Abstract
We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of \(\hbox {SL}_d(K)\), for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical polytope.
Similar content being viewed by others
References
Abramenko, P., Brown, K.S.: Buildings. Theory and Applications. Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)
Cartwright, D., Häbich, M., Sturmfels, B., Werner, A.: Mustafin varieties. Selecta Math. (N.S.) 17(4), 757–793 (2011)
Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)
Dress, A., Terhalle, W.: The tree of life and other affine buildings. In: Proceedings of the International Congress of Mathematicians (Berlin 1998). Doc. Math. Extra Volume ICM Berlin 1998, part III, 565–574 (1998)
Faltings, G.: Toroidal resolutions for some matrix singularities. In: Moduli of Abelian Varieties (Texel Island 1999). Progr. Math., vol. 195, pp. 157–184. Birkhäuser, Basel (2001)
Fontaine, B., Kamnitzer, J., Kuperberg, G.: Buildings, spiders, and geometric Satake. Compos. Math. 149(11), 1871–1912 (2013)
Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Polytopes – Combinatorics and Computation (Oberwolfach 1997). DMV Sem., vol. 29, pp. 43–73. Birkhäuser, Basel (2000)
Hahn, M.A., Li, B.: Mustafin varieties, moduli spaces and tropical geometry (2017). arXiv:1707.01216
Hitzelberger, P.: Non-discrete affine buildings and convexity. Adv. Math. 227(1), 210–244 (2011)
Joswig, M.: Essentials of Tropical Combinatorics (book in preparation). http://page.math.tu-berlin.de/~joswig/etc/index.html
Joswig, M., Sturmfels, B., Yu, J.: Affine buildings and tropical convexity. Albanian J. Math. 1(4), 187–211 (2007)
Kapovich, M., Leeb, B., Millson, J.J.: The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra. Mem. Amer. Math. Soc. 192, # 896. American Mathematical Society, Providence (2008)
Keel, S., Tevelev, J.: Geometry of Chow quotients of Grassmannians. Duke Math. J. 134(2), 259–311 (2006)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)
Pappas, G., Rapoport, M., Smithling, B.: Local models of Shimura varieties, I. Geometry and combinatorics. In: Handbook of Moduli, vol. III. Adv. Lect. Math. (ALM), vol. 26, pp. 135–217. Int. Press, Somerville (2013)
Rincón, F.: Local tropical linear spaces. Discrete Comput. Geom. 50(3), 700–713 (2013)
Supasiti, T.: Serre’s Tree for \(SL_2({\mathbb{F}})\). Honors thesis, University of Melbourne (2008)
Acknowledgements
The author would like to thank the Max Planck Institute for Mathematics in the Sciences for its hospitality while working on this project. He was partially supported by a National Science Foundation Graduate Research Fellowship. The author is grateful to Jacinta Torres, Lara Bossinger, and Madeline Brandt for reading early drafts of this manuscript. He would also like to thank Michael Joswig and Lars Kastner for generous help on writing a polymake extension, Petra Schwer for suggesting a geometric interpretation of Lemma 3.8, and Bernd Sturmfels for much valuable discussion and feedback. Finally, he is grateful to the anonymous reviewers for their thoughtful feedback and many helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, L. Computing Min-Convex Hulls in the Affine Building of \(\hbox {SL}_d\). Discrete Comput Geom 65, 1314–1336 (2021). https://doi.org/10.1007/s00454-020-00223-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-020-00223-x