The Bergman complex of a matroid and phylogenetic trees

Authors:
Federico Ardila;Caroline J. Klivans
Affiliations:
Department of Mathematics, University of Washington, Seattle, WA and Mathematical Sciences Research Institute, Berkeley, CA;Departments of Mathematics and Computer Science, University of Chicago, Chicago, IL and Mathematical Sciences Research Institute, Berkeley, CA
Venue:
Journal of Combinatorial Theory Series B
Year:
2006

Citing 5
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Abstract

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan B˜(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn × R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.