A q-Analog of Foulke's conjecture

TL;DR

This paper introduces a q-analog of Foulkes' conjecture involving Hall-Littlewood polynomials, showing it holds at q=1 and connecting it to the classical case at q=0, with additional evidence and generalizations.

Contribution

It formulates a new q-analog of Foulkes' conjecture using Hall-Littlewood polynomials, extending the classical conjecture and providing initial evidence for its validity.

Findings

The q-analog reduces to Foulkes' conjecture at q=0.

The proposed q-analog holds true at q=1.

Supporting evidence and generalizations are discussed.

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 $H_n(\mathbf{x};q)$ replaces the analogous difference of plethysms of complete homogeneous symmetric functions $h_n(\mathbf{x})$ 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.