TWO CHARACTER FORMULAS FOR $\widehat{\mathfrak{sl}}_2$ SPACES OF COINVARIANTS

Abstract

We consider [Formula: see text] spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra [Formula: see text]. The first one is generated by [Formula: see text], and the second one is generated by [Formula: see text], [Formula: see text] where P(t), [Formula: see text] are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and q-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion product of [Formula: see text]-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.