Isotropical Linear Spaces and Valuated Delta-Matroids

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.