Intertwining vertex operators and certain representations of sl(n)^

TL;DR

This paper constructs exact sequences for principal subspaces of level 1 modules of affine sl(n), deriving recursions that determine their graded dimensions, advancing the understanding of Rogers-Ramanujan-type identities.

Contribution

It introduces a new method using intertwining vertex operators to derive recursions for principal subspaces of affine sl(n) modules, completing the characterization of their graded dimensions.

Findings

Derived exact sequences for principal subspaces

Established recursions for graded dimensions

Confirmed previous results through new methods

Abstract

We study the principal subspaces, introduced by B. Feigin and A. Stoyanovsky, of the level 1 standard modules for $\hat{\goth{sl}(l+1)}$ with $l \geq 2$. In this paper we construct exact sequences which give us a complete set of recursions that characterize the graded dimensions of the principal subspaces of these representations. This problem can be viewed as a continuation of a new program to obtain Rogers-Ramanujan-type recursions, which was initiated by S. Capparelli, J. Lepowsky and A. Milas. In order to prove the exactness of the sequences we use intertwining vertex operators and we supply a proof of the completeness of a list of relations for the principal subspaces. By solving these recursions we recover the graded dimensions of the principal subspaces, previously obtained by G. Georgiev using a different method.