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

Abstract

Abstract. We consider sl2 spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra U (sl2 ⊗ C[t]). The first one is generated by sl2 ⊗ t N , and the second one is generated by e ⊗ P (t), f ⊗ P (t) where P (t), P (t) 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 sl3-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.

A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra Uq(sl2)

“…Here K (k) j,(n,0,...,0) (v) denotes the level restricted Kostka polynomial for sl 2 , see e.g. (2.9) in [6]. We conjecture that the right hand side gives the character of the associated graded space of the induced filtration mentioned above.…”

“…is the graded character of the fusion product of n copies of 2-dimensional irreducible sl 2 -modules, see, e.g., (2.11) in [6].…”

“…In particular, taking the sum over n we obtain the known character formula of the integrable sl 2 -modules of level −1 [16]: is the graded character of the fusion product of n copies of 2-dimensional irreducible sl 2 -modules, see, e.g., (2.11) in [6].…”

“…In general, however, the subspaces defining the filtration are not generated by tensor products of extremal vectors. In the case of integrable sl 2 -modules of level −k, the following fermionic formula is known (the formula (2.14) in [6] in the limit N → ∞): ch v,z V −(k − j)Λ 0 − jΛ 1 = n≡j mod 2 K (k) j,(n,0,...,0…”