Orders and polytropes: matrix algebras from valuations

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 or...

Full description

Saved in:
Bibliographic Details
Published in:Beiträge zur Algebra und Geometrie 2022-09, Vol.63 (3), p.515-531
Main Authors: El Maazouz, Yassine, Hahn, Marvin Anas, Nebe, Gabriele, Stanojkovski, Mima, Sturmfels, Bernd
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Combinatorial analysis
Convex and Discrete Geometry
Convexity
Geometry
Mathematics
Mathematics and Statistics
Original Paper
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0138-4821
2191-0383
DOI:10.1007/s13366-021-00600-4