Fermionic formulas for (k, 3)-admissible configurations

TL;DR

This paper derives fermionic formulas for the characters of (k, 3)-admissible configurations, connecting combinatorial bases of certain subspaces in affine Lie algebra modules with symmetric polynomial spaces.

Contribution

It introduces a new fermionic formula for (k, 3)-admissible configurations and links these to bases of subspaces in affine Lie algebra modules using vertex operators.

Findings

Derived explicit fermionic formulas for (k, 3)-admissible configurations.

Connected combinatorial bases with symmetric polynomial spaces.

Provided a filtration approach to determine graded space components.

Abstract

We obtain the fermionic formulas for the characters of (k, r)-admissible configurations in the case of r=2 and r=3. This combinatorial object appears as a label of a basis of certain subspace $W(\Lambda)$ of level-$k$ integrable highest weight module of $\hat{sl}_{r}$. The dual space of $W(\Lambda)$ is embedded into the space of symmetric polynomials. We introduce a filtration on this space and determine the components of the associated graded space explicitly by using vertex operators. This implies a fermionic formula for the character of $W(\Lambda)$.