On the Normalizer of a Unipotent Root Subgroup in a Chevalley Group

2018 ◽  
Vol 232 (5) ◽  
pp. 717-720
Author(s):  
V. V. Nesterov
Keyword(s):  
Chevalley Group ◽  
Root Subgroup

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.

scholarly journals Lefschetz Operators, Hodge–Riemann Forms, and Representations

2020 ◽  
Author(s):  
Peter Fiebig
Keyword(s):  
Root System ◽  
Bilinear Form ◽  
Chevalley Group ◽  
Simple Root ◽  
Complex Geometry ◽  
Linear Operators ◽  
Vector Spaces ◽  
Tilting Module ◽  
Weight Lattice ◽  
Simply Connected

Abstract For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

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 Quantifying residual finiteness of arithmetic groups

2012 ◽  
Vol 148 (3) ◽  
pp. 907-920 ◽  
Author(s):  
Khalid Bou-Rabee ◽  
Tasho Kaletha
Keyword(s):  
Chevalley Group ◽  
Arithmetic Groups ◽  
Residual Finiteness ◽  
Arithmetic Subgroup ◽  
Higher Rank ◽  
Finite Quotients

AbstractThe normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

scholarly journals The Chevalley group G2(2) of order 12096 and the octonionic root system of E7

2007 ◽  
Vol 422 (2-3) ◽  
pp. 808-823 ◽  
Author(s):  
Mehmet Koca ◽  
Ramazan Koç ◽  
Nazife Ö. Koca
Keyword(s):  
Root System ◽  
Chevalley Group

scholarly journals Identification of matrix generators of a Chevalley group

2007 ◽  
Vol 309 (2) ◽  
pp. 484-496 ◽  
Author(s):  
Alexander J.E. Ryba
Keyword(s):  
Chevalley Group

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.

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