Isotropical Linear Spaces and Valuated Delta-Matroids
Felipe Rinc\'on

TL;DR
This paper develops a tropical geometric framework for isotropic subspaces and delta-matroids, extending tropical linear space theory to Coxeter matroids of type D, with combinatorial characterizations and basis considerations.
Contribution
It introduces tropical Wick vectors and linear spaces, characterizes them via subdivisions of Delta-matroid polytopes, and explores their basis properties, generalizing tropical linear space results.
Findings
Characterization of tropical Wick vectors through subdivisions of Delta-matroid polytopes
Development of a combinatorial theory for tropical isotropic linear spaces
Analysis of the Wick relations as a tropical basis
Abstract
The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.
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