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We give an explicit combinatorial formula for some irreducible components of$GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To thisend we introduce a new path combinatorial object $T_{n,s}$ allowing us to givethe formula directly in terms of Schur functions. This paper also containsformulas written in terms of Schur functions in the $q$ and $t$ variables forspecial cases of $\nabla(e_n)$, $\nabla^r(e_n)$ and $\Delta'_{e_k}(e_n)$. Wealso give an interpretation in term of path to the adjoint dual Pieri ruleapplied on these $GL_k\times \mathbb{S}_n$-characters.

Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics