Journal of the
Korean Mathematical Society JKMS

In this paper, we develop the Gr\"obner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Gr\"obner-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 Gr\"obner-Shirshov pairs and monomial bases for finite dimensional irreducible representations of the simple Lie algebra $S_{l3}$. Each of these monomial bases is in 1-1 correspondence with the set of semistandard Young tableaux with a given shape.

Keywords: Grobner-Shirshov pair, monomial basis, representation, simple Lie algebra, semistandard Young tableau

MSC numbers: 16Gxx