A $q$-Analog of Foulkes' Conjecture

Abstract

We propose a $q$-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials $H_n(\boldsymbol{x};q)$ replaces the analogous difference of plethysms of complete homogeneous symmetric functions $h_n(\boldsymbol{x})$ in Foulkes' conjecture. At $q=0$, we get back the original statement of Foulkes, and we show that our version holds at $q=1$. We discuss further supporting evidence, as well as various generalizations, including a $(q,t)$-version.