Fermionic Formulas for $(k,3)$-admissible Configurations

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 ${\widehat{\mathfrak{sl}}}_r$. 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 $W(\Lambda )$.