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 $-i$?
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asked May 31, 2021 at 12:12
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1$\begingroup$ Can this good node be taken to be the last in one of the $\lambda$-sequences in Definition 4.5 of the third of Kleshchev's papers on modular restriction/induction? If so, maybe this gives a somewhat indirect proof, since I think it's now known that Kleshchev's description of the Mullineux map agrees with Mullineux's original. $\endgroup$– Mark WildonCommented May 31, 2021 at 15:32
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The equivalence of Kleshchev's algorithm and Mullineux's algorithm was proved by Ford and Kleshchev, but the result they prove is slightly weaker than you want. The result you're asking for is Corollary 4.12 in Bessenrodt & Olsson 'On Residue Symbols and the Mullineux Conjecture', J. Algebraic Combin 7.
