Abstract
We study the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. The set of dissimilarity vectors of weighted trees is contained in the tropical Grassmannian, so we describe here the tropical linear space of a dissimilarity vector and its associated family of matroids. This gives a family of complete flags of tropical linear spaces, where each flag is described by a weighted tree. Nous étudions les propriétés combinatoires des arbres pondérés avec le formalisme de la géométrie tropicale et des espaces linéaires tropicaux. L'ensemble de vecteurs de dissimilarité des arbres pondérés est contenu dans la grassmannienne tropicale, donc nous décrivons ici l'espace linéaire tropical d'un vecteur de dissimilarité et sa famille de matroïdes associée. Cela permet d'obtenir une famille de drapeaux complets d'espaces linéaires tropicaux, où chaque drapeau est décrit par un arbre pondéré.
Highlights
1.1 Basic DefinitionsFor every finite set E, defineE m to be the collection of all subsets of E of size m.We adopt the convention that [n] = {1, 2, . . . , n}, and we further define [0] = ∅
Theorem 1.5 (Pachter and Sturmfels [PS05]) The set of 2-dissimilarity vectors of trees is equal to the tropical Grassmannian G2,n
2 This result is in the direction of solving the more general problem: Problem 1.7 Characterize the set of m-dissimilarity vectors of trees, m ≥ 2
Summary
E m to be the collection of all subsets of E of size m. Let dA be the total weight of the subtree of T spanned by the set of leaves A. We will consider instead a tree U with n leaves labeled by the set [n] with n ≥ 1, with a weight function E(U ) −→ω R≥0. The function ω will again extend to a function from the subtrees of U to the nonnegative reals in the natural way, so that we may speak of total weights of subtrees of U and define the m-dissimilarity vector of U in a completely analogous way. We will assume that U is trivalent, rooted, -equidistant (i.e. the total weight of the minimal path from every leaf to the root is a constant ), and that ω induces a metric on L(U ) = [n]. Along with U we consider the poset (V(U ), ≤TO), called the tree-order of U , by which u ≤TO v if the minimal path from the root of U to v contains u
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