Split BN-pair Research Articles

Special Moufang sets coming from quadratic Jordan division algebras

The theory of Moufang sets essentially deals with groups having a split BN-pair of rank one. Every quadratic Jordan division algebra gives rise to a Moufang set such that its root groups are abelian and a certain condition called special is satisfied. It is a major open question if also the converse is true, i.e. if every special Moufang set with abelian root groups comes from a quadratic Jordan division algebra. De Medts and Segev [Amer. Math. Soc. 360 (2008), pp. 5831–5852] proved in Theorem 5.11 that this is the case for special Moufang set satisfying two conditions. In this paper we prove that these conditions are in fact equivalent and hence either of them suffices. Even more, we can replace them by weaker conditions.

THE VERTICES OF THE COMPONENTS OF THE PERMUTATION MODULE INDUCED FROM PARABOLIC GROUPS

We consider the permutation module $k_P{\uparrow^{{\rm GL}_n(p^f)}}$, where $P$ is a parabolic group in the general linear group ${\rm GL}_n(p^f)$ and $k$ is an algebraically closed field of prime characteristic $p$. The vertices of the components of these modules have been calculated in [9] by Tinberg, who studied these modules for all groups with split BN-pairs in characteristic $p$. In this paper we show that the idea of suitability is strong enough to find all $p$-groups that are vertex of some component of $k_P{\uparrow^{{\rm GL}_n(p^f)}}$. Furthermore using a result of Burry and Carlson we show that all components have a different vertex.

A geometric proof of Wilbrink’s characterization of even order classical unitals

Using geometric methods and without invoking deep results from group theory, we prove that a classical unital of even order n≥4 is characterized by two conditions (I) and (II): (I) is the absence of O’Nan configurations of four distinct lines intersecting in exactly six distinct points; (II) is a notion of parallelism. This was previously proven by Wilbrink (1983), where the proof depends on the classification of finite groups with a split BN-pair of rank 1.

Moufang sets of finite Morley rank of odd type

We show that for a wide class of groups of finite Morley rank the presence of a split BN-pair of Tits rank 1 forces the group to be of the form PSL2 and the BN-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian 2-subgroups and show that all such Moufang sets are standard (and thus associated to PSL2(F) for F an algebraically closed field of characteristic not 2) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are L-groups and the root groups are not simple.

On groups of finite Morley rank with a split BN-pair of rank 1

We study groups of finite Morley rank with a split BN-pair of Tits rank 1 in the case where the normal complement to B ∩ N in B is infinite and abelian. For such groups, we give conditions ensuring that the standard action of G on the cosets of B is isomorphic to the natural action of PSL 2 ( F ) on P 1 ( F ) for F an algebraically closed field. In particular, we show that SL 2 ( F ) and PSL 2 ( F ) are the only infinite quasisimple L ⁎ -groups of finite Morley rank possessing a split BN-pair of Tits rank 1 where the normal complement to B ∩ N in B is infinite abelian. Our approach is through the theory of Moufang sets and is tied to work attempting to classify the abelian Moufang sets of finite Morley rank.

Special abelian Moufang sets of finite Morley rank

Moufang sets are split doubly transitive permutation groups, or equivalently, groups with a split BN-pair of rank one. In this paper, we study so-called special Moufang sets with abelian root groups, under the model-theoretic restriction that the groups have finite Morley rank. These groups have a natural base field, and we classify them under the additional assumption that the base field is infinite. The result is that the group is isomorphic to PSL2(K) over some algebraically closed field K.

Geometric characterizations of finite Chevalley groups of type B₂

Finite Moufang generalized quadrangles were classified in 1974 as a corollary to the classification of finite groups with a split BN-pair of rank 2 2 , by P. Fong and G. M. Seitz (1973), (1974). Later on, work of S. E. Payne and J. A. Thas culminated in an almost complete, elementary proof of that classification; see Finite Generalized Quadrangles, 1984. Using slightly more group theory, first W. M. Kantor (1991), then the first author (2001), and finally, essentially without group theory, J. A. Thas (preprint), completed this geometric approach. Recently, J. Tits and R. Weiss classified all (finite and infinite) Moufang polygons (2002), and this provides a third independent proof for the classification of finite Moufang quadrangles. In the present paper, we start with a much weaker condition on a BN-pair of Type B 2 \mathbf {B}_2 and show that it must correspond to a Moufang quadrangle, proving that the BN-pair arises from a finite Chevalley group of (relative) Type B 2 \mathbf {B}_2 . Our methods consist of a mixture of combinatorial, geometric and group theoretic arguments, but we do not use the classification of finite simple groups. The condition on the BN-pair translates to the generalized quadrangle as follows: for each point x x , the stabilizer of all lines through that point acts transitively on the points opposite x x .

Moufang lines defined by (generalized) Suzuki groups

We show that the automorphism group of a geometry defined by the generalized Suzuki groups is contained in the automorphism group of the corresponding Suzuki group. This shows that the study of these groups is equivalent to the study of those geometries. This completes, for the Suzuki groups as split BN-pairs of rank 1, a program set up by Jacques Tits some years ago. We also provide a similar result for the generalized Suzuki–Tits inversive planes related to these groups.

