On the modality of parabolic subgroups of linear algebraic groups

1999 ◽  
Vol 98 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Gerhard Röhrle
Keyword(s):  
Algebraic Groups ◽  
Parabolic Subgroups ◽  
Linear Algebraic Groups

A Semigroup Approach to Linear Algebraic Groups III. Buildings

1986 ◽  
Vol 38 (3) ◽  
pp. 751-768 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Algebraic Groups ◽  
Usual Sense ◽  
Closed Field ◽  
Parabolic Subgroups ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Semigroup Approach ◽  
Image Position

Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , thenare opposite parabolic subgroups of G in the usual sense [1], [28]. However the mapdoes not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

scholarly journals Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

2001 ◽  
Vol 4 ◽  
pp. 135-169 ◽  
Author(s):  
Frank Lübeck
Keyword(s):  
Algebraic Groups ◽  
Small Degree ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Linear Algebraic Groups ◽  
Large Rank ◽  
Projective Representations ◽  
Computer Calculations ◽  
Simply Connected

AbstractThe author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

scholarly journals Closedness of some subgroups in linear algebraic groups

1962 ◽  
Vol 14 (3) ◽  
pp. 272-275
Author(s):  
T. MIYATA ◽  
T. ODA ◽  
K. OTSUKA
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups

scholarly journals Overgroups of regular unipotent elements in simple algebraic groups

2021 ◽  
Vol 8 (25) ◽  
pp. 788-822
Author(s):  
Gunter Malle ◽  
Donna Testerman
Keyword(s):  
Algebraic Groups ◽  
Parabolic Subgroups ◽  
Connected Component ◽  
Unipotent Element

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

scholarly journals Arithmetic purity of strong approximation for semi-simple simply connected groups

2020 ◽  
Vol 156 (12) ◽  
pp. 2628-2649
Author(s):  
Yang Cao ◽  
Zhizhong Huang
Keyword(s):  
Quadratic Form ◽  
Strong Approximation ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Spin Group ◽  
Linear Algebraic Groups ◽  
Simply Connected ◽  
Quadratic Hypersurfaces ◽  
Almost Prime

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

scholarly journals Analytic Dirac approximation for real linear algebraic groups

2010 ◽  
Vol 351 (2) ◽  
pp. 403-410 ◽  
Author(s):  
Christoph Lienau
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups ◽  
Real Linear

Endomorphisms of linear algebraic groups

1968 ◽  
Vol 0 (80) ◽  
pp. 0-0 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups
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