On the Automorphisms of Infinite Chevalley Groups

1969 ◽  
Vol 21 ◽  
pp. 908-911 ◽  
Author(s):  
J. E. Humphreys
Keyword(s):  
Finite Field ◽  
Chevalley Group ◽  
Chevalley Groups ◽  
Positive Roots ◽  
Simple Chevalley Group

In (8, § 3.2) Steinberg proved the following result.THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1”) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and Ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.

scholarly journals On the characters of the Sylow -subgroups of untwisted Chevalley groups

2016 ◽  
Vol 19 (2) ◽  
pp. 303-359 ◽  
Author(s):  
Frank Himstedt ◽  
Tung Le ◽  
Kay Magaard
Keyword(s):  
Root System ◽  
Chevalley Group ◽  
Positive Root ◽  
Chevalley Groups ◽  
Irreducible Characters ◽  
Single Root ◽  
The Family ◽  
Positive Roots ◽  
Root Character

Let$UY_{n}(q)$be a Sylow$p$-subgroup of an untwisted Chevalley group$Y_{n}(q)$of rank$n$defined over $\mathbb{F}_{q}$where$q$is a power of a prime$p$. We partition the set$\text{Irr}(UY_{n}(q))$of irreducible characters of$UY_{n}(q)$into families indexed by antichains of positive roots of the root system of type$Y_{n}$. We focus our attention on the families of characters of$UY_{n}(q)$which are indexed by antichains of length$1$. Then for each positive root$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.

Weyl ǵroups and finite Chevalley ǵroups

1970 ◽  
Vol 67 (2) ◽  
pp. 269-276 ◽  
Author(s):  
R. W. Carter
Keyword(s):  
Finite Group ◽  
Finite Field ◽  
Lie Algebra ◽  
Weyl Groups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Positive Roots ◽  
Image Position ◽  
A Finite Group ◽  
Fundamental Paper

In his fundamental paper (1) Chevalley showed how to associate with each complex simple Lie algebra L and each field K a group G = L(K) which is (in all but four exceptional cases) simple. If K is a finite field GF(q), G is a finite group of orderwhere l is the rank of L, m is the number of positive roots of L and d is a certain integer determined by L and K. The integers m1, m2,…,m1 are determined by L only and satisfy the condition

scholarly journals Large abelian subgroups of Chevalley groups

1979 ◽  
Vol 27 (1) ◽  
pp. 59-87 ◽  
Author(s):  
Michael J. J. Barry
Keyword(s):  
Finite Field ◽  
Chevalley Group ◽  
Abelian Subgroup ◽  
Maximal Order ◽  
Chevalley Groups ◽  
Abelian Subgroups

AbstractFor any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.

Elements of Order Coxeter Number +1 in Chevalley Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 945-951 ◽  
Author(s):  
Bomshik Chang
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Trivial Solution ◽  
Affirmative Answer ◽  
Chevalley Groups ◽  
General Solutions ◽  
Coxeter Number ◽  
Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

scholarly journals Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

2014 ◽  
Vol 17 (1) ◽  
pp. 109-122 ◽  
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Computer Program ◽  
Explicit Formula ◽  
Chevalley Group ◽  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Sylow Subgroups ◽  
Finite Chevalley Group ◽  
Integer Coefficients ◽  
The Third

AbstractLet$G(q)$be a finite Chevalley group, where$q$is a power of a good prime$p$, and let$U(q)$be a Sylow$p$-subgroup of$G(q)$. Then a generalized version of a conjecture of Higman asserts that the number$k(U(q))$of conjugacy classes in$U(q)$is given by a polynomial in$q$with integer coefficients. In [S. M. Goodwin and G. Röhrle,J. Algebra321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of$k(U(q))$. By implementing it into a computer program using$\mathsf{GAP}$, they were able to calculate$k(U(q))$for$G$of rank at most five, thereby proving that for these cases$k(U(q))$is given by a polynomial in$q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of$k(U(q))$for finite Chevalley groups of rank six and seven, except$E_7$. We observe that$k(U(q))$is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write$k(U(q))$as a polynomial in$q-1$, then the coefficients are non-negative.Under the assumption that$k(U(q))$is a polynomial in$q-1$, we also give an explicit formula for the coefficients of$k(U(q))$of degrees zero, one and two.

On root-involutions and root-subgroups of E6(K) for fields K of characteristic two

2019 ◽  
Vol 18 (01) ◽  
pp. 1950017 ◽  
Author(s):  
S. Aldhafeeri ◽  
M. Bani-Ata
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Algebra ◽  
Chevalley Group ◽  
Chevalley Groups ◽  
Type Formula ◽  
Characteristic Two

The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group [Formula: see text] for fields [Formula: see text] of characteristic two. The approach we follow is elementary and self-contained depends on the notion of [Formula: see text]-sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beit. Algebra Geom. 58 (2017) 529–534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among [Formula: see text]-sets, the last one is on Lie algebra.

scholarly journals THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS

2018 ◽  
Vol 6 ◽  
Author(s):  
OLIVER LORSCHEID
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Tropical Geometry ◽  
Linear Representation ◽  
Unified Approach ◽  
Chevalley Groups ◽  
General Linear Groups ◽  
Base Extension ◽  
Field With One Element ◽  
Intrinsic Definition

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category$\text{Sch}_{{\mathcal{T}}}$ comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for ${\mathcal{G}}$ in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of ${\mathcal{G}}$ defines a model $G$ in $\text{Sch}_{{\mathcal{T}}}$ whose Weyl extension is the Weyl group $W$ of ${\mathcal{G}}$. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

scholarly journals PRESENTATIONS OF AFFINE KAC–MOODY GROUPS

2018 ◽  
Vol 6 ◽  
Author(s):  
INNA CAPDEBOSCQ ◽  
KARINA KIRKINA ◽  
DMITRIY RUMYNIN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Chevalley Groups ◽  
Generators And Relations ◽  
Simply Connected ◽  
Finite Presentations

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 $G(\mathbb{F}_{q}[[t]])$.

scholarly journals Commutator width of Chevalley groups over rings of stable rank 1

2019 ◽  
Vol 22 (1) ◽  
pp. 83-101
Author(s):  
Andrei Smolensky
Keyword(s):  
Chevalley Group ◽  
Stable Rank ◽  
Chevalley Groups ◽  
Commutator Width ◽  
Elementary Chevalley Group

Abstract It is shown that each element of the elementary Chevalley group of rank greater than 2 over a ring of stable rank 1 can be expressed as a product of few commutators.

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