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

Abstract

We give an explicit combinatorial formula for some irreducible components of $G{L}_k\times {\mathbb{S}}_n$-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 ${\Delta }_{e_k}^{\prime }(e_n)$. We also give an interpretation in term of path to the adjoint dual Pieri rule applied on these $G{L}_k\times {\mathbb{S}}_n$-characters.

This is the extended version of "Séminaire Lotharingien de Combinatoire: Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana)", (2019), Vol 82B, #71