NONRECURSIVE CANONICAL BASIS COMPUTATIONS FOR LOW RANK KASHIWARA CRYSTALS OF TYPE A

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type A on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type A and rank e=2, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with Λ=aΛ0+aΛ1, we give formulae for the canonical basis elements for all the e-regular multipartitions with defects either k(a−k) or k(a−k)+2a, for 0≤k≤a.