Groups Of Lie Type Research Articles

Character ratios, representation varieties and random generation of finite groups of Lie type

We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.

Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G} . In this paper, we determine the finite simple groups of type E 6 ⁢ ( q ) {E_{6}(q)} and E 6 2 ⁢ ( q ) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

Green functions and Glauberman degree-divisibility

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.

Pseudoblocks of Finite Dimensional Algebras

The notion of pseudoblocks is borrowed from [1] and introduced to finite-dimensional algebras. We determine the pseudoblocks for several known algebras such as the triangular algebra and the cyclic group algebra. Also, we determine the pseudoblocks for the group algebra of the special linear group $SL(2,p)$ in the natural characteristic being the only finite group of Lie type of finite representation type.

A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an earlier theorem of the authors and Short. We also propose two conjectures that relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in $SL_n(q)$, as well as to the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev.

Topological generation of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field k and let C1,…,Ct be non-central conjugacy classes in G. In this paper, we consider the problem of determining whether there exist gi∈Ci such that 〈g1,…,gt〉 is Zariski dense in G. First we establish a general result, which shows that if Ω is an irreducible subvariety of Gt, then the set of tuples in Ω generating a dense subgroup of G is either empty or dense in Ω. In the special case Ω=C1×⋯×Ct, by considering the dimensions of fixed point spaces, we prove that this set is dense when G is an exceptional algebraic group and t⩾5, assuming k is not algebraic over a finite field. In fact, for G=G2 we only need t⩾4 and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case t=2, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random (r,s)-generation of exceptional groups.

On Thompson’s Conjecture for Finite Simple Exceptional Groups of Lie Type

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

Let be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an -Sylow subgroup. The classification assists in describing the -Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an -Sylow subgroup is G itself, the reflection subgroup within the parabolic minimally containing an -Sylow subgroup is the whole parabolic subgroup.

The generalized Gelfand–Graev characters of GLn(Fq)

Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

Flag-Transitive Non-Symmetric 2-Designs with $(r, \lambda)=1$ and Exceptional Groups of Lie Type

This paper determines all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G$ is the almost simple flag-transitive automorphism group of $\mathcal{D}$ with an exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of designs $\mathcal{D}$.

Flag-Transitive Block Designs and Finite Simple Exceptional Groups of Lie Type

In this article, we study 2-designs with $$\gcd (r,\lambda )=1$$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and provide some examples in each of these families.

On heights of characters of finite groups

In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z(G) isomorphic to A6,A7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group, as well as for all blocks of finite general linear or unitary groups. For the proof of the case with symmetric groups, it relates to an open question of Olsson on the existence of t-core partitions, where t≥3 is an integer. As a byproduct, our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the Isaacs-Moretó-Navarro-Tiep Conjecture, claiming that the number of distinct irreducible character degrees of a Sylow p-subgroup P of an arbitrary finite group G is at most one more than the number of irreducible character degrees of G that are multiples of p.

Extending Huppert’s conjecture to almost simple groups of Lie type

Let G be a finite group and cd(G) be the set of all irreducible complex character degrees of G without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if H is an almost simple group of Lie type with cd(G)=cd(H), then there exists an Abelian normal subgroup A of G such that G/A≅H. It is furthermore shown that G is not necessarily the direct product of H and A. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.

Bounds on the dimension of Ext for finite groups of Lie type

Let G be a finite group of Lie type defined in characteristic p, and let k be an algebraically closed field of characteristic r>0. We will assume that r≠p (so, we are in the non-defining characteristic case). Let V be a finite-dimensional irreducible left kG-module. In 2011, Guralnick and Tiep found bounds on the dimension of H1(G,V) in non-defining characteristic, which are independent of V. The aim of this paper is to generalize the work of Gurlanick and Tiep. We assume that G is split and use methods of modular Harish-Chandra theory to find bounds on the dimension of ExtkG1(Y,V), where Y and V are irreducible kG-modules. We then use Dipper and Du's algorithms to illustrate our bounds in a series of examples.

Computing Green functions in small characteristic

Let G(q) be a finite group of Lie type over a field with q elements, where q is a prime power. The Green functions of G(q), as defined by Deligne and Lusztig, are known in almost all cases by work of Beynon–Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type E62, E7, E8 in small characteristics. We propose a general method for dealing with these cases, which proceeds by a reduction to the case where q is a prime and then uses computer algebra techniques. In this way, all open cases in type E62, E7 are solved, as well as at least one particular open case in type E8.

An asymptotic cell category for cyclic groups

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué, Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.

A Miniature CCA Public Key Encryption Scheme Based on Non-abelian Factorization Problem in Finite Groups of Lie Type

Abstract 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.

A New Algorithm for Fast Generalized DFTs

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups G . The new algorithm uses O (∣ G ∣ ω /2 + o (1) ) operations to compute the generalized DFT over finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 2. Previously, “exponent one” algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known, even under the assumption ω = 2, for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for SL 2 (F q ) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group and exponent one if ω = 2. Our algorithm also yields an improved exponent for computing the generalized DFT over general finite groups G , which beats the longstanding previous best upper bound for any ω. In particular, assuming ω = 2, we achieve exponent √ 2, while the previous best was 3/2.

Explicit Equations for Exterior Square of the General Linear Group

Several explicit systems of equations defining the exterior square of the general linear group ⋀2 GLn as affine group scheme are presented. Algebraic ingredients of the equations, so called exterior numbers, are translated to the language of weight diagrams corresponding to a Lie group of type An−1 in the representation with the highest weight ϖ2.

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.