Finite Fields, Root Systems, and Orbit Numbers of Chevalley Groups

Radical 3-subgroups of the finite groups of Lie type E6

Let X ( r ) X(r) and G ( q ) G(q) be finite groups of Lie type and r r and q q be coprime. If G ( q ) G(q) is embedded in X ( r ) X(r) , then the Landazuri-Seitz-Zalesskii theorem implies that G ( q ) G(q) is small relative to X ( r ) X(r) . We formalize this observation and illustrate how it can be used with some applications.

Abstract. The computation of the cohomology for finite groups of Lie type in the describing characteristicis a challenging and difficult problem. In [BNP], the authors constructed an induction functor which takesmodules over the finite group of Lie type, G(F q ), to modules for the ambient algebraic group G. In particularthis functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in[BNP] to determine the first non-trivial cohomology class when the underlying root system is of type A n orC n . In this paper the authors extend these results toward locating the first non-trivial cohomology classesfor the remaining finite groups of Lie type (i.e., the underlying root system is of type B n , C n , D n , E 6 , E 7 ,E 8 , F 4 , and G 2 ) when the prime is larger than the Coxeter number. 1. Introduction1.1. Let Gbe a simple algebraic group scheme over a field kof prime characteristic pwhich is definedand split over the prime field F p , and F : G→ Gdenote the Frobenius map. The fixed points of therth iterate of the Frobenius map, denoted G(F

With the development of Lie theory, Lie groups have attained profound significance in several branches of Mathematics and Physics. In Lie theory, the matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as the finite analogue of Lie groups, finite groups of Lie type have potential applications in cryptography due to their unique mathematical structures. In this paper, we first put forward a novel idea of designing cryptosystems based on Lie theory. First of all, combing with discrete logarithm problem and group factorization problem, we proposed several new intractable assumptions based on the matrix exponential in finite groups of Lie type. Subsequently, in analog with Boyen’s scheme (Asiacrypt 2007), we designed a public-key encryption scheme based on the non-abelian factorization problem in finite groups of Lie type. Finally, our proposal was proved to be indistinguishable against adaptively chosen-ciphertext attack in the random oracle model. It is encouraging that our scheme also has the potential to resist against Shor’s quantum algorithm attack.

Finite groups of Lie type encompass most of the finite simple groups. Their representations and characters have been studied intensively for half a century, though some key problems remain unsolved. This is the first comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic. As a subtheme, the relationship between ordinary and modular representations is explored, in the context of Deligne-Lusztig characters. One goal has been to make the subject more accessible to those working in neighbouring parts of group theory, number theory, and topology. Core material is treated in detail, but the later chapters emphasize informal exposition accompanied by examples and precise references.

1. Symmetries of vector spaces: 1.1. What is a symmetry? 1.2. Distance is fundamental 1.3. Groups of symmetries 1.4. Bilinear forms and symmetries of spacetime 1.5. Putting the pieces together 1.6. A broader view: Lie groups 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers 2.2. Quaternions 2.3. The geometry of rotations of R3 2.4. Putting the pieces together 2.5. A broader view: octonions 3. Linearization: 3.1. Tangent spaces 3.2. Group homomorphisms 3.3. Differentials 3.4. Putting the pieces together 3.5. A broader view: Hilbert's fifth problem 4. One-parameter subgroups and the exponential map: 4.1. One-parameter subgroups 4.2. The exponential map in dimension one 4.3. Calculating the matrix exponential 4.4. Properties of the matrix exponential 4.5. Using exp to determine L(G) 4.6. Differential equations 4.7. Putting the pieces together 4.8. A broader view: Lie and differential equations 4.9. Appendix on convergence 5. Lie algebras: 5.1. Lie algebras 5.2. Adjoint maps { big `A' and small `a' 5.3. Putting the pieces together 5.4. A broader view: Lie theory 6. Matrix groups over other fields: 6.1. What is a field? 6.2. The unitary group 6.3. Matrix groups over finite fields 6.4. Putting the pieces together 6.5. A broader view of finite groups of Lie type and simple groups Appendix I. Linear algebra facts Appendix II. Paper assignment used at Mount Holyoke College Appendix III. Opportunities for further study Solutions to selected problems Bibliography.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

In an earlier paper [P1]; we studied self-dual complex representations of a finite group of Lie type. In this paper; we make an analogous study in thep-adic case. We begin by recalling the main result of that paper. LetG(F) be the group of F rational points of a connected reductive algebraic group G over a finite field F. Fix a Borel subgroup B of G; defined over F; which always exists by Lang’s theorem. LetBDTU be a Levi decomposition of the Borel subgroupB. Suppose that there is an element t02 T(F) that operates byi1 on all the simple roots in U with respect to the maximal split torus inT. It can be seen thatt 2 belongs to the center ofG. We proved in [P1] thatt 2 operates by 1 on a self-dual irreducible complex representation …; which has a Whittaker model; if and only if… carries a symmetricG-invariant bilinear form. If … carries a symmetric G-invariant bilinear form; then we call … an orthogonal representation. If … is an irreducible admissible self-dual representation of a p-adic group G; then there exists a nondegenerateG-invariant bilinear formB: …£…! C. This form is unique up to scalars by a simple application of Schur’s lemma; and it is either symmetric or skew-symmetric. The aim of this paper is to provide for a criterion to decide which of the two possibilities holds in the context of p-adic groups similar to our work in the finite field case in [P1]. This partly answers a question raised by Serre (see [P1]). We are able to say nothing about representations not admitting a Whittaker model. Our method; however; works in some cases even when there is no element inT which operates on each simple root byi1. This is the case; for instance; in some cases of SLn andSpn; where we work instead with the similitude group where such an element exists; then we deduce information on SLn andSpn.

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G. The multiplicative semigroup .4 (F) , where F is a finite field, is a particular example. Using HarishChandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G. We go on to show that if F is any field and S any multiplicative subsemigroup of .4 (F) , then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = .4 (F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of .4 (F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.

Tame component of the Schur multipliers of finite groups of Lie type

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

The Divisibility Graph of finite groups of Lie type

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

The workshop was set up in order to stimulate the interaction between (finite and algebraic) geometries and groups. Five areas of concentrated research were chosen on which attention would be focused, namely: diagram geometries and chamber systems with transitive automorphism groups, geometries viewed as incidence systems, properties of finite groups of Lie type, geometries related to finite simple groups, and algebraic groups. The list of talks (cf. page iii) illustrates how these subjects were represented during the workshop. The contributions to these proceedings mainly belong to the first three areas; therefore, (i) diagram geometries and chamber systems with transitive automorphism groups, (ii) geometries viewed as incidence systems, and (iii) properties of finite groups of Lie type occur as section titles. The fourth and final section of these proceedings has been named graphs and groups; besides some graph theory, this encapsules most of the work related to finite simple groups that does not (explicitly) deal with diagram geometry. A few more words about the content: (i). Diagram geometries and chamber systems with transitive automorphism groups. As a consequence of Tits' seminal work on the subject, all finite buildings are known. But usually, in a situation where groups are to be characterized by certain data concerning subgroups, a lot less is known than the full parabolic picture corresponding to the building.