Rost invariants of simply connected algebraic groups

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.

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

Compositio Mathematica ◽

10.1112/s0010437x20007617 ◽

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.

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

Journal of Group Theory ◽

10.1515/jgth-2017-0019 ◽

2018 ◽

Vol 21 (1) ◽

pp. 1-20 ◽

Cited By ~ 3

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].

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On rational subgroups of reductive algebraic groups over integral domains

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073059 ◽

1995 ◽

Vol 117 (2) ◽

pp. 203-212 ◽

Cited By ~ 2

Author(s):

Yu Chen

Keyword(s):

Integral Domain ◽

Algebraic Groups ◽

Root Systems ◽

Chevalley Groups ◽

Integral Domains ◽

Irreducible Components ◽

Simply Connected ◽

Reductive Algebraic Groups

Let G and G′ be reductive algebraic groups defined over infinite fields k and k′ respectively. The purpose of this paper is to show that G and G′ have isomorphic root systems if their rational subgroups G(R) and G′(R′), where R and R′ are integral domains with R ⊇ k and R′ ⊇ k′, are isomorphic to each other, except in one particular case (see Theorem 3·4). This has been proved by R. Steinberg in [6, theorem 31] for simple Chevalley groups over perfect fields. In particular, when G and G′ are semisimple and adjoint, every isomorphism between G(R) and G′(R′) induces an isomorphism between their irreducible components (see Proposition 3·3). These results imply that, when G and G′ are semisimple k-groups and when both are either simply connected or adjoint, then they are isomorphic to each other as algebraic groups if and only if their rational subgroups over an integral domain that contains k are isomorphic to each other, except in one particular case (see Corollary 3·5).

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Projective simplicity of groups of rational points of simply connected algebraic groups defined over number fields

Banach Center Publications ◽

10.4064/-26-2-455-466 ◽

1990 ◽

Vol 26 (2) ◽

pp. 455-466 ◽

Cited By ~ 3

Author(s):

G. Tomano

Keyword(s):

Algebraic Groups ◽

Number Fields ◽

Rational Points ◽

Simply Connected

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The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000597 ◽

2018 ◽

Vol 62 (2) ◽

pp. 523-552 ◽

Cited By ~ 1

Author(s):

Jay Taylor

Keyword(s):

Algebraic Groups ◽

Reductive Groups ◽

Unpublished Manuscript ◽

Reductive Algebraic Group ◽

Regular Embedding ◽

Isomorphism Classes ◽

Reduction Techniques ◽

Frobenius Endomorphism ◽

Derived Subgroup ◽

Simply Connected

AbstractWe investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

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