We obtain the fermionic formulas for the characters of (k, r)-admissible conﬁgurations in the case of r = 2 and r = 3. This combinatorial object appears as a label of a basis of certain subspace W(Λ) of level-k integrable highest weight module of sl^r. The dual space of W(Λ) is embedded into the space of symmetric polynomials. We introduce a ﬁltration on this space and determine the components of the associated graded space explicitly by using vertex operators. This implies a fermionic formula for the character of W(Λ).
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Evgeny Mukhin, Boris Feigin, M. Jimbo, Tetsuji Miwa, Yoshihiro Takeyama, Fermionic Formulas for (<em>k</em>, 3)-admissible Conﬁgurations. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 125–162DOI 10.2977/PRIMS/1145475968