On a generalization of Dipper--James--Murphy's Conjecture

TL;DR

This paper extends Dipper--James--Murphy's conjecture to Ariki--Koike algebras with multiple parameters, establishing a correspondence between restricted multipartitions and Kleshchev multipartitions, and characterizing these sets based on the root of unity status of q.

Contribution

It generalizes the conjecture to higher r and clarifies the relationship between restricted, Kleshchev, and ladder multipartitions in this broader context.

Findings

Any (Q,e)-restricted r-multipartition is Kleshchev in ${K}_r(n)$.

For e>1, multi-core multipartitions in ${K}_r(n)$ are (Q,e)-restricted.

If e=0, ${K}_r(n)$ equals the set of (Q,e)-restricted and ladder r-multipartitions.

Abstract

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 $n$.