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

Abstract

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.

ACKNOWLEDGMENTS

Both authors have been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the program “Mathematical structures” and by the Belarus Basic Research Foundation, Project F98–180. The first author has been also supported by Alexander von Humboldt Foundation.