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arXiv · 2107.00503

Orders and Polytropes: Matrix Algebras from Valuations

Yassine El Maazouz·Marvin Anas Hahn·Gabriele Nebe·Mima Stanojkovski·Bernd Sturmfels
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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.

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Yassine El Maazouz, Marvin Anas Hahn, Gabriele Nebe, Mima Stanojkovski, Bernd Sturmfels. 2021-08-26. Orders and Polytropes: Matrix Algebras from Valuations. https://doi.org/10.1007/s13366-021-00600-4

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