Multiplicity-free classification conjecture for symmetric and alternating Kronecker squares

Let $n\in\mathbb N$ and let $\lambda\vdash n$. A character is multiplicity-free when every irreducible constituent occurs with multiplicity at most one. The partitions listed in parts (1) and (3) of Proposition~ are the partitions identified there. Multiplicity-free classification conjecture. These are all partitions for which either the symmetric part $S^2([\lambda])$ or the alternating part $A^2([\lambda])$ is multiplicity-free. The proposition gives the currently known families, while computational data motivate this stronger classification; the conjecture remains open.