Stiefel tropical linear spaces. (English) Zbl 1321.15044
Summary: The tropical Stiefel map associates to a tropical matrix \(A\) its tropical Plücker vector of maximal minors, and thus a tropical linear space \(L(A)\). We call the \(L(A)\)s obtained in this way Stiefel tropical linear spaces. We prove that they are dual to certain matroid subdivisions of polytopes of transversal matroids, and we relate their combinatorics to a canonically associated tropical hyperplane arrangement. We also explore a broad connection with the secondary fan of the Newton polytope of the product of all maximal minors of a matrix. In addition, we investigate the natural parametrization of \(L(A)\) arising from the tropical linear map defined by \(A\).
MSC:
| 15A80 | Max-plus and related algebras |
| 14T05 | Tropical geometry (MSC2010) |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
Keywords:
tropical linear spaces; tropical hyperplane arrangements; matchings; matroid polytope subdivisions; transversal matroids; mixed subdivisions; Stiefel mapReferences:
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