncations of certain matroids. As an application of the theory it is shown that the degree of the minimal extension field of GF(q) needed to represent the first Dilworth truncation of PG(r-1, q) is greater than 2r-4. The kth Dilworth truncation, denoted D,(M), is a canonical construction , first realised in [S] which assigns to any matroid M a new matroid whose ground set is the set of flats of M with rank k + 1 (in this paper subsets of E(M) with cardinality k + 1). If M and M' are matroids sharing a common ground set then M' is a quotient of M if every flat of M' is also a flat of M. If the rank of M and M' differ by one then M' is an elementary quotient of M. Elementary quotients are determined by modular cuts of M. In this paper we show that if (D,(M))' is an elementary quotient of D,(M) determined by a modular cut of D,(M) whose minimal members are connected then (D,(M))' = D,(W) where M' is an elementary quotient of M. The modular cut of A4 determining M' is specified. As special cases we are able to characterise principal truncations by connected flats of D,(M) and contractions of D,(M). As an application of the theory it is shown that the degree of the minimal extension field of GF(q) needed to represent the first Dilworth truncation of PG(r-1, q) is greater than 2r-4. This improves a bound of Brylawski [2].