Mullineux map: $d$-balanced partitions and $d$-runner matrices

Let $1<d<e$ be two coprime integers and let $m_e$ denote the Mullineux map, which for $e$ prime describes tensor products of the irreducible modules of symmetric groups with the sign in characteristic $e$. We prove that if $\lambda$ is an $e$-regular partition such that $d$ divides the arm length of any rim hook of $\lambda$ of size divisible by $e$, then $m_e(\lambda)'$ is a partition such that the arm length of any of its rim hooks of size divisible by $e$ is congruent to $-1$ modulo $d$. We introduce a new parameter for partitions called the $d$-runner matrix and show that if $\lambda$ is as above, then the $d$-runner matrices of $\lambda$ and $m_e(\lambda)'$ agree. This determines $m_e(\lambda)'$ uniquely. We approach the whole problem combinatorially and take advantage of a new Abacus Mullineux Algorithm introduced in this paper. We also establish equivalent descriptions of the above partitions which provide an alternative version of the main result about the Mullineux map, which becomes particularly strong when $d=2$.