Fayers–Mathas conjecture on irreducible doubly-singular Specht modules in positive characteristic

Let $\mathbb{F}$ be a field of characteristic $p>0$, let $\mathcal{H}_n=\mathcal{H}_{\mathbb{F},-1}(\mathfrak{S}_n)$, and let $\lambda$ be a doubly-singular partition of $n$. A partition is doubly-singular when both it and its conjugate $\lambda'$ are $2$-singular. For such a partition, define integers $a$, $b$, and $c$ by

$$a=\max\{i:\lambda_i-\lambda_{i+1}\geqslant 2\},\qquad b=\max\{i:\lambda_i=\lambda_{i+1}\geqslant 1\},\qquad c=\max\{j:\lambda_{a+j}>0\}.$$

The partition $\lambda$ is an FM-partition if all of the following hold: $\lambda_i-\lambda_{i+1}\leqslant 1$ for $i\ne a$; $\lambda_b\geqslant a-1\geqslant b$; $\lambda_1>\cdots>\lambda_c$; if $c=0$, all addable nodes except possibly those in the first row and first column have the same residue; and if $c>0$, all addable nodes have the same residue. A partition is a $2p$-core if none of its hook lengths is divisible by $2p$.

Fayers–Mathas conjecture. The $\mathcal{H}_n$-module $S^\lambda$ is irreducible if and only if $\lambda$ is a $2p$-core and $\lambda$ or $\lambda'$ is an FM-partition.

This conjecture proposes the classification of irreducible Specht modules labelled by doubly-singular partitions in positive characteristic. The source explains that non-$2p$-cores give reducible modules and that only finitely many FM-partitions remain to be checked for each prime $p$; the proposed complete classification is based on computer calculations.