I. D. Suprunenko

MINIMAL INDUCTIVE SYSTEMS OF MODULAR REPRESENTATIONS FOR NATURALLY EMBEDDED ALGEBRAIC AND FINITE GROUPS OF TYPE A

The article is devoted to the classification of the minimal and minimal nontrivial inductive systems of modular representations for naturally embedded algebraic and finite groups of type A and related locally finite groups. It occurs that the minimal systems consist of the trivial representations for the relevant groups and the minimal nontrivial ones are connected with Frobenius twists of the standard representations and their duals. These results are applied to the description of the maximal ideals in group algebras of the locally finite groups SL ∞ and SU ∞ in describing characteristic. It is also proved that for an arbitrary classical algebraic group, the restriction of an irreducible module with highest weight large enough to a naturally embedded finite Chevalley group of the same type, but a smaller rank contains the regular module.

MODULAR BRANCHING RULES FOR 2-COLUMN DIAGRAM REPRESENTATIONS OF GENERAL LINEAR GROUPS

In this paper branching rules for the polynomial irreducible representations of the general linear groups in positive characteristic with highest weights labeled by partitions of the form (2a, 1b, 0c) and their restrictions to the special linear groups are found. The submodule structure of the restrictions of the corresponding irreducible modules for the group GLn(F) (or SLn(F)) to the naturally embedded subgroup GLn-1(F) (or SLn-1(F)) is determined. As a corollary, inductive systems of irreducible representations for GL∞(F) and SL∞(F) that consist of representations indicated above, are classified. The submodule structure of the relevant Weyl modules is refined.

RESTRICTIONS OF LARGE IRREDUCIBLE REPRESENTATIONS OF THE CLASSICAL GROUPS TO NATURALLY EMBEDDED SMALL SUBGROUPS CANNOT BE SEMISIMPLE

It is proved that the restriction of a p-restricted representation of a classical algebraic group G of rank r in characteristic p > 0 to a naturally embedded semisimple subgroup cannot be completely reducible (semisimple) if the subgroup has a simple component of rank m small enough with respect to r and the highest weight is large enough with respect to p. It suffices to assume that r ≥ 2m and that the highest weight is equal to ∑ r i=1 ai ω i with ∑ r i=1 ai ≥ 2p − 1 if p ≠ 2 or G ≠ Cr (K) and ∑ r i=1 ai ≥ 4 for p = 2 and G = Cr (K).