The lattice of flats and its underlying flag matroid polytope
The paper is concerned with the notions of flag matroid and flag matroid polytope. For a given sequence of strictly ascending numbers \(k_1,\dots, k_m\), a flag on the ground set \(\{1,\dots, n\}\) for this given index is a sequence (flag) \(A_1\subseteq\cdots\subseteq A_m\) of subsets of \(\{1,\dots,n\}\) such that \(| A_i|= k_i\). A set of flags of a given index is called a flag matroid if for each \(k_i\) the sets of cardinality \(k_i\) occurring in the flags form the set of bases of a matroid. The flag matroid polytope for a set of flags of a given index is the polytope spanned in real \(n\)-space by all linear combinations \(\sum c_i e_i\), \(e_i\) being the \(i\)th vector in the standard basis, for a fixed flag \((A_1,\dots, A_m)\) of type \((k_1,\dots, k_m)\) and \(c_i\) the number of occurrences of \(i\) in the sets \(A_j\). It follows from a result by \textit{I. M. Gelfand} and \textit{V. V. Serganova} [Usp. Mat. Nauk 42, No. 2, 107-134 (1987; Zbl 0629.14035); translation in Russ. Math. Surv. 42, No. 2, 133-168 (1987)] that the sets of flags which form a flag matroid can be classified in terms of the geometry of the flag matroid polytope. The main results of the paper are the classification of the adjacency relation for vertices in the flag matroid polytope and a bijection between the maximal chain in the lattice of flats of a matroid \(M\) and certain facets of the corresponding flag matroid polytope.
scientific article; zbMATH DE number 1309761