The q-Foulkes conjecture

For each positive integer $n$, let $H_n(\mathbf{x};q)$ be the Hall–Littlewood polynomial used in the paper, and for $0<a\leq b$ define

$$F_{a,b}(\mathbf{x};q):=\frac{H_b[H_a]-H_a[H_b]}{1-q}.$$

Let $f_{a,b}=h_b[h_a]-h_a[h_b]$. The q-Foulkes conjecture. The Schur expansion of $F_{a,b}(\mathbf{x};q)$ has coefficients in $\mathbb{N}[q]$, and specializes at $q=0$ to $f_{a,b}$.

This is a proposed $q$-analogue of Foulkes's conjecture: the specialization at $q=0$ recovers the classical statement, while the paper establishes the corresponding positivity at $q=1$ and gives computational evidence for the conjecture.