Bergman fans and their subdivisions

In this talk we first introduce another way of constructing new matroids from a given one M, by minimizing the linear functional \(w \cdot x\) over \(x\) in the matroid polytope of \(M\), for a fixed \(w \in \mathbb{R}^n\). We then use this to define the Bergman fan of a given matroid, which are spaces of weights \(w\) whose corresponding matroids do not contain loops. To understand the topology of the Bergman fan, we introduce a combinatorial gadget called phylogenetic trees which can be used to parametrize the Bergman fan, and give a nice subdivision of it.

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Date published: Wednesday, October 9, 2024