Subgroups of Chevalley groups containing a maximal torus

1993 ◽  
pp. 59-100 ◽  
Author(s):  
N. A. Vavilov
Keyword(s):  
Maximal Torus ◽  
Chevalley Groups

scholarly journals On the adjoint quotient of Chevalley groups over arbitrary base schemes

2010 ◽  
Vol 9 (4) ◽  
pp. 673-704 ◽  
Author(s):  
Pierre-Emmanuel Chaput ◽  
Matthieu Romagny
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Maximal Torus ◽  
Adjoint Action ◽  
Group Scheme ◽  
Base Change ◽  
Chevalley Groups ◽  
Base Scheme ◽  
Arbitrary Base

AbstractFor a split semisimple Chevalley group scheme G with Lie algebra $\mathfrak{g}$ over an arbitrary base scheme S, we consider the quotient of $\mathfrak{g} by the adjoint action of G. We study in detail the structure of $\mathfrak{g} over S. Given a maximal torus T with Lie algebra $\mathfrak{t}$ and associated Weyl group W, we show that the Chevalley morphism π : $\mathfrak{t}$/W → $\mathfrak{g}/G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient $\mathfrak{g} → \mathfrak{g}//G commutes, or does not commute, with base change on S.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals Orthogonal groups containing a given maximal torus

2003 ◽  
Vol 266 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Rosali Brusamarello ◽  
Pascale Chuard-Koulmann ◽  
Jorge Morales
Keyword(s):  
Maximal Torus ◽  
Orthogonal Groups

scholarly journals 1-Cohomology of Chevalley groups

1989 ◽  
Vol 127 (2) ◽  
pp. 353-372 ◽  
Author(s):  
Helmut Völklein
Keyword(s):  
Chevalley 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.

scholarly journals Maximal subgroups and automorphisms of Chevalley groups

1977 ◽  
Vol 71 (2) ◽  
pp. 365-403 ◽  
Author(s):  
N. Burgoyne ◽  
Robert Griess ◽  
Richard Lyons
Keyword(s):  
Maximal Subgroups ◽  
Chevalley Groups
Sign in / Sign up

Export Citation Format

Share Document