Decomposing Lie Algebra Representations Using Crystal Graphs

We use the theory of crystal graphs to give a simple graph-theoretical algorithm for determining the branching rule for decomposing a representation of a simple Lie algebra when restricted to a simple subalgebra. We also describe a computer package for determining such decomposi-tions graphically. 0.1 Introduction When modeling elementary particle interactions and symmetry breaking in physics it becomes important to understand how tensor products of representations of simple Lie algebras decompose into irreducible sub-representations and how irreducible representations decompose when restricted to simple subalgebras. The classiication of irreducible nite dimensional representations of simple Lie algebras is well-understood ((2]). In principal, this classiication yields straightforward, if cumbersome , algorithms for decomposing 1) tensor products of such representations and 2) representations when restricted to simple subalgebras. The purpose of this paper is to present simpler algorithms for solving these problems using crystal graphs ((5]). We, further, describe an implementation of these algorithms in a Maple package ((3],,4]). 0.2 Crystal Graphs Let G be a simple Lie algebra with highest weight. Let denote a nite dimensional representation of G. DEFINITION 2.1: The crystal graph of the representation (G) is a