A b s t r ac t . We study the fan structure of Dressians Dr (d, n) and local Dressians Dr(M) for a given matroid M. In particular we show that the fan structure on Dr(M) given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope P (M). As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse. arXiv:1809.08965v1 [math.CO] 24 Sep 2018 Cambridge University Press, Cambridge, 1986.I n s t i t u t f ü r M at h e m at i k , F U B e r l i n , A r n i m a l l e e 2 , 1 4 1 9 5 B e r l i n , G e r m a n y , E -m a i l : o l a rt e @ z e dat . f u -b e r l i n . d e I n s t i t u t f ü r M at h e m at i k , T U B e r l i n , S t r . d e s 1 7 . J u n i 1 3 6 , 1 0 6 2 3 B e r l i n , G e r m a n y , E -m a i l : pa n i z z u t @ m at h . t u -b e r l i n . d e D e pa rt m e n t o f M at h e m at i c a l S c i e n c e s , B i n g h a m t o n U n i v e r s i t y , B i n g h a m t o n , N Y 1 3 9 0 2 , U S A , E -m a i l : s c h ro e t e r @ m at h . b i n g h a m t o n . e d u

We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a valuated generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian.

Given 𝑑 row vectors of 𝑛 tropical numbers, 𝑑 < 𝑛, the tropical Stiefel map constructs a version of their row space, whose Plücker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint of matroid theory, the tropical Stiefel map defines a generalization of transversal matroids in the valuated context, and our results are the valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on the set of all set families that present a given transversal matroid. We show that a connected valuated matroid is transversal if and only if all its connected initial matroids are transversal. The duals of our results describe complete stable intersections of tropical linear spaces via valuated strict gammoids.

We study valuated matroids, their tropical incidence relations, flag matroids, and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra. Further, we get a characterization of those subdivisions arising from positive valuated flag matroids.

In 1974 Chvátal conjectured that no intersecting family F in a downset can be larger than the largest star. In the same year Kleitman and Magnanti proved the conjecture when F is contained in the union of two stars, and Sterboul when rank(F ) ≤ 3. We give short self-contained proofs of these two statements.