External Littelmann paths for crystals of type A

TL;DR

This paper establishes a direct, non-recursive correspondence between certain multipartitions and Littelmann paths in affine Lie algebra type A, revealing structured geometric properties of these paths.

Contribution

It introduces a novel non-recursive construction linking residue-homogeneous multipartitions to unidirectional Littelmann paths in affine type A crystals.

Findings

Multipartitions correspond to unidirectional Littelmann paths.

Paths project into specific quadrants, separated by oscillating segments.

Construction uses only integer data from multipartitions.

Abstract

For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight $\Lambda$, 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 since Kashiwara demonstrated that the crystals are isomorphic, there is a bijection between the labels. We choose a multicharge $(k_1,\dots,k_r)$, with $0 \leq k_1\leq k_2....\leq k_r \leq e-1$. We put $k_i$ in the node at the upper left corner of partition $i$ of the multipartition 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 partitions with the same corner residue have first rows of the same parity. It is strongly residue homogeneous if each partition ends in a triangle of whose side has length one less than the first row of the next partition. In this paper we show that each such 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 oscillating paths with a fixed integer oscillator. The path corresponding to such a multipartition can be constructed non-recursively using only integers describing the structure of the multipartition.