Let $K$ be a field and $q\in K^{ \times}$. Let $e$ be the multiplicative order of $q$; or 0 if $q$ is not a root of unity. Let $\bQ:=(q^{ v_1},...,q^{ v_r})$. Let ${ K}_r(n)$ be the set of Kleshchev $r$-multipartitions with respect to $(e;\bQ)$. In this paper, we consider an extention of Dipper--James--Murphy's Conjecture to the Ariki--Koike algebra $H_{ r,n}(q;\bQ)$ with $r>2$. We show that any $(\bQ,e)$-restricted $r$-multipartition of $n$ is a Kleshchev multipartition in ${ K}_r(n)$; and if $e>1$, then any multi-core $\ulam=(\lam^{ (1)},...,\lam^{ (r)})$ in ${ K}_r(n)$ is a $(\bQ,e)$-restricted $r$-multipartition. As a consequence, we show that if $e=0$ (i.e., $q$ is not a root of unity), then ${ K}_r(n)$ coincides with the set of $(\bQ,e)$-restricted $r$-multipartitions of $n$ and also coincides with the set of ladder $r$-multipartitions of $n$.
Categories: math.QA math.RT