Lie Type Research Articles

Short laws for finite groups of Lie type

Short laws for finite groups of Lie type

Application and diagnostic points of carotid artery ultrasound in the detection of internal carotid artery absence: Analysis of 16 cases and literature review.

To summarize the ultrasound diagnostic key points and imaging characteristics of absence of the internal carotid artery (ICA), and to explore the application value of carotid artery ultrasound in the diagnosis of ICA absence. Sixteen patients diagnosed with ICA absence at our hospital were retrospectively analyzed. Clinical data, ultrasound images, and other relevant imaging data were reviewed. The ultrasound diagnostic key points and related imaging characteristics of ICA absence were summarized. Combined with a literature review, the application of carotid artery ultrasound in the diagnosis of ICA absence was discussed. Among the 16 patients, there were 10 males and 6 females ages ranging from 13 to 80 years. Among them, 5 patients had a history of cerebral infarction, 1 patient had a history of subarachnoid hemorrhage, and 10 patients had no history of stroke. Among the 16 patients, there were 10 left-sided lesions, 5 right-sided lesions, and 1 bilateral lesion. Three patients had concomitant aneurysms. Carotid ultrasound examinations in all 16 ICA absent patients revealed that a significantly slender common carotid artery (CCA) diameter compared to the contralateral (3.91 ± 0.73 vs. 6.92 ± 1.42, p < 0.001). On the side of ICA absence, the bifurcation of the carotid artery was not detected, with the CCA directly continuing as a single artery (external carotid artery, ECA). The temporal superficial artery percussion test was positive in all cases, and the blood flow spectrum of the single artery (ECA) on the side of the absent ICA showed a serrated pattern during diastole. On the side where the ICA was absent, the blood flow spectrum morphology of the CCA and ECA was consistent. Based on other relevant imaging examinations, the patients were classified into Lie types: A (10 cases), B (2 cases), C (1 case), D (1 case), and 2 cases were unclassified. Carotid artery ultrasound has the advantages of easy operation, affordability, non-invasiveness, and capable of dynamic and high repeatability. Compared to other imaging examinations, it can provide more evidence for the diagnosis of ICA absence, and is suitable for widespread application and promotion.

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) .

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

Equivariant K$K$‐theory of flag Bott manifolds of general Lie type

AbstractThe aim of this paper is to describe the equivariant and ordinary Grothendieck ring and the equivariant and ordinary topological ‐ring of flag Bott manifolds of the general Lie type. This will generalize the results on the equivariant and ordinary cohomology of flag Bott manifolds of the general Lie type due to Kaji, Kuroki, Lee, and Suh.

The isomorphism problem for cominuscule Schubert varieties

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.

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 .

Characterization of nonlinear Lie type derivations on von Neumann algebras by local action

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R , we prove that if a map δ : R → R satisfies δ ( P n ( A 1 , A 2 , A 3 , … , A n ) ) = ∑ i = 1 n P n ( A 1 , … , A i − 1 , δ ( A i ) , A i + 1 , … , A n ) for any A 1 , A 2 , A 3 , … , A n ∈ R with A 1 A 2 A 3 = 0 , then δ ( A + B ) − δ ( A ) − δ ( B ) ∈ Z ( R ) for any A , B ∈ R . In particular, if R is a von Neumann algebra with no central summands of type I 1 or a factor, then δ ( x ) = d ( x ) + τ ( x ) for all x ∈ R , where d : R → R is an additive derivation and τ : R → Z ( R ) is a map vanishing on each ( n − 1 ) -th commutator p n ( A 1 , A 2 , A 3 , … , A n ) with A 1 A 2 A 3 = 0 .

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.

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

Nonlinear skew Lie type higher derivations on some operator algebras

Nonlinear skew Lie type higher derivations on some operator algebras

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 &gt; 0 \epsilon &gt; 0 there exists N &gt; 0 N &gt; 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.