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Two character formulas for $\hat{sl_2}$ spaces of coinvariants

B. Feigin·M. Jimbo·S. Loktev·T. Miwa
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Abstract

We consider $\hat{sl_2}$ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra $U(sl_2\otimes\C[t])$. The first one is generated by $sl_2\otimes t^N$, and the second one is generated by $e\otimes P(t), f\otimes R(t)$ where $P(t), R(t)$ 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 $sl_3$-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.

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B. Feigin, M. Jimbo, S. Loktev, T. Miwa. 2002-11-22. Two character formulas for $\hat{sl_2}$ spaces of coinvariants. https://doi.org/10.1142/s0217751x04020361

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