Orders and polytropes: matrix algebras from valuations

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, vol. 63, pp. 515–531

Abstract

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.

Authors 5

1. Yassine El Maazouz

   University of California, Berkeley

   Affiliation as printed

   UC Berkeley, Berkeley, USA

   UC Berkeley., Berkeley, USA#TAB#

2. Marvin Anas Hahn

   Max Planck Institute for Mathematics in the Sciences

   Affiliation as printed

   MPI-MiS Leipzig, Leipzig, Germany

3. Gabriele Nebe Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen, Aachen, Germany

   RWTH Aachen, Aachen, , Germany

4. Mima Stanojkovski corresponding Aachen

   RWTH Aachen University · Max Planck Institute for Mathematics in the Sciences

   Affiliation as printed

   MPI-MiS Leipzig, Leipzig, Germany

   RWTH Aachen, Aachen, Germany

5. Bernd Sturmfels

   University of California, Berkeley · Max Planck Institute for Mathematics in the Sciences

   Affiliation as printed

   MPI-MiS Leipzig, Leipzig, Germany

   UC Berkeley, Berkeley, USA