Semidirect Sums of Matroids

For matroids M and N on disjoint sets S and T , a semidirect sum of M and N is any matroid K on S ∪ T that, like the direct sum and the free product, has the restrictionK|S equal toM and the contractionK/S equal toN . We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension ofM on the set S ∪ T with the direct sum ofN and the rank-0 matroid on S is a semidirect sum of M and N . We study principal sums in depth; these are such matroid unions where the extension of M has each element of T added either as a loop or freely on a fixed flat of M . A second construction of semidirect sums, defined by a Higgs lift, also specializes to principal sums. We also explore what can be deduced if M and N , or certain of their semidirect sums, are transversal or fundamental transversal matroids. To James Oxley on his 60th birthday 1. BLOCK UPPER-TRIANGULAR MATRICES AND SEMIDIRECT SUMS A simple way to combine two matricesA andB over a field F is to have them be blocks of a block-diagonal matrix. A richer collection of matrices results by putting a third matrix U over F in the upper right corner to obtain a block upper-triangular matrix (A,B;U), where (A,B;U) = (