External Vertices for Crystals of Affine Type A

We demonstrate that for a fixed dominant integral weight and fixed defect d , there are only a finite number of Morita equivalence classes of blocks of cyclotomic Hecke algebras, by combining some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over Kac-Moody algebras of affine type A. This is an extension of a proof for symmetric groups of a conjecture known as Donovan’s conjecture. We fix a dominant integral weight Λ. The blocks of cyclotomic Hecke algebras H n Λ for the given Λ correspond to the weights P (Λ) of a highest weight representation with highest weight Λ. We connect these weights into a graph we call the reduced crystal P ̂ ( Λ ) , in which vertices are connected by i -strings. We define the hub of a weight and show that a vertex 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 i -external in at least one i -string, lying at the high degree end of the i -string. For e = 2, we calculate an approximation to this bound.