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Uniform Semimodular Lattice, Valuated Matroid, and Euclidean Building

Speaker

Hiroshi Hirai, University of Tokyo

Time

2018.07.12 15:30-16:30

Venue

Middle Lecture Room, Math Building

Abstract

In this talk, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices (=atomistic semimodular lattices). We introduce a class of semimodular lattices, called uniform semimodular lattices, and establish a cryptomorphic equivalence between integer-valued valuated matroids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incorporates the Dress-Terhalle completion process of valuated matroids, and establishes a smooth connection with Euclidean buildings of type A.