Flag matroids: algebra and geometry

Amanda Cameron, Rodica Dinu, Mateusz Michałek, and Tim Seynnaeve

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Submission date: 06. Nov. 2018
Pages: 37
Bibtex
MSC-Numbers: 05B35, 52B40, 14M15, 14M25, 19E08
Keywords and phrases: flag matroids, Tutte polynomial, K-theory of flag varieties
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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.