Flag matroids: algebra and geometry

TL;DR

This paper explores the deep connections between flag matroids, algebraic geometry, and K-theory, generalizing classical matroid concepts to flag varieties and introducing an analogue of the Tutte polynomial.

Contribution

It extends the geometric and algebraic framework of matroids to flag matroids and flag varieties, including a new Tutte polynomial analogue.

Findings

Established a geometric description of flag matroids.

Generalized matroid constructions to polymatroids.

Introduced an analogue of the Tutte polynomial for flag matroids.

Abstract

Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the $K$-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study the class of flag matroids and their relations to flag varieties. In this way, we obtain an analogue of the Tutte polynomial for flag matroids.