Dressians, Tropical Grassmannians, and Their Rays
Sven Herrmann, Michael Joswig, and David Speyer
TL;DR
This paper investigates the combinatorial structure of the Dressian and tropical Grassmannians, focusing on their rays and degenerations, and introduces a new rigidity concept for tropical point configurations.
Contribution
It characterizes the rays of Dressians for arbitrary k and n, introduces a new rigidity concept for tropical point configurations, and computes the entire fan Dr(3,8).
Findings
Characterization of rays of Dr(k,n) for general k and n.
Introduction of a new rigidity concept for tropical configurations.
Complete computation of the fan Dr(3,8).
Abstract
The Dressian Dr(k,n) parametrizes all tropical linear spaces, and it carries a natural fan structure as a subfan of the secondaryfan of the hypersimplex \Delta(k,n). We explore the combinatorics of the rays of Dr(k,n), that is, the most degenerate tropical planes, for arbitrary k and n. This is related to a new rigidity concept for configurations of n-k points in the tropical (k-1)-torus. Additional conditions are given for k=3. On the way, we compute the entire fan Dr(3,8).
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