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.