Unique Node Extremal Subgroups in Chevalley Groups

Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.     

Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field; namely, orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.     

Subgroups normalized by the elementary Levi subgroup

Subgroups of the unipotent radical of a maximal parabolic subgroup of a Chevalley group over a field K, which are normalized by the commutator subgroup of the Levi subgroup, are described. It is shown that in the typical case, such subgroups are in one-to-one correspondence with the closed subsets of {1, 2, ..., n} for a natural n. In the exceptional cases the classification also involves additive subgroups of K. See the table in the paper for a detailed list of possibilities. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 199–215.     

Multiplicative Semigroup of a Residue Class Ring

In this paper, we consider the structure of the multiplicative semigroup of a residue class ring R/I of a commutative ring R with identity modulo its nonzero ideal I. For the general case, we investigate the H-classes, maximal subgroups and the structure of Reg(R/I) which is the set of regular elements of R/I. If R is any integral domain and if I is a product of powers of invertible maximal ideals, we show that R/I is an epigroup, every H^*-class of R/I is a nil-extension of a group (:unipotent epigroup) and that R/I is a complete lattice of unipotent epigroups.     

On Maximal Unipotent Subgroups of a Special Linear Group Over Commutative Ring

Ukrainian Mathematical Journal - It is proved that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive...     

Discreteness criterion for subgroups of products of SL(2)

Let G be a finite product of SL(2, K _i )’s for local fields K _i of characteristic zero. We present a discreteness criterion for nonsolvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular, such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL₂( \mathbbR \mathbb{R} )’s.     

Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

Irreducible Subgroups in an Orthogonal Group

On the basis of the author’s results, the irreducible subgroups of an orthogonal group over an algebraically closed field that are generated by short root unipotent subgroups are described. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 187–194.     

Subgroups of a spinor group containing a maximal split torus. I

This paper studies subgroups of the split spinor group G=Spin (n, k),n 6, over a field K containing a maximal split torus T=T(n,K)assuming that charK2and ||K ||7.The main result of the first pan of the paper is the reduction of the proof that subgroups containing T are standard to the study of subgroups not containing root unipotent elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 49–75, 1991.     

On the groups SL2(ℤ[x]) and SL2(k[x,y])

This paper studies free quotients of the groups SL₂(ℤ[x]) and SL₂(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary or unipotent elements.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Maps behaving like exponentials and maximal unipotent subgroups of groups of Lie type

Let U be a maximal unipotent subgroup of a finite classical group in good characteristic. We prove the existence of a bijection between U and the associated Lie algebra preserving centralizers. As a consequence, we obtain information on the sizes of the conjugacy classes of U. Similar results are proved in the exceptional cases.     

Algèbres amassées et algèbres préprojectives : le cas non simplement lacé

We generalize to the non simply-laced case results of Geiß, Leclerc and Schröer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case. To cite this article: L. Demonet, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Cluster algebras in algebraic lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac–Moody groups. We also review a quantized version of these results.     

Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups

Images of regular unipotent elements from subsystem subgroups of type A₂ and B₂ in irreducible modular representations of classical groups are studied. For images of such elements and representations with locally small highest weights, all sizes of Jordan blocks of one and the same parity are found. Bibliography: 17 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 159–178.     

Galois Cohomology of Unipotent Algebraic Groups and Field Extensions

In this note we discuss, in the case of unipotent groups over nonperfect fields k of characteristic p, an analog of a theorem of Steinberg (formely a Serre's conjecture) for unipotent algebraic group schemes, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of k of degree divisible by p.