Combinatorics of the $\widehat{\mathfrak{sl}}_2$ spaces of coinvariants: loop Heisenberg modules and recursion

Abstract.

The spaces of coinvariants are quotient spaces of integrable $\widehat{\mathfrak{sl}}_2$ modules by subspaces generated by the actions of certain subalgebras labeled by a set of points on a complex line. When all the points are distinct, the spaces of coinvariants essentially coincide with the spaces of conformal blocks in the WZW conformal field theory and their dimensions are given by the Verlinde rule.¶ We describe monomial bases for the $\widehat{\mathfrak{sl}}_2$ spaces of coinvariants. In particular, we prove that the spaces of coinvariants have the same dimensions when all the points coincide. We establish recurrence relations satisfied by the monomial bases and the corresponding characters of the spaces of coinvariants. For the proof we use filtrations of the $\widehat{\mathfrak{sl}}_2$ modules. The adjoint graded spaces are certain modules on the loop Heisenberg algebra. The recurrence relation is established by using filtrations on these modules.¶This paper is the continuation of [FKLMM].