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

## TL;DR

This paper provides explicit combinatorial formulas for certain irreducible components of multivariate diagonal harmonic modules, introducing new path objects and Schur function expressions for key symmetric functions.

## Contribution

It introduces a new path combinatorial object and derives explicit formulas for irreducible components of multivariate diagonal harmonic modules in terms of Schur functions.

## Key findings

- Explicit formulas for irreducible components of multivariate diagonal harmonics.
- Introduction of the path combinatorial object $T_{n,s}$.
- Formulas for $
abla(e_n)$, $
abla^r(e_n)$, and $ riangle'_{e_k}(e_n)$ in Schur functions.

## Abstract

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.