Related papers: Presentation of affine Kac-Moody groups over rings
For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was…
Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups…
To any generalised Cartan matrix (GCM) $A$ and any ring $R$, Tits associated a Kac-Moody group $\mathfrak{G}_A(R)$ defined by a presentation \`a la Steinberg. For a domain $R$ with field of fractions $\mathbb{K}$, we explore the question of…
For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when…
The main result of the present paper is bounded elementary generation of the Steinberg groups $\mathrm{St}(\Phi,R)$ for simply laced root systems $\Phi$ of rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded…
Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\leq 2$, where for nullity 2 there is only one known such presentation. We…
Let $\mathcal{G}$ be a Kac-Moody group functor in the sense of Tits, with associated Coxeter system $(W,S)$. For any field $F$, the group $\mathcal{G}(F)$ is finitely generated iff $F$ is finite. We are interested in the question when $G =…
High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of…
A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that roughly speaking these groups are determined by their local structure. This result was later extended to Kac-Moody groups by P.~Abramenko and B.~M\"uhlherr.…
Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete…
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…
An analog of the Tits building is defined and studied for commutative rings. We prove a Solomon-Tits theorem when $R$ either satisfies a stable range condition, or is the ring of $S$-integers of a global field. We then define an analog of…
For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when…
We prove that every finite direct product of crystallographic groups arising from an irreducible root system (in the sense of Lie theory) is profinitely rigid (equiv. first-order rigid). This is a generalization of recent proofs of…
We introduce and study the class of groups graded by root systems. We prove that if {\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\Phi}, then under certain natural conditions on the grading,…
How many generators and relations does ${\mathrm SL}_n({\mathbb F}_q[t, t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac-Moody groups over…
Tits has defined Kac-Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require…
Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a…
Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most…
We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…