Spaces of coinvariants and fusion product II. [formula omitted] character formulas in terms of Kostka polynomials

Abstract

In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work [B. Feigin et al., math.QA/0205324, 2002]. We describe the sl n fusion products for symmetric tensor representations following the method of [B. Feigin, E. Feigin, math.QA/0201111, 2002], and show that their Hilbert polynomials are A n−1 -supernomials. We identify the fusion product of arbitrary irreducible sl n -modules with the fusion product of their restriction to sl n−1 . Then using the equivalence theorem from [B. Feigin et al., math.QA/0205324, 2002] and the results above for sl 3 we give a fermionic formula for the Hilbert polynomials of a class of sl 2 coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [B. Feigin et al., Transfom. Groups 6 (2001) 25–52; math.QA/0009198, 2000; math.QA/0012190, 2000]. Our formula differs from the fermionic formula established in [B. Feigin et al., Transfom. Groups 6 (2001) 25–52; math.QA/0009198, 2000; math.QA/0012190, 2000] and implies the alternating sum formula conjectured in [B. Feigin, S. Loktev, math.QA/9812093, 1998; Amer. Math. Sci. Transl. 194 (1999) 61–79] for this case.