On Chari–Loktev bases for local Weyl modules in type A

摘要

AbstractThis paper is a study of the bases introduced by Chari–Loktev in 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. The stability conjecture has been proved in in the rank one case.]]>