On Chari-Loktev bases for local Weyl modules in type $A$

TL;DR

This paper investigates Chari-Loktev bases for local Weyl modules in type A, introducing partition overlaid patterns (POPs) as a new parametrization tool, and explores their combinatorial and representation-theoretic properties.

Contribution

It introduces POPs as a new combinatorial parametrization of Chari-Loktev bases and studies their properties, including a maximal area pattern and a conjecture on basis stability.

Findings

Identification of a unique maximal area pattern for given parameters

Establishment of a bijection between colored partitions and POPs

Discussion of the stability conjecture for Chari-Loktev bases

Abstract

This paper is a study of the bases introduced by Chari-Loktev for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for short---whose introduction is one of the aims of this paper---form convenient parametrizing sets of these bases. They play a role analogous to that played by (Gelfand-Tsetlin) patterns in the representation theory of the special linear Lie algebra. The notion of a POP leads naturally to the notion of area of a pattern. We observe that there is a unique pattern of maximal area among all those with a given bounding sequence and given weight. We give a combinatorial proof of this and discuss its representation theoretic relevance. We then state a conjecture about the "stability", i.e., compatibility in the long range, of Chari-Loktev bases with respect to inclusions of local Weyl modules. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which may be of interest in itself.