External vertices for crystals of type A

Abstract

We show that a vertex in the reduced crystal is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect d are external in at least one i-string. Combining this with the Chuang-Rouquier categorification for the simple modules of the cyclotomic Hecke algebras of type A and rank e, this would imply a version of Donovan's Conjecture for the cyclotomics. For e=2, we calculate an approximation to this bound.