## 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 $$n$$.