Matroids from hypersimplex splits

Abstract

A b s t r ac t . A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians.2010 Mathematics Subject Classification. 52B40 (05B35, 14T05).

Ehrhart polynomials of rank two matroids

“…In [12] Joswig and the third author introduce the class of split matroids which provides the same separation property in arbitrary rank. This class strictly contains paving matroids and thus include the loopless matroids of rank 2.…”

On local Dressians of matroids

“…The statement above generalizes Theorem 4 by Chatelain and Ramírez [4] which deals with sequences of weakly compatible hyperplane splits. While the article by Joswig and Schröter [15] provides the case of sequences of strongly compatible hyperplane splits and the matroid polytopes that occur in these matroid subdivisions. We refer to Herrmann and Joswig [12] for the definitions.…”

“…This fact is even reflected in their dimensions. The dimension of Dr(d, n) is of order n d−1 for fixed d, while the dimension of TGr p (d, n) grows linear in n, see [15,Corollary 32].…”

Matroids and their Dressians

“…Since then, many questions about Dressians have been studied. Bounds on the dimension of Dressians were given in [JS17,HJJS09]. Rays of the Dressian have been studied in [JS17,HJS14].…”

“…Bounds on the dimension of Dressians were given in [JS17,HJJS09]. Rays of the Dressian have been studied in [JS17,HJS14].…”