Canonical Bases for Fundamental Modules of Quantized Enveloping Algebras of Type A

Let g be a finite-dimensional simple Lie algebra over C of type An, and let U be the q-analogue of its universal enveloping algebra defined by Drinfel’d [4] and Jimbo [6]. According to [9, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight λ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U -module V (λ) with highest weight λ. Kashiwara [7] and Lusztig [8] have independently shown the existence of a certain canonical basis B(λ) for V (λ). Fix r ∈ I = {1, 2, . . . , n}, let ωr be the r-th fundamental weight and let Vr be the corresponding fundamental module V (ωr) (we use the same numbering as [2, Planche I]). Let W r be the set of distinguished left coset representatives in the Weyl group W of g, with respect to the parabolic subgroup generated by all of the fundamental generators s1, s2, . . . , sn of W except sr.