Chevalley Groups Research Articles

On Bruhat-Tits theory over a higher dimensional base

Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies certain tameness assumptions \eqref{tameness}. Let $\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ and set $K_{_n} := \text{Fract}~\cO_{_n}$. Let $G$ be an almost-simple, simply-connected affine Chevalley group scheme with a maximal torus $T$ and a Borel subgroup $B$. Given a $n$-tuple ${\bf f} = (f_{_1}, \ldots, f_{_n})$ of concave functions on the root system of $G$ as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we define {\it {\tt n}-bounded subgroups ${\tt P}_{_{\bf f}}\subset G(K_{_n})$} as a direct generalization of Bruhat-Tits groups for the case $n=1$. We show that these groups are {\it schematic}, i.e. they are valued points of smooth {\em quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it adapted to the divisor with normal crossing $z_1 \cdots z_n =0$} in the sense that the restriction to the generic point of the divisor $z_i=0$ is given by $f_i$ (resp. sums of concave functions given by points of the apartment). This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In \S\ref{mixedstuff}, under suitable assumptions on $k$ \S \ref{charassum}, we extend all these results for a $n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the root system of $G$ replacing $\mathcal O_{_n}$ by ${\cO} \llbracket x_{_1},\cdots,x_{_n} \rrbracket$ where $\cO$ is a complete discrete valuation ring with a perfect residue field $k$ of characteristic $p$. In the last part of the paper, we give applications in char zero to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of {\tt 2-parahoric} group schemes on minimal resolutions of surface singularities that arose in \cite{balaproc}.

On pairs of root groups and reflections associated to the Chevalley group E 6(K), in characteristic two

This paper is concerned with computing the number of orbits under the action of the algebraic group E = E 6 ( K ) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL ( A ) has been fixed right at the onset.

Fixed point subgroups of a supertight automorphism

Let G be an infinite simple group of finite Morley rank and α a supertight automorphism of G so that the fixed point subgroup P n : = C G ( α n ) is pseudofinite for all n ∈ N ∖ { 0 } . It is know (using CFSG) that the socle S n : = Soc ( P n ) is a (twisted) Chevalley group over a pseudofinite field. We prove that there is r ∈ N ∖ { 0 } so that for each n we have [ P n : S n ] < r and that there is no m ∈ N ∖ { 0 } so that for each n the sizes of the Sylow 2-subgroups of S n are bounded by m. We also note that in the recent identification result of G under the assumption pr 2 ( G ) = 1 , the use of CFSG is not needed.

Automorphisms of a Chevalley Group of Type G2 Over a Commutative Ring R with 1/3 Generated by the Invertible Elements and 2R

Automorphisms of a Chevalley Group of Type G2 Over a Commutative Ring R with 1/3 Generated by the Invertible Elements and 2R

On the top-dimensional cohomology of arithmetic Chevalley groups

Let K \mathbb {K} be a number field with ring of integers O \mathfrak {O} and let G \mathcal {G} be a Chevalley group scheme not of type E 8 \mathtt {E}_8 , F 4 \mathtt {F}_4 or G 2 \mathtt {G}_2 . We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H v c d ( G ( O ) ; Q ) = 0 H^{{vcd}}(\mathcal {G}(\mathfrak {O}); \mathbb {Q}) = 0 if O \mathfrak {O} is Euclidean.

Centrality of K2 for Chevalley groups: a pro-group approach

Centrality of K2 for Chevalley groups: a pro-group approach

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1.We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.

Galois cohomology and profinitely solitary Chevalley groups

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.

Certain linear isomorphisms for hyperalgebras relative to a Chevalley group

Let [Formula: see text] be a simply connected and simple algebraic group defined and split over a finite prime field [Formula: see text] of [Formula: see text] elements. In this paper, using an [Formula: see text]-linear map splitting Frobenius endomorphism on a hyperalgebra relative to [Formula: see text], we obtain some [Formula: see text]-linear isomorphisms induced by multiplication in the hyperalgebra.

Simple Hurwitz groups and eta invariant

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely $84(g - 1)$ automorphisms, where $g$ is the genus of the surface. Our starting point is a result on the smallest Hurwitz group $\mathrm{PSL}(2,\mathbb{F}_{7})$ which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the $\eta$-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.

A Schur ring approach to supercharacters of groups associated with finite radical rings

We consider the central Schur ring associated with the standard supercharacters of the adjoint group G ( A ) of a finite radical ring A , and define supercharacters of the subgroup C G ( A ) ( σ ) consisting of elements fixed by an involution of G ( A ) naturally defined by an (anti-)involution of A . In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups.

Automorphisms of Chevalley groups over commutative rings

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.

ON MODEL-THEORETIC CONNECTED GROUPS

AbstractWe introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

Bounded reduction for Chevalley groups of types $$E_6$$ and $$E_7$$

Bounded reduction for Chevalley groups of types $$E_6$$ and $$E_7$$

Finite index subgroups in Chevalley groups are bounded: An addendum to “On Bi-Invariant Word Metrics”

We prove that finite index subgroups in [Formula: see text]-arithmetic Chevalley groups are bounded.

On construction of maximal parabolic centralizers of root elements and the Weyl group of type F4 in E6(K) for fields K of characteristic 2

The purpose of this paper is to give an elementary and explicit construction of maximal parabolic centralizers of root elements in the Chevalley group E6(K) and to show that the centralizer of a Seigel involution in the Weyl group W of type E6(K) is the Weyl group of type F4, or equivalently it is the stabilizer of a totaly singular line L in W, using properties of the generalized quadrangle (Ω,L) of type O6−(2).The construction here of the maximal parabolic centralizers of root elements in E6K for fields K of characteristic 2, is somewhat novel, beginning as it does, with root bases of Levi-type and root bases of unipotent type, giving an explicit construction of the group generated by these two types.

Regular bi-interpretability of Chevalley groups over local rings

Regular bi-interpretability of Chevalley groups over local rings

The character table of the finite Chevalley group F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} for q a power of 2

Let q be a prime power and F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} be the Chevalley group of type F4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4$$\\end{document} over a finite field with q elements. Marcelo and Shinoda (Tokyo J Math 18:303–340, 1995) determined the values of the unipotent characters of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of 2, we explain how to construct the complete character table of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document}.

Bounded generation and commutator width of Chevalley groups: function case

Bounded generation and commutator width of Chevalley groups: function case

Conjugation-invariant norms on SL2(R) for rings of S-algebraic integers with infinitely many units

This paper studies conjugation-invariant norms on for R a ring of S-algebraic integers with infinitely many units in a number field. We will show that barring some differences in the proofs the properties of these norms strongly resemble related properties of conjugation-invariant norms on arithmetic Chevalley groups of higher rank. Specifically, we prove two properties: First, we show that the diameter of each word norm on defined by finitely many conjugacy classes has an upper bound linear in the number of conjugacy-classes. Second, we show that all non-discrete conjugation-invariant norms on have a profinite norm-completion. Both of these properties of conjugation-invariant norms are already known for higher rank arithmetic Chevalley groups.