On a generalisation of the Dipper–James–Murphy conjecture

Let r,n be positive integers. Let e be 0 or an integer bigger than 1. Let v1,…,vr∈Z/eZ and Kr(n) be the set of Kleshchev r-partitions of n with respect to , where Q:=(v1,…,vr). The Dipper–James–Murphy conjecture asserts that Kr(n) is the same as the set of -restricted bipartitions of n if r=2. In this paper we consider an extension of this conjecture to the case where r>2. We prove that any multi-core λ=(λ(1),…,λ(r)) in Kr(n) is a -restricted r-partition. As a consequence, we show that in the case e=0, Kr(n) coincides with the set of -restricted r-partitions of n and also coincides with the set of ladder r-partitions of n.