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We construct a family of irreducible representations of the quantum continuous $gl_\infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type.
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B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama, Fermionic formulas for (k, 3)-admissible configurations
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M. Kasatani, Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GL n {\rm GL}_{n} at t k + 1 q r − 1 = 1 t^{k+1}q^{r-1}=1
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