Unidirectional Littelmann Paths for Crystals of Type A and Rank 2

Abstract

For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight Λ, there is a labeling by multipartitions and by piecewise-linear paths in the real weight space called Littelmann paths. Both labelings are constructed recursively, but the crystals are isomorphic, so there is a bijection between the labels. We choose a multicharge $(k_{1},\dots ,k_{r})$, with 0 ≤ k₁ ≤ k₂.... ≤ k_r ≤ e − 1. We put k_i in the node at the upper left corner of partition i and let the residues from $\mathbb {Z}/ e \mathbb {Z}$ increase across rows and decrease down columns. For e = 2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and if partitions with the same corner residue have first rows of the same parity. The multipartition is called strongly residue-homogeneous if each partition ends in a right triangle of whose side has length one less than the first row of the next partition. We give explicit examples, including many of the important case of symmetric groups. We show that such a multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by short paths between fixed integers encoding the number of rows and addable nodes in the multipartition. The path corresponding to a strongly residue-homogeneous multipartition can be constructed non-recursively using only integers describing the multipartition.