Orders and Polytropes: Matrix Algebras from Valuations

TL;DR

This paper explores the application of tropical geometry to matrix algebras over valued fields, focusing on polytropes and their role in classifying graduated orders and their ideal class structures.

Contribution

It introduces the concept of ideal class polytropes, advancing the geometric combinatorics of endomorphism rings in affine buildings.

Findings

Classification of graduated orders via polytrope regions

Introduction of ideal class polytropes for graduated orders

Examples and computations illustrating the theory

Abstract

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.