Gorsky, Negut, Oblomkov, Rasmussen, Rozansky and Shende have recently interpreted the Khovanov-Rozansky homology in terms of algebraic geometry using the Hilbert Scheme. Shortly after, in 2017, Hogancamp proved that the KR homology of the $(n,nr+1)$ torus knot has a combinatorial interpretation related to the characters of diagonal harmonics. The diagonal harmonic modules and their characters had been studied for many years. In particular, Garsia and Haiman showed that the Frobenius transformation of the graded characters of the diagonal harmonic modules can be expressed as $\nabla(e_n)$, where $\nabla$ is the Macdonald eigenoperator introduced by Bergeron and Garsia.

In this talk we discus the multivariate diagonal harmonics. In a generalized rectangular context, the characters of these $GL_k\times \mathbb{S}_n$-modules are denoted by $\mathcal{E}_{rn,n}^{\langle k\rangle}$. In our setting, $\nabla^r(e_n)$ will correspond to the special case where $k=2$. We will 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. We will also interprets the application of the Pieri rule on our formula, directly in terms of paths. If time allows it we will also discus the restriction to $k=2$ which is related to $\langle \nabla^r(e_n), s_{\mu}\rangle$ and $\langle \Delta'_{e_k}(e_n),e_n\rangle$. This gives a new combinatorial formula in terms of hook shaped Schur functions in $q$ and $t$, where the shape of the hooks are determined by the major index.

This work was supported by NSERC doctoral scholarship