Papers › Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians
Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians
Dominik Bendle, Janko Boehm, Yue Ren, Benjamin Schröter
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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr₀(3,8), and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr^+(3,8) and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$.
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