The Bergman complex of a matroid and phylogenetic trees

2003/11/21 by Federico Ardila, Ardila, Federico, Carly Klivans +1 · 10 citations
Computer Science · Mathematics · #05B35 #52B05 #92B10 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation #Topological and Geometric Data Analysis #math.CO #msc:05B35 #msc:52B05 #msc:92B10

paper · pdf · doi:10.48550/arxiv.math/0311370

15 pages, 6 figures. Reorganized paper and updated references. To appear in J. Combin. Theory Ser. B

openalex publication_date 2003/11/21 · arxiv created 2005/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 its lattice of flats. In addition, we show that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n is homeomorphic to the space of phylogenetic trees T_n.

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