New runner removal theorems for the cyclotomic Hecke algebras of type $G(r,1,n)$

For the Iwahori-Hecke algebras ${\mathcal{H}}_q({\mathfrak{S}}_n)$ of the symmetric group ${\mathfrak{S}}_n$ at a primitive $e$-th root of unity, James and Mathas proved a theorem which relates $v$-decomposition numbers $d_{\lambda \mu }^e(v)$ for different values of $e$, by adding empty runners'' to the abacus display for the labelling partitions $λ,μ$. Fayers proved a similar theorem, which involves adding full’’ runners to these abacus displays. In this paper we use Uglov’s map from the set of $r$-abaci for $r$-partitions to the set of $e$-abaci for partitions to extend these theorems to the cyclotomic Hecke algebras of type $G(r,1,n)$.