On a generalisation of the Dipper-James-Murphy conjecture (Q616441)

From the author's summary: Let \(r\), \(n\) be positive integers. Let \(e\) be 0 or an integer bigger than 1. Let \(v_1,v_2, \ldots, v_r \in \mathbb{Z}/e\mathbb{Z}\) and \(\mathcal K_r(n)\) be the set of Kleshchev \(r\)-partition of \(n\) with respect to \((e;\mathbf Q)\), where \(\mathbf Q:=(v_1,v_2, \ldots, v_r )\). The Dipper-James-Murphy conjecture asserts that \(\mathcal K_r(n)\) is the same as the set of \((\mathbf Q,e)\)-restricted bipartition of \(n\) if \(r=2\). In this paper an extension of this conjecture to the case where \(r>2\) is considered. It is proved that any multi-core \(r\)-partition in \(\mathcal K_r(n)\) is a \((\mathbf Q,e)\)-restricted \(r\)-partition. As a consequence, the authors show that in the case \(e=0\), \(\mathcal K_r(n)\) coincides with the set of \((\mathbf Q,e)\)-restricted \(r\)-partition of \(n\) and also coincides with the set of ladder \(r\)-partition of \(n\).