An Example of Homomorphisms from the Guay's affine Yangians to non-rectangular $W$-algebras

We construct a non-trivial homomorphism from the Guay’s affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ to the universal enveloping algebra of the $W$-algebra associated with a Lie algebra $\mathfrak{g}\mathfrak{l}(m+n)$ and its nilpotent element whose Jordan block of type $(2^n,1^{m-n})$ for $m>n$.