Addendum to `Bosonic formulas for $(k,l)$-admissible partitions'

TL;DR

This paper clarifies the relationship between two combinatorial models of (k,l)-admissible partitions and provides an explicit combinatorial parameterization of the affine Weyl group of sl_l.

Contribution

It establishes the connection between earlier combinatorial studies of (k,l)-admissible partitions and Primc's basis labels, and offers a new combinatorial parameterization of the affine Weyl group.

Findings

Clarified the relation between two models of (k,l)-admissible partitions.

Provided an explicit combinatorial parameterization of the affine Weyl group of sl_l.

Enhanced understanding of the combinatorial structure underlying affine Lie algebra representations.

Abstract

In our earlier paper we made a combinatorial study of (k,l)-admissible partitions. This object appeared already in the work of M. Primc as a label of a basis of level k-integrable modules over $\hat{sl}_l$. We clarify the relation between these two works. As a byproduct we obtain an explicit parameterization of the affine Weyl group of $\hat{sl}_l$ by a simple combinatorial set.