Orders and polytropes: matrix algebras from valuations

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 graduated orders introduced by Plesken and Zassenhaus. These are classified polytrope region. advance ideal theory introducing their class polytropes. This article emphasizes examples computations. It offers first steps in geometric combinatorics endomorphism rings configurations affine buildings.