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# Journal of Combinatorics

## Volume 3 (2012)

### Number 3

### The combinatorics of the HMZ operators applied to Schur functions

Pages: 401 – 450

DOI: https://dx.doi.org/10.4310/JOC.2012.v3.n3.a6

#### Authors

#### Abstract

Haglund, Morse, and Zabrocki introduced a family of symmetric function operators $\{\mathbb{B}_{m}\}_{m \geq 1}$ and $\{\mathbb{C}_{m}\}_{m \geq 1}$ which are closely related to operators of Jing \cite{Jing}. Hanglund, Morse, and Zabrocki used these operators to refine the shuffle conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov which gives a combinatorial interpretation of the coefficient of the monomial symmetric function in the Frobenius image of the character generating function of the ring of diagonal harmonics. In this paper, we give combinatorial interpretations of the coefficients that arise in Schur function expansion of $\mathbb{B}_{m}s_\la[X]$ and $\mathbb{C}_{m}s_\la[X]$ where $s_\la[X]$ is the Schur function associated to the partition $\lambda$. We then use such combinatorial interpretations to give a new recursion for the Kostka-Foulkes polynomials $K_{\lambda,\mu}(q)$.

#### Keywords

HMZ operators, Schur function expansion, plethysm, diagonal harmonics, cocharge

#### 2010 Mathematics Subject Classification

Primary 05E05. Secondary 05E10.

Published 19 February 2013