Nonrecursive canonical basis computations for low rank Kashiwara crystals of type A

For symmetric Kashiwara crystals of 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 non-recursive manner. In particular, for a symmetric crystal with $Λ=a Λ_0+a Λ_1$, 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 $0 \leq k \leq a$.

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