The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces
Kevin Kuehn, Arne Kuhrs

TL;DR
This paper introduces a signed analogue of the Goldman-Iwahori space over real closed fields, linking it to real tropical geometry and providing a combinatorial interpretation via oriented matroids.
Contribution
It defines the signed Goldman-Iwahori space, establishes its relation to real tropical linear spaces, and connects it to realizable oriented matroids and real Bergman fans.
Findings
The signed Goldman-Iwahori space is the limit of all real tropicalized linear embeddings.
It provides a combinatorial interpretation via realizable oriented matroids.
Explicit description of the space over the real numbers and its relation to real Bergman fans.
Abstract
The Goldman-Iwahori space of seminorms on a finite-dimensional vector space over a non-Archimedean field is a non-Archimedean analogue of a symmetric space. If, in addition, is real closed, we define a signed analogue of the Goldman-Iwahori space consisting of signed seminorms. This new space can be seen as the linear algebraic version of the real analytification of projective space over . We study this space with methods from real tropical geometry by constructing natural real tropicalization maps from the signed Goldman-Iwahori space to all real tropicalized linear spaces. We prove that this space is the limit of all real tropicalized linear embeddings. We give a combinatorial interpretation of this result by showing that the signed Goldman-Iwahori space is the real tropical linear space associated to the universal realizable oriented matroid. In the constant coefficient case…
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Taxonomy
TopicsAdvanced Topics in Algebra · Holomorphic and Operator Theory · Mathematics and Applications
