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

TL;DR

This paper introduces a non-recursive method for computing canonical basis elements in symmetric Kashiwara crystals of type A with rank 2, focusing on external elements associated with weights on the crystal's outer boundary.

Contribution

It provides explicit formulas for canonical basis elements of all e-regular multipartitions with specific defects in symmetric crystals of type A, advancing computational techniques.

Findings

Explicit non-recursive formulas for basis elements

Applicable to all e-regular multipartitions with certain defects

Enhances understanding of crystal structure and basis computation

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 $0 \leq k \leq a$.