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 v 1 , … , v r ∈ Z / e Z and K r ( n ) be the set of Kleshchev r-partitions of n with respect to ( e ; Q ) , where Q : = ( v 1 , … , v r ) . The Dipper–James–Murphy conjecture asserts that K r ( n ) is the same as the set of ( Q , e ) -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 K r ( n ) is a ( Q , e ) -restricted r-partition. As a consequence, we show that in the case e = 0 , K r ( n ) coincides with the set of ( Q , e ) -restricted r-partitions of n and also coincides with the set of ladder r-partitions of n.