Faces in Crystals of Affine Type A and the Shape of Canonical Basis Elements

For a dominant integral weight $\Lambda$ in a Lie algebra of affine type A and rank $e$, and an interval $I_0$ in the residue set $I$, we define the face for the interval $I_0$ to be the subgraph of the block-reduced crystal $\widehat P(\Lambda)$ that is generated by $f_i$ for $i \in I_0$. We show that such a face has an automorphism that preserves defects. For an interval of length $2$, we also give a non-recursive construction of the $e$-regular multipartitions with weights in the face, as well as a formula for the number of $e$-regular multipartitions at each vertex of the face. For an affine Lie algebra of type $A$ we define and investigate the shape of canonical basis elements, a sequence counting the number of multipartitions with a given coefficient. For finite faces generated by $\Lambda$ with $|I_0|=1,2$, we give a non-recursive closed formula for the canonical basis elements.