scholarly journals On the Jordan block structure of images of some unipotent elements in modular irreducible representations of the classical algebraic groups

2004 ◽  
Vol 273 (2) ◽  
pp. 586-600 ◽  
Author(s):  
A.A. Osinovskaya ◽  
I.D. Suprunenko
Keyword(s):  
Block Structure ◽  
Algebraic Groups ◽  
Jordan Block ◽  
Irreducible Representations

scholarly journals Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

2018 ◽  
Vol 21 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Donna M. Testerman ◽  
Alexandre E. Zalesski
Keyword(s):  
Algebraic Groups ◽  
Jordan Block ◽  
Linear Algebraic Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Jordan Normal Form ◽  
Unipotent Element ◽  
Content Type ◽  
Simply Connected ◽  
Graphic Content

AbstractLetGbe a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed fieldFof characteristic{p\geq 0}, and let{u\in G}be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation ofG. Then the Jordan normal form of{\phi(u)}contains at most one non-trivial block if and only ifGis of type{G_{2}},uis a regular unipotent element and{\dim\phi\leq 7}. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

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 HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

2020 ◽  
pp. 1-24
Author(s):  
MATTHEW WESTAWAY
Keyword(s):  
Tensor Product ◽  
Algebraic Groups ◽  
Preceding Paper ◽  
Irreducible Representations ◽  
Product Theorem ◽  
Enveloping Algebras ◽  
Frobenius Kernel ◽  
Frobenius Kernels ◽  
Lie Algebra Representations ◽  
Semisimple Algebraic Groups

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

scholarly journals Representations of Algebraic Groups

1963 ◽  
Vol 22 ◽  
pp. 33-56 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Vector Spaces ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Field ◽  
Infinite Degree ◽  
Semisimple Algebraic Groups

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

scholarly journals Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

Sign in / Sign up

Export Citation Format

Share Document