Combinatorics of the $\hat{sl}_2$ Spaces of Coinvariants

Abstract

We consider two types of quotients of the integrable modules of $\hat{sl}_2$. These spaces of coinvariants have dimensions described in terms of the Verlinde algebra of level-$k$. We describe monomial bases for the spaces of coinvariants, which leads to a fermionic description of these spaces. For $k=1$, we give the explicit formulas for the characters. We also present recursion relations satisfied by the characters and the monomial bases.