---
_id: '66304'
abstract:
- lang: eng
  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."
author:
- first_name: Desmond
  full_name: Coles, Desmond
  last_name: Coles
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Coles D, Ulirsch M. Towards the tropicalization of reductive groups. <i>arXiv:250321654</i>.
    Published online 2025.
  apa: Coles, D., &#38; Ulirsch, M. (2025). Towards the tropicalization of reductive
    groups. In <i>arXiv:2503.21654</i>.
  bibtex: '@article{Coles_Ulirsch_2025, title={Towards the tropicalization of reductive
    groups}, journal={arXiv:2503.21654}, author={Coles, Desmond and Ulirsch, Martin},
    year={2025} }'
  chicago: Coles, Desmond, and Martin Ulirsch. “Towards the Tropicalization of Reductive
    Groups.” <i>ArXiv:2503.21654</i>, 2025.
  ieee: D. Coles and M. Ulirsch, “Towards the tropicalization of reductive groups,”
    <i>arXiv:2503.21654</i>. 2025.
  mla: 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).
date_created: 2026-07-08T06:19:22Z
date_updated: 2026-07-08T06:19:31Z
external_id:
  arxiv:
  - '2503.21654'
language:
- iso: eng
publication: arXiv:2503.21654
status: public
title: Towards the tropicalization of reductive groups
type: preprint
user_id: '82981'
year: '2025'
...