Algebraic Group Of Type Research Articles

A Geometric Characterization of the Hjelmslev–Moufang Planes

AbstractHjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces.

More on rank-2 tori and its cohomological applications

Let k be a field of characteristic different from 2 and 3. In this paper, we study the maximal k-tori T in the connected simple algebraic groups of type A 2 and G 2. We refer to these tori as unitary tori. We study the cohomology and deduce how the presence of maximal k-tori with effects the structure of these groups.

On the Saturation Conjecture for Spin(2n)

In this article we examine the saturation conjecture on decompositions of tensor products of irreducible representations for complex semisimple algebraic groups of type D (the even spin groups: Spin for an integer), extending work done by Kumar–Kapovich–Millson on Spin(8). Our main theorem asserts that the saturation conjecture holds for Spin(10) and Spin(12): for all triples of dominants weights such that is in the root lattice, and for any N > 0, if and only iffor or . Some related results for groups of other types are listed as well.

On the Jordan Block Structure of a Product of Long and Short Root Elements in Irreducible Representations of Algebraic Groups of Type B r

The behavior of a product of commuting long and short root elements of the group of type B r in p-restricted irreducible representations is investigated. For such representations with certain local properties of highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. For a p-restricted representation with highest weight a1ω1 +· · ·+arωr, this fact is proved when aj ≠ p − 1 for some j < r − 1 and one of the following conditions holds: (1) $$ {a}_r\ne p-1\kern0.75em and\kern0.5em {\displaystyle \sum_{i=1}^{r-2}{a}_i\ge p-1;} $$ and (2) $$ 2{a}_{r-1}+{a}_r<p,{\displaystyle \sum_{i=1}^{r-2}r-3}{a}_i\ne 0\;for\;2{a}_{r-1}+{a}_r=p-2\; or\;p-1\; and\;{\displaystyle \sum_{i=1}^{r-3}{a}_i\ne 0} $$ or (r−3) (p-1) for a r = p−1.

Isomorphism classes of k-involutions of algebraic groups of type $${{\mathrm{E}}}_6$$ E 6

Automorphisms of order 2 are studied in order to understand generalized symmetric spaces. The groups of type \({{\mathrm{E}}}_6\) we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as the identity component of the automorphism group of a class of structurable algebras known as Brown algebras. We will classify the k-involutions of these groups of type \({{\mathrm{E}}}_6\) using aspects of both descriptions.

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$p&gt;0$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$\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.

Examples of algebraic groups of type G 2 having the same maximal tori

Answering a question of A. Rapinchuk, we construct examples of non-isomorphic semisimple algebraic groups H1 and H2 of type G2 having coherently equivalent systems of maximal k-tori.

Algebraic groups of type $D_4$, triality, and composition algebras.

Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D_4 are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G_2 ; for the other type they are simple of type A_2 when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras.

Representations of algebraic groups of type C n with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C n in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]4,where [n]4 is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 l . For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.

Restrictions of Modular Representations of Special Linear Groups to A1×A1 Subgroups

We study the restrictions of infinitesimally irreducible modular representations of algebraic groups of type An to subsystem subgroups of type A1 × A1. Under some restrictions on the highest weight of a representation we find their composition factors.

Groups and Geometries

The workshop \textit{Groups and Geometries} was one of a series of Oberwolfach workshops which takes place every 3 years. It focused on finite simple groups, Lie-type groups and their interactions with geometry. There were 47 participants and 25 talks. The themes of the latter centered around the classification of the finite simple groups and its applications, simple algebraic groups and group-theoretic applications of Moufang buildings. There were several 1-hour talks describing recent major results concerning these topics. For instance, D. Segal gave a talk about a proof of a long standing conjecture on finitely generated pro-finite groups which uses the classification. The topic of B. Martin's was the answer to a question of Serre about the notion of complete reducibility for algebraic groups. R. Weiss determined the Whitehead group for certain algebraic groups of type E_6 and E_7 as an application of his new approach to exceptional Moufang quadrangles. There had been also important contributions by young participants. For instance, by D. Bundy on the C(G,T) -theorem, by P.-E. Caprace on the solution of the isomorphism problem for Kac-Moody groups, by T. De Medts on rank-1 groups and by R. Gramlich on local recognition. The conference showed that the theory of simple groups and their geometries is a very active area and that there is a lot of interaction with an increasing number of other areas. There is in particular growing impact of algebraic groups on the theory of finite groups. One of the aims of the conference was, to bring together people from these different areas, so that they can speak with each other and possibly work together on common problems. This interaction was indeed very lively and thus the meeting stands in the tradition of very successful meetings on \textit{Groups and Geometries} at Oberwolfach.

Splitting fields for E 8 -torsors

We show that every algebraic group of type E 8 over any field becomes split over some field extension of degree dividing 2 6 ⋅ 3 2 ⋅5=2880 . This improves a bound by Tits and, in fact, is optimal.

IRREDUCIBLE CHARACTERS FOR ALGEBRAIC GROUPS IN CHARACTERISTIC THREE. II

ABSTRACT In this note, we determine the irreducible characters for the simple algebraic groups of type and over an algebraically closed field K of characteristic 3, by using a theorem of Xi Nanhua [1] and the MATLAB software.