Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution M on a partition Q which is a union of a regular partition P with an added good node of residue $i$, the result will be the Mullineux involution M(P) applied to P to which we add a good node with residue $-i$?

Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added good node of residue $i$, the result will be the Mullineux involution $M(P)$ applied to $P$ to which we add a good node with residue $-i$?

Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution M on a partition Q which is a union of a regular partition P with an added good node for P of residue $i$, the result will be the Mullineux involution M(P) applied to P to which we add a good node for the partition which is the transpose of M(P) with residue $-i$ with respect to M(P)?

Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution M on a partition Q which is a union of a regular partition P with an added good node of residue $i$, the result will be the Mullineux involution M(P) applied to P to which we add a good node with residue $-i$?