On Chari-Loktev bases for local Weyl modules in type $A$
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.
Cite
@article{arxiv.1606.01191,
title = {On Chari-Loktev bases for local Weyl modules in type $A$},
author = {K. N. Raghavan and B. Ravinder and Sankaran Viswanath},
journal= {arXiv preprint arXiv:1606.01191},
year = {2017}
}
Comments
The exposition has been shortened. The final section (of the previous version), which was fairly independent of the rest, has been removed and will appear separately. The main conjecture has since been proven by one of us (arxiv: 1612.01484)