A simple algorithm for computing the global crystal basis of an irreducible U q (sl n )

We describe a simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of U q (sl n ). R'esum'e Nous d'ecrivons un algorithme simple pour calculer la base cristalline globale de Kashiwara d'une repr'esentation irr'eductible de dimension finie de U q (sl n ). 1 Introduction The problem of finding a distinguished basis with favourable properties in each finite dimensional irreducible representation of the complex Lie algebra sl n goes back to the seminal article of Gelfand and Tsetlin in 1959 [4]. In the 1980's, De Concini and Kazhdan [3], and Gelfand and Zelevinsky [5] have explored new directions, introducing the notions of special bases and good bases, respectively. More recently, the development of the theory of quantum groups has lead to the discovery by Lusztig [13] and Kashiwara [7] independently of a distinguished basis of U \Gamma q (g) for any complex simple Lie algebra g. This basis projects onto a canonical basis ..