Split BN-Pairs of Rank at Least 2 and the Uniqueness of Splittings

Let (G, B, N) be a group with an irreducible spherical BN-pair of rank at least 2, and let U be a nilpotent normal subgroup of B such that B = U (B ⋂ N). We show that U is unique with respect to B. As a corollary, we obtain a complete classification of all irreducible spherical split BN-pairs of rank at least two.

Half-moufang generalized hexagons

If Γ is a half-Moufang generalized hexagon, then Γ is Moufang. We also give a very short proof that a generalized hexagon admitting a split BN-pair is a Moufang hexagon.

Split BN-pairs of rank 2: the octagons

Let G be a group with an irreducible spherical BN-pair of rank 2 where B contains a normal nilpotent subgroup U with B= U( B∩ N). Then G is essentially a group of Lie type. This completes the classification of split BN-pairs of rank 2, generalizing the corresponding result for finite groups due to Fong and Seitz.

On the commutator formula of a split BN-pair

The purpose of this note is to prove in an elementary way and with geometric considerations that the Levi decomposition of a finite group with a split BN-pair of characteristic p (a prime integer) implies the commutator formula.

Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

Abstract We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

Green Vertex Theory, Green Correspondence, and Harish-Chandra Induction

In modular representation theory of a finite group the theory of Green vertices plays an important role. In the representation theory of finite groups with split BN-pairs Harish-Chandra induction has become one of the most important tools in the last decades. This paper will show how the combination of both gives new insight into the modular representation theory of these groups.

Contravariant forms on representations of finite groups with split BN-pairs

Contravariant forms on representations of finite groups with split BN-pairs

Brauer morphism between modular Hecke algebras

Noticeable progress in the knowledge of modular representations of finite groups has been accomplished by considering (papers of Broue, Puig and Scott) the most general situations where one can define the relative traces ideal and the Brauer morphism (see [4]). Our aim in this paper is to describe what happens in the case of p-permutation modules and certain derived modules (most of our matter comes from unpublished results of L. Puig), then to apply that to representations of split BN-pairs in natural characteristic. These two subjects have been treated previously in two papers. The first one, by Scott [26], points out the possibility of setting up for permutation modules a theory analogous to Brauer’s for group algebra and which contains it; the results he obtains are analogues of Brauer’s first and second main theorems and Brauer’s theorems on defect 0 and 1. The other paper, by Tinberg [29], presents the results of Curtis and Richen and Sawada and Green on irreducible modules for split &V-pairs in natural characteristic and certain associated indecomposable modules. Tinberg’s paper contains two further results: an induction formula (leading to very precise information about the dimension of the involved modules) and a computation of vertex, the latter result using Scott’s paper. Our study of the link between these subjects permits us first to simplify (Part B) several proofs of the Curtis-Richen-Sawada theory (existence of “weights,” structure coefficients for the modular “Hecke algebra” End,,(indG, k)) and after that to obtain certain results about indecomposable direct summands of indG, k (Part C). In particular we check that irreducible kc-modules (G is a split BN-pair represented in natural characteristic) are in l-l correspondence with G-conjugacy classes of pairs (I’, rr), where V is a p-subgroup of G and 7c an irreducible kJlr,( V)/V-module of defect 0.

A remark on finite groups having a split BN-pair of rank one with characteristic two

* This paper contains an essential part of the author’s talk delivered on the occasion of “Higmanfest” held at the Mathematical Institute in Oxford on July 13, 1984. A part of this work was carried out while the author held an SERC Visiting Fellowship at the Mathematical Institute. During his stay at Oxford, the author was a Visiting Fellow of University College. At various early stages, this research was supported by grants from the National Science Foundation. The author extends his thanks for this support and for the hospitality of the Institute and of the organizers of the Higmanfest.

Principal series representation of finite groups with split bn pairs∗

Principal series representation of finite groups with split bn pairs∗

Finite groups with a split BN-pair of rank 1. II

Finite groups with a split BN-pair of rank 1. II

Finite groups with a split BN-pair of rank 1. I

In Part I of this paper (Hering, Kantor, and Seitz [SO]), 2-transitive groups of even degree were classified when the stabilizer of a point has a normal subgroup regular on the remaining points. The identification with groups of known type was made by finding a 2-Sylow subgroup and then applying the deep classification theorems of Alperin, Brauer and Gorenstein [l, 21 and Walter [39]. The purpose of the present continuation of [50] is to point out that the proof of the main result of [50] can be completed without using [I] and [2]. 1Ioreover, Walter’s classification theorem [39] and the Gorenstein-Walter Theorem [49] are not required in [50], although the end of Walter [53] seems to be needed. Our arguments are natural continuations of those of [50, Sections 4, 8, and 91. Much use is also made of character-theoretic information contained in Brauer [46] and [47]. Our goal is to show that a minimal counteresamplc has a cyclic two points stabilizer G,,, and then apply a result of Kantor, O’Nan and Seitz [22, Theorem 1.1 or Section 5, Case D]. Re first show that G,, is metacyclic, and then “transfer out field automorphisms” in order to prove that G,, is cyclic. This transfer argument yielded an unexpected dividend: in the course of examining a similar argument in Suzuki [34, Section 211, an error was found. This has been corrected, and, in fact, the entire transfer argument is stated for odd and even degree groups simultaneously. ‘Phc numbering of both the sections and the references will be continued from [50].