Subgroups Of Finite Classical Groups Research Articles

Irreducible restrictions of representations of symmetric and alternating groups in small characteristics

Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such a classification is known when the characteristic of the ground field is greater than 3, but the small characteristics cases require a substantially more delicate analysis and new ideas. Our results fit into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

Irreducible restrictions of representations of alternating groups in small characteristics: Reduction theorems

We study irreducible restrictions from modules over alternating groups to proper subgroups, and prove reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This problem had been solved when the characteristic of the ground field is greater than33, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

Orbit method for p-Sylow subgroups of finite classical groups

For the p-Sylow subgroups U of the finite classical groups of untwisted Lie type, p an odd prime, we construct a monomial CU-module M which is isomorphic to the regular representation of CU by a modification of Kirillov's orbit method called monomial linearisation. We classify a certain subclass of orbits of the U-action on the monomial basis of M consisting of so called staircase orbits and show, that every orbit module in M is isomorphic to a staircase one. Finally we decompose the André-Neto supercharacters of U into a sum of U-characters afforded by staircase orbit modules contained in M.

Irreducible restrictions of representations of symmetric groups in small characteristics: reduction theorems

We study irreducible restrictions of modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the characteristic of the ground field is greater than 3, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher–Scott program on maximal subgroups of finite classical groups.

Overgroups of root groups in classical groups

We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.

On the Invariants of Root Subgroups of Finite Classical Groups

We show that the invariant fields F q (X 1 , . . . ,X n ) G are purely transcendental over F q if G are root subgroups of finite classical groups. The key step is to find good similar groups of our groups. Moreover, the invariant rings of the root subgroups of special linear groups are shown to be polynomial rings and their corresponding Poincaré series are presented.

Cross-characteristic representations of [formula omitted] and their restrictions to proper subgroups

We classify all pairs (V,H), where H is a proper subgroup of G=Sp6(q), q even, and V is an ℓ-modular representation of G for ℓ≠2 which is absolutely irreducible as a representation of H. This problem is motivated by the Aschbacher–Scott program on classifying maximal subgroups of finite classical groups.

On Invariants of Some Maximal Subgroups of Finite Classical Groups

The maximal subgroups of the finite classical groups are divided into nine classes by Aschbacher's theorem. In this paper, we give explicit transcendental bases of the invariant subfields of those maximal subgroups of classical groups of linear, symplectic and unitary cases that lie in the first two of these classes. Also, we show that the invariant rings of the maximal subgroups of the finite linear groups that lie in the first class are polynomial rings.

Automorphic realization of residual Galois representations

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification of irreducible subgroups of finite classical groups with certain sorts of generators.

Representations of the general linear groups which are irreducible over subgroups

We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field ${\Bbb F}$ of characteristic prime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\downarrow_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups in finite classical groups.

On restrictions of modular spin representations of symmetric and alternating groups

Let $\mathbb F$ be an algebraically closed field of characteristic $p$ and $H$ be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups $G$ of $H$ and $\mathbb F H$-modules $V$ such that the restriction $V{\downarrow }_G$ is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $H$ is the Schur’s double cover $\hat A_n$ or $\hat S_n$.

Representations of the Alternating Group which are Irreducible Over Subgroups

Let F be an algebraically closed field of characteristic p ⩾ 0 and An be the alternating group on n letters. The main goal of this paper is to describe the pairs (G, E), where E is an irreducible FAn-module and G < An is a proper subgroup of An such that the restriction E↓G is irreducible. We are able to give a list of all such pairs, provided p > 3. The case p = 0 has been treated by J. Saxl. The problem is important for the classification of maximal subgroups in finite classical groups. 2000 Mathematical Subject Classification: 20C20, 20C30, 20B35, 20B20.

Representations of the symmetric group are reducible over simply transitive subgroups

If the characteristic of F is zero, this problem has been solved by Saxl [16]. An important feature of Saxl’s result is that a groupG as in the problem above is either 2-transitive or fixes a point, i.e. is contained in some Σn−1. The caseG ≤ Σn−1 can then be settled using the branching rule and induction. On the other hand, an explicit list of k-transitive groups (for k ≥ 2) is available, which can be used to complete the proof in characteristic zero, see [16] for more details. From now on we assume that p > 0. In this case the problem is important for determining maximal subgroups of finite classical groups [1],[13]. However, the situation is nowmore complicated. For example, to determine the pairs (G,D) as above withG = An one needs the Mullineux conjecture [7], [2]. IfG is intransitive then, up to a conjugation, it is contained in a standard Young subgroup of the form Σn−k ×Σk. The irreducible restrictions from Σn toΣn−1 have been described in [14], see also [11], [6]. In [11] it is also

Twisted tensor product subgroups of finite classical groups

If Cln(qr) denotes a classical group with natural module W of dimensioa n over Fqr , then the twisted tensor product module is realised over Fq, and yields an embedding . These embeddings play a significant role in the subgroup structure of classical groups; for example, Seitz [18] shows that any maximal absolutely irreducible subgroup defined over a proper extension field of Fq is of this form. In this paper we study the precise nature of these embeddings, and go on to investigate their maximality or otherwise. We show that the normaliser of Cln(qr) is usually maximal, with an explicit list of just 4 families of exceptions.

Second class of maximal subgroups in finite classical groups

Second class of maximal subgroups in finite classical groups

A class of maximal subgroups in finite classical groups

A class of maximal subgroups in finite classical groups