The Ariki--Koike algebras and Rogers--Ramanujan type partitions

TL;DR

The paper analyzes Ariki–Koike algebras at roots of unity, focusing on the q = -1 specialization to connect simple modules (labeled by Kleshchev multipartitions) with generalized Rogers–Ramanujan partitions. It provides an analytic proof of the Ariki–Mathas generating function in the q = -1 case with a specified set of $Q$-parameters and develops a detailed partition-residue framework to study simple modules in fixed blocks. For the case m = 2, it derives explicit bivariate generating functions for the residue statistic and proves corollaries via Jacobi’s triple product, linking block sizes to fixed-residue counts. The main result extends to general $a,m$ through a symmetry principle, a q-binomial multisum transformation, and connections to Andrews and Kim–Yee identities, yielding a product–multisum correspondence that generalizes Andrews–Gordon type identities and highlights deep ties to partition theory and representation theory of cyclotomic Hecke algebras.

Abstract

In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the so-called Kleshchev multipartitions. This together with Ariki's categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl--Kac character formula. In this paper, we revisit this generating function for the $q=-1$ case. This $q=-1$ case is particularly interesting, for the corresponding Kleshchev multipartitions have a very close connection to generalized Rogers--Ramanujan type partitions when $Q_1=\cdots=Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Our second objective is to investigate simple modules of the Ariki--Koike algebra in a fixed block. It is known that these simple modules in a fixed block are labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities when $m=2$.

Figures (3)

• Figure 1: $Y_{(5, 4, 4, 2)}$
• Figure 2: Residues mod $3$
• Figure 3: The strict partition $\pi=(9, 7, 6, 3)$

Theorems & Definitions (40)

• Theorem 1.1: Ariki--Mathas AM
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• proof
• Theorem 3.1: berkovich, Theorem 4.1
• Theorem 3.2: berkovich, zeng
• proof