Groups Of Lie Type Research Articles

Short laws for finite groups of Lie type

Short laws for finite groups of Lie type

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair (G,Γ), and we prove that the last coincides with the Lott L2 analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case G=H, the Heisenberg group.

On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic ℓ≠p has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for GLn.

Cohomology and Ext for blocks whose Brauer trees are lines or stars

We compute the dimensions of Ext G n ( V , W ) for all irreducible modules V, W lying in r-blocks of cyclic defect whose Brauer tree is either a star or a line (open polygon), which in the line case extends a result of Dudas. We also make use of this information to aid in completing the same computation for all cross characteristic r-blocks of cyclic defect in rank one groups of Lie type.

On the Uniqueness Theorem

The aim of this paper is to investigate whether the Uniqueness Theorem introduced by Feit and Thompson holds in finite simple groups of Lie type of Lie rank one. We also introduce and examine the Strong Uniqueness Theorem. We define the concept of containment minimality and prove that a non-Abelian finite simple group is minimal if and only if it is containment minimal. Finally, we inquire when a minimal non-soluble group has a faithful doubly transitive representation.

Probabilistic generation of finite almost simple groups

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x∈G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to ∞). This is new even if G is simple. Together with results of Lucchini and Burness–Guralnick–Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.

The maximal Tori of finite exceptional groups of Lie type

Let G(q) be a twisted or untwisted finite exceptional group of Lie type over GF ( q ) , either adjoint or simply connected, and let W be the Weyl group of G(q). This paper compiles an extensive data set on the maximal tori of G(q). A systematic labeling is given, via the corresponding conjugacy classes of W, together with a complete list of the structures of the maximal tori in terms of their torsion coefficient decomposition. Additionally the associated admissible graphs and the structure or shape of centralizers of elements in W are given as well as compatible computer files. Also various apparent inconsistencies in the current literature on simply connected groups are discussed and resolved.

On algebraic normalisers of maximal tori in simple groups of Lie type

Abstract Let 𝐺 be a finite simple group of Lie type and let 𝑇 be a maximal torus of 𝐺. It is well known that if the defining field of 𝐺 is large enough, then the normaliser of 𝑇 in 𝐺 is equal to the algebraic normaliser N ⁢ ( G , T ) N(G,T) . We identify explicitly all the cases when N G ⁢ ( T ) N_{G}(T) is not equal to N ⁢ ( G , T ) N(G,T) .

Collineation groups of octonionic and split-octonionic planes

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley–Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type [Formula: see text] and [Formula: see text].

Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type

Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type

On the labelling of characters of Weyl groups of type $$F_4$$

AbstractIn the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type $$F_4$$ F 4 . We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system , we spotted an error in Spaltenstein’s tables for the generalised Springer correspondence in type $$E_7$$ E 7 .

Solubilizers in profinite groups

The solubilizer of an element x of a profinite group G is the set of the elements y of G such that the subgroup of G generated by x and y is prosoluble. We propose the following conjecture: the solubilizer of x in G has positive Haar measure if and only if x centralizes ‘almost all’ the non-abelian chief factors of G. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive c such that, for every finite simple group S and every (a,b)∈(Aut(S)∖{1})×Aut(S), the number of s is S such that 〈a,bs〉 is insoluble is at least c|S|. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when S is a simple group of Lie type. In this paper we prove the conjecture for alternating groups.

Special Unipotent Representations of Simple Linear Lie Groups of Type A

Special Unipotent Representations of Simple Linear Lie Groups of Type A

The group factorization problem in finite groups of Lie type

With the development of Lie theory, Lie groups have profound significance in many branches of mathematics and physics. In Lie theory, matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as finite analogues of Lie groups, finite groups of Lie type also have wide application scenarios in mathematics and physics owning to their unique mathematical structures. In this context, it is meaningful to explore the potential applications of finite groups of Lie type in cryptography. In this paper, we firstly built the relationship between matrix exponential and discrete logarithmic problem (DLP) in finite groups of Lie type. Afterwards, we proved that the complexity of solving non-abelian factorization (NAF) problem is polynomial with the rank n of the finite group of Lie type. Furthermore, combining with the Algebraic Span, we proposed an efficient algorithm for solving group factorization problem (GFP) in finite groups of Lie type. Therefore, it's still an open problem to devise secure cryptosystems based on Lie theory.

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩

Let p p be a prime. In this paper we investigate finite K { 2 , p } \mathcal K_{\{2,p\}} -groups G G which have a subgroup H ≤ G H \le G such that K ≤ H = N G ( K ) ≤ Aut ⁡ ( K ) K \le H = N_G(K) \le \operatorname {Aut}(K) for K K a simple group of Lie type in characteristic p p , and | G : H | |G:H| is coprime to p p . If G G is of local characteristic p p , then G G is called almost of Lie type in characteristic p p . Here G G is of local characteristic p p means that for all nontrivial p p -subgroups P P of G G , and Q Q the largest normal p p -subgroup in N G ( P ) N_G(P) we have the containment C G ( Q ) ≤ Q C_G(Q)\le Q . We determine details of the structure of groups which are almost of Lie type in characteristic p p . In particular, in the case that the rank of K K is at least 3 3 we prove that G = H G = H . If H H has rank 2 2 and K K is not PSL 3 ⁡ ( p ) \operatorname {PSL}_3(p) we determine all the examples where G ≠ H G \ne H . We further investigate the situation above in which G G is of parabolic characteristic p p . This is a weaker assumption than local characteristic p p . In this case, especially when p ∈ { 2 , 3 } p \in \{2,3\} , many more examples appear. In the appendices we compile a catalogue of results about the simple groups with proofs. These results may be of independent interest.

Remarks on the inverse Galois problem over function fields

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ-adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible systems.

Products of normal subsets

In this paper we consider which families of finite simple groups G G have the property that for each ϵ > 0 \epsilon > 0 there exists N > 0 N > 0 such that, if | G | ≥ N |G| \ge N and S , T S, T are normal subsets of G G with at least ϵ | G | \epsilon |G| elements each, then every non-trivial element of G G is the product of an element of S S and an element of T T . We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form P S L n ( q ) \mathrm {PSL}_n(q) where q q is fixed and n → ∞ n\to \infty . However, in the case S = T S=T and G G alternating this holds with an explicit bound on N N in terms of ϵ \epsilon . Related problems and applications are also discussed. In particular we show that, if w 1 , w 2 w_1, w_2 are non-trivial words, G G is a finite simple group of Lie type of bounded rank, and for g ∈ G g \in G , P w 1 ( G ) , w 2 ( G ) ( g ) P_{w_1(G),w_2(G)}(g) denotes the probability that g 1 g 2 = g g_1g_2 = g where g i ∈ w i ( G ) g_i \in w_i(G) are chosen uniformly and independently, then, as | G | → ∞ |G| \to \infty , the distribution P w 1 ( G ) , w 2 ( G ) P_{w_1(G),w_2(G)} tends to the uniform distribution on G G with respect to the L ∞ L^{\infty } norm.

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSLn(q) and PSp2n(q)

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSLn(q)(n≥4), PSL3(q)(q>2), or PSp2n(q)(n≥3), is isomorphic to a group algebra.