On a generalization of Dipper--James--Murphy's Conjecture

Original · EN

Let K be a field and q∈ K×. Let e be the multiplicative order of q; or 0 if q is not a root of unity. Let:=(qᵛ¹,...,qᵛʳ). Let Kᵣ(n) be the set of Kleshchev r-multipartitions with respect to (e;). In this paper, we consider an extention of Dipper--James--Murphy's Conjecture to the Ariki--Koike algebra Hᵣ,ₙ(q;) with r>2. We show that any (,e)-restricted r-multipartition of n is a Kleshchev multipartition in Kᵣ(n); and if e>1, then any multi-core =(⁽¹⁾,...,⁽ʳ⁾) in Kᵣ(n) is a (,e)-restricted r-multipartition. As a consequence, we show that if e=0 (i.e., q is not a root of unity), then Kᵣ(n) coincides with the set of (,e)-restricted r-multipartitions of n and also coincides with the set of ladder r-multipartitions of n.