On the bases for local Weyl modules in type A [HBNI Th89]

Abstract

This thesis is a study of the Chari-Pressley-Loktev (CPL) bases for local Weyl modules of the current algebra Sl r+1 [t]. As convenient parametrizing sets of these bases, we introduce the notion of partition overlaid patterns (POPs), which 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 prove the ''Stability'', ie., compatibility in the long range, of CPL bases with respect to inclusions of local Weyl modules in the case r = 1 and state it as a conjecture for r > 1. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which is of interest in itself.

Irreducible representations of the special linear Lie algebra occur as grade zero pieces of the corresponding local Weyl modules. The CPL basis being homogeneous, those basis elements that are of grade zero form a basis for the irreducible representation space. We prove a triangular relationship between this basis and the classical Gelfand-Tsetlin basis.