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PRESENTATIONS OF AFFINE KAC–MOODY GROUPS

Part of: Special aspects of infinite or finite groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  26 October 2018

INNA CAPDEBOSCQ ,
KARINA KIRKINA  and
DMITRIY RUMYNIN
INNA CAPDEBOSCQ
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK; I.Capdeboscq@warwick.ac.uk, kazina3@yahoo.com
KARINA KIRKINA
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK; I.Capdeboscq@warwick.ac.uk, kazina3@yahoo.com
DMITRIY RUMYNIN
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK; I.Capdeboscq@warwick.ac.uk, kazina3@yahoo.com Associated member of Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Russia; D.Rumynin@warwick.ac.uk

Abstract

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How many generators and relations does $\text{SL}\,_{n}(\mathbb{F}_{q}[t,t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac–Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac–Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G(\mathbb{F}_{q}[t,t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G(\mathbb{F}_{q}[[t]])$.

MSC classification

Primary: 20G44: Kac-Moody groups
Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e21
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

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