{"user_id":"82981","_id":"66304","language":[{"iso":"eng"}],"date_updated":"2026-07-08T06:19:31Z","year":"2025","status":"public","title":"Towards the tropicalization of reductive groups","author":[{"full_name":"Coles, Desmond","first_name":"Desmond","last_name":"Coles"},{"full_name":"Ulirsch, Martin","last_name":"Ulirsch","first_name":"Martin","id":"114697"}],"type":"preprint","external_id":{"arxiv":["2503.21654"]},"date_created":"2026-07-08T06:19:22Z","abstract":[{"text":"Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of $G$.\r\n  Our first suggestion is the extended affine building associated to $G$. This perspective makes makes use of Berkovich's embedding of the extended affine building into the Berkovich analytic space $G^{\\textrm{an}}$ and expands on work of Mumford by associating a toroidal bordification of $G$ to the choice of stacky fan in the building. We show that the natural retraction onto the building is compatible with the tropicalization map associated to a toroidal bordification.\r\n  Our second suggestion is a Weyl chamber of $G$, a special instance of spherical tropicalization, where we think of $G$ as a spherical $G\\times G$-variety with respect to left-right-multiplication. We show that the spherical tropicalization map may be identified with the toroidal tropicalization map associated to a wonderful compactification of $G$. This map also has a moduli-theoretic interpretation expanding on the compactifications of $G$ as moduli spaces of framed $\\mathbb{G}_m$-equivariant principal bundles on chains of projective lines introduced by Martens and Thaddeus.","lang":"eng"}],"publication":"arXiv:2503.21654","citation":{"mla":"Coles, Desmond, and Martin Ulirsch. “Towards the Tropicalization of Reductive Groups.” <i>ArXiv:2503.21654</i>, 2025.","bibtex":"@article{Coles_Ulirsch_2025, title={Towards the tropicalization of reductive groups}, journal={arXiv:2503.21654}, author={Coles, Desmond and Ulirsch, Martin}, year={2025} }","ama":"Coles D, Ulirsch M. Towards the tropicalization of reductive groups. <i>arXiv:250321654</i>. Published online 2025.","ieee":"D. Coles and M. Ulirsch, “Towards the tropicalization of reductive groups,” <i>arXiv:2503.21654</i>. 2025.","apa":"Coles, D., &#38; Ulirsch, M. (2025). Towards the tropicalization of reductive groups. In <i>arXiv:2503.21654</i>.","chicago":"Coles, Desmond, and Martin Ulirsch. “Towards the Tropicalization of Reductive Groups.” <i>ArXiv:2503.21654</i>, 2025.","short":"D. Coles, M. Ulirsch, ArXiv:2503.21654 (2025)."}}