Faces in crystals of affine type a and the shape of canonical basis elements

For a dominant integral weight (Formula presented.) in a Lie algebra of affine type A and rank e, and an interval (Formula presented.) in the residue set I, we define the (Formula presented.) -face for the interval (Formula presented.) to be the subgraph of the block-reduced crystal that is generated by (Formula presented.) for (Formula presented.). We show that such a face has an automorphism that preserves defects. For an interval of length 2 in a crystal with (Formula presented.), 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. We give a conjecture about the proper form of such a path for an interval of arbitrary length t, with (Formula presented.). 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 with (Formula presented.) and (Formula presented.), we give a non-recursive closed formula for the canonical basis elements.