%0 Generic %1 ArdilaKlivansWilliams2004 %A Ardila, Federico %A Klivans, Caroline %A Williams, Lauren %D 2004 %K Ardila Bergman Klivans Las_Vergnas Williams associahedron complex face fan graph lattice matroid orientation poset tropics variety %T The Positive Bergman Complex of an Oriented Matroid %U http://arxiv.org/abs/math/0406116 %X We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M_I, the positive tropical variety associated to I is equal to the fan over B+(M_I). Our main result is that a certain "fine" subdivision of B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph K_n, we show that the face poset of the "coarse" subdivision of B+(K_n) is dual to the face poset of the associahedron A_n-2, and we give a formula for the number of fine cells within a coarse cell.

@misc{ArdilaKlivansWilliams2004, abstract = { We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M_I, the positive tropical variety associated to I is equal to the fan over B+(M_I). Our main result is that a certain "fine" subdivision of B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph K_n, we show that the face poset of the "coarse" subdivision of B+(K_n) is dual to the face poset of the associahedron A_{n-2}, and we give a formula for the number of fine cells within a coarse cell. }, added-at = {2008-12-23T16:06:17.000+0100}, author = {Ardila, Federico and Klivans, Caroline and Williams, Lauren}, biburl = {https://www.bibsonomy.org/bibtex/206220af442774efbecabdf23496013ba/fwd13}, description = {The Positive Bergman Complex of an Oriented Matroid}, interhash = {a5588ae05a401eaaf4fae95676974984}, intrahash = {06220af442774efbecabdf23496013ba}, keywords = {Ardila Bergman Klivans Las_Vergnas Williams associahedron complex face fan graph lattice matroid orientation poset tropics variety}, note = {cite arxiv:math/0406116 Comment: 15 pages, 8 figures}, timestamp = {2008-12-23T17:31:15.000+0100}, title = {The Positive Bergman Complex of an Oriented Matroid}, url = {http://arxiv.org/abs/math/0406116}, year = 2004 }