Grobner-Shirshov bases for representation theory

Abstract

In this paper, we develop the Gr6bner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Gr6bner-Shirshov pairs. Our result can be applied to solve the reduction problem in representation theory and to construct monomial bases of representations of associative algebras. As an illustration, we give an explicit construction of Gr6bner-Shirshov pairs and monomial bases for finite dimensional ir:-educible representations of the simple Lie algebra Sl3' Each of these monomial bases is in 1-1 corr3spondence with the set of semistandard Young tableaux with a given shape.