Dependencies among dependencies in matroids

James Oxley¹ ; Suijie Wang²

¹Louisiana State University
²Institute of Mathematics, Institute of Mathematics

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

In 1971, Rota introduced the concept of derived matroids to investigate "dependencies among dependencies" in matroids. In this paper, we study the derived matroid $\delta M$ of an ${\mathbb F}$-representation of a matroid $M$. The matroid $\delta M$ has a naturally associated ${\mathbb F}$-representation, so we can define a sequence $\delta M$, $\delta^2 M$, \dots . The main result classifies such derived sequences of matroids into three types: finite, cyclic, and divergent. For the first two types, we obtain complete characterizations and thereby resolve some of the questions that Longyear posed in 1980 for binary matroids. For the last type, the divergence is estimated by the coranks of the matroids in the derived sequence.

Détail

DOI : 10.37236/8742

Classification : 05B35