On ChariLoktev bases for local Weyl modules in type $A$
Abstract
This paper is a study of the bases introduced by ChariLoktev for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for shortwhose introduction is one of the aims of this paperform convenient parametrizing sets of these bases. They play a role analogous to that played by (GelfandTsetlin) 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 ChariLoktev 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.01191
 Bibcode:
 2016arXiv160601191R
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 17B67;
 05E10
 EPrint:
 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)