Exceptional Algebraic Groups Research Articles

On the involution fixity of exceptional groups of Lie type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

On Non-Generic Finite Subgroups of Exceptional Algebraic Groups

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups $\text{Alt}_{n}$, $\text{Sym}_{n}$ for $n \ge 10$, do not occur as Lie primitive subgroups of an exceptional algebraic group. A subgroup of $G$ is called $G$-completely reducible if, whenever it lies in a parabolic subgroup of $G$, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-$G$-completely reducible, non-generic simple subgroups. As an intermediate result, for each simply connected $G$ of exceptional type, and each non-generic finite simple group $H$ which embeds into $G/Z(G)$, we derive a set of feasible characters, which restrict the possible composition factors of $V \downarrow S$, whenever $S$ is a subgroup of $G$ with image $H$ in $G/Z(G)$, and $V$ is either the Lie algebra of $G$ or a non-trivial Weyl module for $G$ of least dimension. This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for $n \ge 10$, $\text{Alt}_n$ and $\text{Sym}_n$, as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.

One-class genera of exceptional groups over number fields

We show that exceptional algebraic groups over number fields do not admit one-class genera of parahoric groups, except in the case G 2 . For the group G 2 , we enumerate all such one-class genera for the usual seven-dimensional representation.

Symmetric invariants related to representations of exceptional simple groups

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an Abelian ideal, form a polynomial ring.

On the Motivic Class of the Classifying Stack of $G_2$ and the Spin Groups

We compute the class of the classifying stack of the exceptional algebraic group $G_2$ and of the spin groups $\mathrm{Spin}_7$ and $\mathrm{Spin}_8$ in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks $\mathscr{B}\mathrm{Spin}_n$ can be reduced to the computation of the classes of $\mathscr{B} \Delta_n$, where $\Delta_n\subset \mathrm{Pin}_n$ is the extraspecial $2$-group, the preimage of the diagonal matrices under the projection $\mathrm{Pin}_n\to \mathrm{O}_n$ to the orthogonal group.

Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

We find the algebras of unipotent invariants of the Cox rings for all double flag varieties of complexity and for the exceptional simple algebraic groups; namely, we obtain presentations of these algebras in terms of generators and relations. It is well known that such an algebra is free in the case of complexity . In this paper, we show that, in the case of complexity , the algebra in question is either a free algebra or a hypersurface. A similar result for classical groups was previously obtained by the author. Knowing the structure of this algebra enables one to decompose tensor products of some irreducible representations effectively into irreducible summands and to obtain some branching rules. Bibliography: 10 titles.

On the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups

We solve the question about the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups over the algebraic closures of finite fields of positive characteristic.

On the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups

Решен вопрос о расщепляемости нормализатора максимального тора в исключительных линейных алгебраических группах над алгебраическими замыканиями конечных полей положительной характеристики. Библиография: 8 наименований.

On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilizers

Let$G$be a simple simply connected exceptional algebraic group of type$G_{2}$,$F_{4}$,$E_{6}$or$E_{7}$over an algebraically closed field$k$of characteristic$p>0$with$\mathfrak{g}=\text{Lie}(G)$. For each nilpotent orbit$G\cdot e$of$\mathfrak{g}$, we list the Jordan blocks of the action of$e$on the minimal induced module$V_{\text{min}}$of$\mathfrak{g}$. We also establish when the centralizers$G_{v}$of vectors$v\in V_{\text{min}}$and stabilizers$\text{Stab}_{G}\langle v\rangle$of$1$-spaces$\langle v\rangle \subset V_{\text{min}}$are smooth; that is, when$\dim G_{v}=\dim \mathfrak{g}_{v}$or$\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.

Irreducible [formula omitted] subgroups of exceptional algebraic groups

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible A1 subgroups of exceptional algebraic groups G. Consequences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A1 subgroups are determined by their composition factors on the adjoint module of G.

Simple irreducible subgroups of exceptional algebraic groups

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups G which are connected, closed and simple of rank at least 2. Consequences are given concerning the representations of such subgroups on various G-modules: for example, with one exception, the conjugacy classes of irreducible simple connected subgroups of rank at least 2 are determined by their composition factors on the adjoint module of G.

Descent of affine buildings - I. Large minimal angles

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This first part lays the foundations for our approach and deals with the 'large minimal angle' case.

Non-G-Completely Reducible Subgroups of the Exceptional Algebraic Groups

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H<G with root system X.

Generating sets for coordinate rings of character varieties

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical matrix groups G. This result together with the method of Gröbner basis gives an algorithm for describing character varieties by explicit polynomial equations.Additionally, we describe finite sets of generators of the fields of rational functions on G-character varieties for all exceptional algebraic groups G.

Simple modules of exceptional groups with normal closures of maximal torus orbits

Let G be an exceptional simple algebraic group, and let T be a maximal torus in G. In this paper, for every such G, we find all simple rational G-modules V with the following property: for every vector v ∈ V, the closure of its T-orbit is a normal affine variety. To solve this problem, we use a combinatorial criterion of normality formulated in terms of weights of simple G-modules. This paper continues the works of the second author in which the same problem was solved for classical linear groups.

Irreducible disconnected subgroups of exceptional algebraic groups

This paper studies the irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. In this paper we give a classification of all triples ( G , Y , V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0 , G is a closed disconnected positive dimensional subgroup of Y and V is a nontrivial rational irreducible KY -module on which G acts irreducibly.

A classification of certain finite double coset collections in the exceptional groups

Let G be an exceptional algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using a reduction to finite groups and character theory. Infiniteness is proven using a dimension criterion that involves root systems.

Unipotent classes in maximal subgroups of exceptional algebraic groups

Let G be a simple algebraic group of exceptional type over an algebraically closed field, and M be a maximal closed connected reductive subgroup of G. For each unipotent M-class we determine the G-class in which it lies.

Zero Cycles on a Twisted Cayley Plane

AbstractLet k be a field of characteristic not 2, 3. Let G be an exceptional simple algebraic group over k of type F4, 1E6 or E7 with trivial Tits algebras. Let X be a projective G-homogeneous variety. If G is of type E7, we assume in addition that the respective parabolic subgroup is of type P7. The main result of the paper says that the degree map on the group of zero cycles of X is injective.

Groups of outer type E6 with trivial Tits algebras

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?