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

Abstract

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 $\Lambda =a{\Lambda }_0+a{\Lambda }_1$, 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 $0\le k\le a$.

21 pages, 1 figure; typos corrected