Bergman complexes, Coxeter arrangements, and graph associahedra

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Roughly speaking, a Bergman complex of a matroid is a matroidal analogue of a tropical variety, and a positive Bergman complex of an oriented matroid is an oriented matroid analogue of a positive tropical variety. It turns out that the positive Bergman complex of an oriented matroid has a nice description in terms of the Las Vergnas face lattice; this implies that the positive Bergman complex is homeomorphic to a sphere. When we consider the Bergman complex and the positive Bergman complex of (the oriented matroid of) a Coxeter arrangement, we get some especially nice results: we get surprising connections to nested set complexes and graph associahedra. The results of this talk come from joint work with Federico Ardila and Carly Klivans, and from joint work with Federico Ardila and Victor Reiner.



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