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

Ariki-Koike 代数的块与层级-秩对偶

Blocks of Ariki-Koike algebras and level-rank duality

本文通过广义核概念研究 Ariki-Koike 代数的块，在层级-秩对偶背景下解释仿射对称群的作用并分析其轨道。

Specht Modules and Branching Rules for Ariki-Koike Algebras

Specht modules for an Ariki-Koike algebra have been investigated recently in the context of cellular algebras. Thus, these modules are defined as quotient modules of certain ``permutation'' modules, that is, defined as ``cell modules'' via cellular bases. We shall introduce in this paper Specht modules for an Ariki-Koike algebra as submodules of those ``permutation'' modules and investigate their basic properties such as Standard Basis Theorem and the ordinary Branching Theorem, generalizing several classical constructions for type $A$. The second part of the paper moves on looking for Kleshch…

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.

Homomorphisms Between Specht Modules of the Ariki Koike Algebra

In this paper we generalize a theorem due to Lyle, extending its application to the setting of the Ariki-Koike algebra, and in doing so establish an analogue of the kernel intersection theorem. This in turn provides us with a means towards constructing homomorphisms between Specht modules for the Ariki-Koike algebra.

An algorithm for the computation of the decomposition matrices for Ariki-Koike algebras

We give a purely combinatorial algorithm for the computation of the decomposition matrices for Ariki-Koike algebras when the parameters are powers of the same root of unity. It generalizes the LLT algorithm.

Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra

Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $\bar S^p$. It is shown that $S^p$ is a standardly based algebra and $\bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.

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

We investigate the representation theory of certain specializations of the Ariki-Koike algebras, obtained by setting $q=0$ in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe restrictions, induction products, Cartan invariants and decomposition matrices. This allows us to identify the Grothendieck rings of the towers of algebras in terms of certain graded Hopf algebras known as the Mantaci-Reutenauer descent algebras, and Poirier Quasi-symmetric functions.

On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity

Following ideas of Geck and Rouquier, we show that there exists a ``canonical basic set'' of Specht modules in bijection with the simple modules of Ariki-Koike algebras at roots of unity. Moreover, we determine the parametrization of this set and we give the consequences of these results on the representation theory of Ariki-Koike algebras.

The representation type of Ariki-Koike algebras and cyclotomic q-Schur algebras

We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.

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.

Graded decomposition numbers of Ariki-Koike algebras for blocks of small weight

We present some blocks of Ariki-Koike algebras $\mathcal{H}_{n,r}$ for which the decomposition matrices are independent of the characteristic of the underlying field. We complete the description of the graded decomposition numbers for blocks of Ariki-Koike algebras of weight at most two, which consists of analysing the indecomposable core blocks at level $r = 3$, and give a closed formula for the decomposition numbers in this case.

A cancellation-free formula for the Schur elements of the Ariki-Koike algebra

Schur elements play a powerful role in the representation theory of symmetric algebras. In the case of the Ariki-Koike algebra, Schur elements are Laurent polynomials whose factors determine when Specht modules are projective irreducible and whether the algebra is semisimple. Formulas for the Schur elements of the Ariki-Koike algebra have been independently obtained first by Geck, Iancu and Malle, and later by Mathas. The aim of this note is to give a cancellation-free formula for these polynomials, so that their factors can be easily read and programmed.

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。

Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras

After reformulating the representation theory of 0-Hecke algebras in an appropriate family of Yang-Baxter bases, we investigate certain specializations of the Ariki-Koike algebras, obtained by setting q=0 in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe restrictions, induction products, Cartan invariants and decomposition matrices. This allows us to identify the Grothendieck rings of the towers of algebras in terms of certain graded Hopf algebras known as the Mantaci-Reutenauer descent algebras, and Poirier quasi-symmetric fu…

Ariki-Koike代数块的范围等价性

Scopes equivalence for blocks of Ariki-Koike algebras

研究Ariki-Koike代数块的范围等价性，给出其相关多划分无给定剩余可加节点的充要条件，依金字塔数和Scopes向量分类其Scopes等价类，证明独立于Richards的1级结果。

On the number of constituents of induced modules of Ariki-Koike algebras

We examine the crystal graph of the $\widehat{\mathfrak{sl}}_e$-module arising from an $\widehat{\mathfrak{sl}}_e$-categorification to study the defining endo-functors of the categorification. This yields lower bounds on the number of irreducible constituents of certain objects. We use Ariki's categorification result on Ariki-Koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules. In particular this will imply reducibility of every induced module.

Rouquier blocks of the cyclotomic Ariki-Koike algebras

The definition of Rouquier for the families of characters of Weyl groups in terms of blocks of the associated Iwahori-Hecke algebra has made possible the generalization of this notion to the complex reflection groups. Here we give an algorithm for the determination of the "Rouquier blocks" of the cyclotomic Ariki-Koike algebras.

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…

Schur elements for the Ariki-Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0.

Schur-Sergeev duality for Ariki-Koike algebras

Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.

