Donna Testerman

A construction of certain maximal subgroups of the algebraic groups E6 and F4

Keywords: maximal closed connected subgroup ; exceptional algebraic ; groups ; irreducible embeddings of semisimple algebraic ; groups ; rational modules ; irreducible ; representations ; embeddings of parabolic subgroups Reference CTG-ARTICLE-1989-002doi:10.1016/0021-8693(89)90218-4 Record created on 2008-12-16, modified on 2017-05-12

Irreducible geometric subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed eld K of characteristic p 0 with natural module W . Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor-indecomposable p-restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify all such triples (G;H;V ), where H is a maximal closed disconnected positive-dimensional subgroup of G and H preserves a natural geometric structure on W .

Irreducible almost simple subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

Irreducibility in algebraic groups and regular unipotent elements

We study (connected) reductive subgroups G of a reductive algebraic group H, where G contains a regular unipotent element of H. The main result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H = SL(n, F), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element.

Completely reducible (2)-homomorphisms

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms Φ: SL 2 → G whose image is geometrically G-completely reducible-or G-cr-in the sense of Serre; the description resembles that of irreducible modules given by Steinbergs tensor product theorem. In case K is algebraically closed and G is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of Φ to be geometrically G-cr; this plays an important role in our proof.

A note on composition factors of weyl modules

Keywords: simple algebraic group ; reducibility ; Weyl modules Reference CTG-ARTICLE-1989-001doi:10.1080/00927878908823771 Record created on 2008-12-16, modified on 2017-05-12

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

Abstract Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p ≥ 0 {p\geq 0} , and let u ∈ G {u\in G} be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ ⁢ ( u ) {\phi(u)} contains at most one non-trivial block if and only if G is of type G 2 {G_{2}} , u is a regular unipotent element and dim ⁡ ϕ ≤ 7 {\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].

Completely reducible

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms Φ: SL 2 → G whose image is geometrically G-completely reducible-or G-cr-in the sense of Serre; the description resembles that of irreducible modules given by Steinbergs tensor product theorem. In case K is algebraically closed and G is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of Φ to be geometrically G-cr; this plays an important role in our proof.

\operatorname{SL}(2)

The aim here is to achieve a classification of semisimple algebraic groups in terms of combinatorial data. It is clear from the previous section that the set of roots plays an essential role in the structure of reductive groups. We now formalize this concept. Root systems Let G be a connected reductive group and T ≤ G a maximal torus. Then associated to this we have a finite set of roots Φ ⊂ X ≔ X(T) with the finite Weyl group W acting faithfully on X , preserving Φ (see Proposition 8.4). Recall the group Y = Y(T) of cocharacters of T and the pairing 〈, 〉 : X × Y → ℤ defined in Section 3.2. We identify X and Y with subgroups of E ≔ X ⊗ ℤ ℝ and E ∨ ≔ Y ⊗ ℤ ℝ, respectively, and denote the induced pairing on E × E ∨ also by 〈, 〉. The actions of W on X and on Y may be extended to actions on E and E ∨ . Recall the reflections s α ∈ W introduced in Section 8.4. We first axiomatize the combinatorial properties satisfied by these data. Definition 9.1 A subset Φ of a finite-dimensional real vector space E is called an (abstract) root system in E if the following properties are satisfied: (R1) Φ is finite, 0 ∉ Φ, 〈Φ〉 = E ; (R2) if c ∈ ℝ is such that α, c α ∈ Φ, then c = ±1; (R3) […]

-homomorphisms

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups, and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.