Strong generators of the subregular W-algebra and combinatorial description at critical level

We construct explicitly strong generators of the affine $\mathcal{W}$-algebra $\mathcal{W}^{K-N}(\mathfrak{sl}_N, f_{sub})$ of subregular type $A$. Moreover, we are able to describe the OPEs between them at critical level. We also give a description the affine $\mathcal{W}$-algebra $\mathcal{W}^{-N}(\mathfrak{sl}_N, f_{sub})$ in terms of certain fermionic fields, which was conjectured by Adamovi\'{c}.