Groups of Exceptional Type, Coxeter Groups and Related Geometries

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

Surface subgroups of Coxeter and Artin groups

In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the defining graph), and the involution which maps each standard generator to its inverse. We observe that the techniques used here to study automorphisms carry over easily to the Coxeter group situation. We thus obtain a classification of the CLTTF type Coxeter groups up to isomorphism and a description of their automorphism groups analogous to that given for the Artin groups.

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

Let be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐ modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient . For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

A theorem of Cassidy states that any Zariski closed differential algebraic subgroup of a simple linear algebraic group, defined over a differential field, is either the whole group or is conjugate to the subgroup of constant matrices. An arithmetic analogue of this theorem is proved in which usual derivations on fields are replaced by certain nonlinear operators on p -adic rings, called " p -derivations."

We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Superstability and central extensions of algebraic groups

Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(\pi,1) -conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland’s stability conditions provide new realisations of the corresponding K(\pi,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(\pi,1) -conjecture.

For a central simple algebra A, the group SK 1(A) is defined to be the quotient \(\{a\in A^{\times}\mid \operatorname {\mathit{Nrd}}_{A}(a) = 1\}\big/ [A^{\times},A^{\times}]\), where \(\operatorname {\mathit{Nrd}}_{A}\) is the reduced norm and [A ×,A ×] is the commutator group of A ×. This group SK 1(A) is a subtle invariant of A × that has been of longstanding interest to those working with central simple algebras, algebraic groups, and K-theory. Since SK 1 is a Brauer class invariant, it suffices to compute SK 1 for division algebras. We prove here nearly all the known formulas for SK 1(D) for D a tame central division algebra over a Henselian field. We do this by first obtaining in §11.1 formulas for SK 1 in the graded setting, where calculations are substantially easier and more transparent. We then recover in §11.2 the corresponding formulas in the valued setting by proving in Th. 11.21 that \(\mathit{SK}_{1}(D) \cong \mathit{SK}_{1}(\operatorname {\mathsf {gr}}(D))\) for F=Z(D) Henselian and D tame over F.

Generators, relations and coverings of algebraic groups, II

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2. We extend a well‐known result that Coxeter groups are to a class of Shephard groups that have ‘enough’ finite parabolic subgroups. We also show that in this setting, if the associated Coxeter group is type (FC), then the Shephard group acts properly and cocompactly on a cube complex. As part of our proof of the former result, we introduce a new criteria for a complex made of simplices to be .

On Engel words on simple algebraic groups