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