Primitive parabolic permutation representations for finite symplectic groups

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite symplectic groups are defined on cosets with respect to maximal parabolic subgroups.     

Primitive parabolic permutation representations for finite simple orthogonal groups in odd dimensions

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite simple orthogonal groups in odd dimensions are defined on cosets with respect to maximal parabolic subgroups.     

Minimal permutation representations of finite simple classical groups. Special linear,symplectic, and unitary groups

This work was supported by the Russian Foundation for Fundamental Research, grant 93-011-1501.     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

Minimal permutation representations of finite simple orthogonal groups

In the paper, nontrivial permutation representations of minimal degree are studied for finite simple orthogonal groups. For them, we find degrees, ranks, subdegrees, point stabilizers and their pairwise intersections.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 603–627, November–December, 1994.     

Orthogonal, symplectic and unitary representations of finite groups

Let be a field, a finite group, and a linear representation on the finite dimensional -space . The principal problems considered are:

I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms which are -invariant.

II. If is such a form, enumerate the equivalence classes of representations of into the corresponding group (orthogonal, symplectic or unitary group).

III. Determine conditions on or under which two orthogonal, symplectic or unitary representations of are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically' equivalent.

This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) -module components of their spaces are equivalent.

We assume throughout that the characteristic of does not divide .

Solutions to I and II are given when is a finite or local field, or when is a global field and the representation is ``split'. The results for III are strongest when the degrees of the absolutely irreducible representations of are odd - for example if has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that is a local or global field, when the representations are split.

     

Categorical actions on unipotent representations of finite unitary groups

Publications mathématiques de l'IHÉS - Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent...     

Minimal permutation representations of finite simple exceptional twisted groups

A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups. Supported by RFFR grant No. 96-01-01893, through the program “Universities of Russia”, and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 17–35, January–February, 1998.     

On the finite dimensional unitary representations of Kazhdan groups

We use A. Weil's criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field.

     

On finite groups isospectral to simple linear and unitary groups

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

Pure quantitative characterization of finite projective special unitary groups

We prove that each projective special unitary group G can be characterized using only the set of element orders of G and the order of G.     

Primitive permutation groups containing ap-cycle

Project supported in part by National Natural Science Foundation of China and STFG