---
_id: '66316'
abstract:
- lang: eng
  text: Using the notion of a root datum of a reductive group $G$ we propose a tropical
    analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$,
    i.e. the case of vector bundles. Here we give a characterization of vector bundles
    in terms of multidivisors and use this description to prove analogues of the Weil--Riemann--Roch
    theorem and the Narasimhan--Seshadri correspondence. We proceed by studying the
    process of tropicalization. In particular, we show that the non-Archimedean skeleton
    of the moduli space of semistable vector bundles on a Tate curve is isomorphic
    to a certain component of the moduli space of semistable tropical vector bundles
    on its dual metric graph.
author:
- first_name: Andreas
  full_name: Gross, Andreas
  last_name: Gross
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
- first_name: Dmitry
  full_name: Zakharov, Dmitry
  last_name: Zakharov
citation:
  ama: 'Gross A, Ulirsch M, Zakharov D. Principal bundles on metric graphs: the $\mathrm{GL}_n$
    case. <i>arXiv:220610219</i>. Published online 2022. doi:<a href="https://doi.org/10.1016/j.aim.2022.108775">10.1016/j.aim.2022.108775</a>'
  apa: 'Gross, A., Ulirsch, M., &#38; Zakharov, D. (2022). Principal bundles on metric
    graphs: the $\mathrm{GL}_n$ case. <i>ArXiv:2206.10219</i>. <a href="https://doi.org/10.1016/j.aim.2022.108775">https://doi.org/10.1016/j.aim.2022.108775</a>'
  bibtex: '@article{Gross_Ulirsch_Zakharov_2022, title={Principal bundles on metric
    graphs: the $\mathrm{GL}_n$ case}, DOI={<a href="https://doi.org/10.1016/j.aim.2022.108775">10.1016/j.aim.2022.108775</a>},
    journal={arXiv:2206.10219}, author={Gross, Andreas and Ulirsch, Martin and Zakharov,
    Dmitry}, year={2022} }'
  chicago: 'Gross, Andreas, Martin Ulirsch, and Dmitry Zakharov. “Principal Bundles
    on Metric Graphs: The $\mathrm{GL}_n$ Case.” <i>ArXiv:2206.10219</i>, 2022. <a
    href="https://doi.org/10.1016/j.aim.2022.108775">https://doi.org/10.1016/j.aim.2022.108775</a>.'
  ieee: 'A. Gross, M. Ulirsch, and D. Zakharov, “Principal bundles on metric graphs:
    the $\mathrm{GL}_n$ case,” <i>arXiv:2206.10219</i>, 2022, doi: <a href="https://doi.org/10.1016/j.aim.2022.108775">10.1016/j.aim.2022.108775</a>.'
  mla: 'Gross, Andreas, et al. “Principal Bundles on Metric Graphs: The $\mathrm{GL}_n$
    Case.” <i>ArXiv:2206.10219</i>, 2022, doi:<a href="https://doi.org/10.1016/j.aim.2022.108775">10.1016/j.aim.2022.108775</a>.'
  short: A. Gross, M. Ulirsch, D. Zakharov, ArXiv:2206.10219 (2022).
date_created: 2026-07-08T06:42:43Z
date_updated: 2026-07-08T09:35:05Z
doi: 10.1016/j.aim.2022.108775
external_id:
  arxiv:
  - '2206.10219'
language:
- iso: eng
publication: arXiv:2206.10219
status: public
title: 'Principal bundles on metric graphs: the $\mathrm{GL}_n$ case'
type: journal_article
user_id: '82981'
year: '2022'
...