Representation theory of the 0-Ariki-Koike-Shoji algebras

On the one dimensional representations of Ariki-Koike algebras at roots of unity

We study the natural labeling of the one dimensional representations for Ariki-Koike algebras at roots of unity. For Hecke algebras of types A and B, some of these representations can be identified with the socle of the Steinberg representation of a finite reductive group. We here give closed formulas for them. This uses, in particular, several results concerning crystal isomorphisms and the Mullineux involution.

Core blocks of Ariki-Koike algebras II: the weight of a core block

We study combinatorial blocks of multipartitions, exploring further the notions of weight, hub and core block introduced by the author in earlier papers. We answer the question of which pairs (w,theta) occur as the weight and hub of a block, and we examine the action of the affine Weyl group on the set of blocks.

The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras

This article is a comprehensive review of the representation theory of the Ariki-Koike algebras and the cyclotomic Schur algebras.

Ariki-Koike代数的满列移除定理

Full runner removal theorem for Ariki-Koike algebras

本文研究Ariki-Koike代数的表示理论，通过定义多重分区算盘显示中添加满列珠子的操作，结合Fayers的LLT型算法，建立了不同$e$值下$e$-多重正则多重分区的$q$-分解数之间的联系。

Moving vectors I：Ariki-Koike代数的块的表示型

Moving vectors I: Representation type of blocks of Ariki-Koike algebras

本文提出Ariki-Koike代数块的新不变量“块移动向量”，据此分类表示有限块，并给出非导出等价与不同轨道的导出等价块实例，同时确定了cyclotomic $q$-Schur代数块的表示型。

Stuttering blocks of Ariki-Koike algebras

We study a shift action defined on multipartitions and on residue multisets of their Young diagrams. We prove that the minimal orbit cardinality among all multipartitions associated to a given multiset depends only on the orbit cardinality of the multiset. Using abaci, this problem reduces to a convex optimisation problem over the integers with linear constraints. We solve it by proving an existence theorem for binary matrices with prescribed row, column and block sums. Finally, we give some applications to the representation theory of the Hecke algebra of the complex reflection group $G(r,p,n…

Gieseker 空间中的不动点与 Ariki-Koike 代数的块

Fixed points in Gieseker spaces and blocks of Ariki-Koike algebras

本文利用 A 型 McKay 箭图上的 Nakajima 箭图簇刻画 Gieseker 簇不动点轨迹，将其不可约分量维数联系到 Ariki-Koike 代数块权重，并给出带电荷多划分核心的多重荷计算及核心块的几何解释。

Ariki–Koike代数与Rogers–Ramanujan型分拆

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

本文针对q=-1情形的Ariki–Koike代数Kleshchev多分拆生成函数，结合其与广义Rogers–Ramanujan型分拆的关联给出解析证明，并借助已有结果得到m=2时固定块单模对应的两个双变量生成函数恒等式。

Transition matrices between Young's natural and seminormal representations

We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.

Tilting modules for cyclotomic Schur algebras

This paper investigates the tilting modules of the cyclotomic q-Schur algebras, the Young modules of the Ariki-Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations -- which are in general more powerful than Specht filtrations -- play only a secondary role.

有限Kleshchev双分拆与$q$-三项式系数

Finite Kleshchev bipartitions and $q$-trinomial coefficients

通过限制双分拆的最大部分，研究Kleshchev双分拆的计数函数，并证明其与Andrews和Baxter引入的两族$q$-三项式系数相关。

Standard multipartitions and a combinatorial affine Schur-Weyl duality

We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations complet…