On a generalisation of the Dipper–James–Murphy Conjecture

Jun Hu

Abstract

Let \(r,\, n\) be positive integers. Let \(e\) be \(0\) or an integer bigger than \(1\). Let \(v_1,\cdots,v_r \in \mathbb{Z}/e\mathbb{Z}\) and \(\mathcal{K}_r(n)\) be the set of Kleshchev \(r\)-partitions of \(n\) with respect to \((e;Q)\), where \(Q:=(v_1,\cdots,v_r)\). The Dipper–James–Murphy conjecture asserts that \(\mathcal{K}_r(n)\) 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 in \(\mathcal{K}_r(n)\) is a \((Q,e)\)-restricted \(r\)-partition. As a consequence, we show that in the case \(e=0\), \(\mathcal{K}_r(n)\) coincides with the set of \((Q,e)\)-restricted \(r\)-partitions of \(n\) and also coincides with the set of ladder \(r\)-partitions of \(n\).

Keywords: Crystal basis, Fock spaces, Kleshchev multipartitions, ladder multipartitions, ladder nodes, Lakshimibai–Seshadri paths.