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

RESUMO

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 $\lambda,\mu$. 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 $G(r,1,n)$.