The self-extension vanishing conjecture for irreducible symmetric-group representations

Let ${\sf S}_n$ be the symmetric group and let $\mathbb{F}$ be an algebraically closed field of characteristic $p\geq 0$. Let {\mathscr P}^{\text{\rm p-res}}_n denote the set of $p$-regular partitions of $n$, and for each \lambda\in{\mathscr P}^{\text{\text{\rm p-res}}}_n let $D^\lambda$ be the corresponding irreducible $\mathbb{F}{\sf S}_n$-representation. Self-extension vanishing conjecture. If $p\geq 3$, then for every \lambda\in{\mathscr P}^{\text{\rm p-res}}_n, $D^\lambda$ has no non-trivial self-extension; equivalently,

$$\operatorname{Ext}^1(D^\lambda,D^\lambda)=0.$$

This conjecture concerns the modular representation theory of symmetric groups. The paper reports partial results and proves the conjecture in specific cases, while the general assertion remains open.