Two character formulas for s-fraktur sign and l-fraktur sign₂ spaces of coinvariants

We consider s-fraktur sign and l-fraktur sign₂ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra U (s-fraktur sign and l-fraktur sign₂ ⊗ ℂ[t]). The first one is generated by s-fraktur sign and l-fraktur sign₂ ⊗ t^N, and the second one is generated by e ⊗ P(t), f ⊗ P̄(t) where P(t), P̄(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 s-fraktur sign and l-fraktur sign₃-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.