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。

Ariki-Koike代数块的v-分解矩阵的等价性

Equivalence of $v$-decomposition matrices for blocks of Ariki-Koike algebras

本文证明了Ariki-Koike代数中模限制诱导多分拆之间自然对应的充分条件，并由此建立Scopes等价下不同块之间v-分解数的等价性。

Ariki-Koike代数的移动向量与核心块

Moving vectors and core blocks of Ariki-Koike algebras

本文利用移动向量分类Ariki-Koike代数的核心块，给出Scopes等价的充要条件，并用Kostka数表示核心块中单模个数，进而联系A型Iwahori-Hecke代数的分次分解数。

Ariki-Koike 代数的 Rouquier 块

Rouquier blocks for Ariki-Koike algebras

本文将对称群代数和 A 型 Hecke 代数中重要的 Rouquier 块定义推广至由多分区索引 Specht 模的 Ariki-Koike 代数，并探究了这些块的性质。

Generalised canonical basic sets for Ariki-Koike algebras

Let H be a non semi-simple Ariki-Koike algebra. According to [18] and [14], there is a generalisation of Lusztig's a-function which induces a natural order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have proved that this order makes unitriangular the decomposition matrix of H. The algebra H is then said to admit a "canonical basic set". We fully classify which values of m afford a canonical basic set for H and which do not. When this is the case, we describe these sets in terms of "twisted Uglov" or "twisted Kleshchev" multipartitions.

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…

Ariki-Koike代数中的空跑者移除定理

Empty runner removal theorem for Ariki-Koike algebras

本文将James和Mathas关于Iwahori-Hecke代数中v分解数关系的定理扩展至Ariki-Koike代数，建立类似的关系。

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 代数的特征标

Characters of Ariki--Koike algebras

该研究通过量子超群代数与 Ariki - Koike 代数的 Schur - Sergeev 互反性，证明 Ariki - Koike 代数特征标的 Regev 公式，并由此得出 G(m,1,n)型复反射群特征标的 Regev 公式，实现了相关公式的推广。

多分划的核与权以及Ariki-Koike代数的块

Cores and weights of multipartitions and blocks of Ariki-Koike algebras

该研究通过Uglov映射定义了任意多电荷下多分划的e-核与e-权，由此导出Ariki-Koike代数块的对应定义及Nakayama猜想类似结论，还推广了[w:k]-对概念并得到其Scopes等价的充分条件。

Ariki-Koike代数的Rouquier块的分解数 I

Decomposition numbers for Rouquier blocks of Ariki-Koike algebras I

本文研究了特征$p\geq 0$域上Ariki-Koike代数的分次分解数，针对同属Rouquier块且具有公共多核的多分划，给出了$p=0$时的闭式公式。

The Ariki-Terasoma-Yamada tensor space and the blob-algebra

We show that the Ariki-Terasoma-Yamada tensor module and its permutation submodules $ M(λ) $ are modules for the blob algebra when the Ariki-Koike algebra is a Hecke algebra of type $B$. We show that $ M(λ)$ and the standard modules $ Δ(λ) $ have the same dimensions, the same localization and similar restriction properties and are equal in the Grothendieck group. Still we find that the universal property for $ Δ(λ) $ fails for $ M(λ) $, making $ M(λ) $ and $ Δ(λ) $ different modules in general. Finally, we prove that $ M(λ) $ is isomorphic to the dual Specht module for the Ariki-Koike algebra.

关于Ariki-Koike代数范畴作用的范围等价注记

A Note on Scopes Equivalences for Ariki--Koike Algebras as Categorical Actions

本文探讨了Ariki-Koike代数中范畴作用的范围等价转换方法，通过抽象结果与具体应用的结合，揭示了权重空间的Morita等价类划分。

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.

修正的Ariki-Koike代数与Yokounuma-Hecke型关系

Modified Ariki-Koike algebra and Yokounuma-Hecke like relations

本文给出修正的Ariki-Koike代数的新表示，证明其与Yokonuma-Hecke代数关系类似，并由此得到迹形式、对称群作用及固定子代数的基与生成元。

JMMO Fock space and Geck-Rouquier classification of simple modules for Hecke algebras

This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.

On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras

Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\Sc(\vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike alg…

Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups G(r,p,n)

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Broué and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible …

Admissibility Conditions for Degenerate Cyclotomic BMW Algebras

We study admissibility conditions for the parameters of degenerate cyclotomic BMW algebras. We show that the u-admissibility condition of Ariki, Mathas and Rui is equivalent to a simple module theoretic condition.

On decomposition numbers of the cyclotomic q-Schur algebras

Let $S(Λ)$ be the cyclotomic q-Schur algebra associated to the Ariki-Koike algebra $H$. We construct a certain subalgebra $S^0(Λ)$ of $S(Λ)$, and show that it is a standardly based algebra in the sense of Du and Rui. $S^0(Λ)$ has a natural quotient $\bar{S^0}(Λ)$, which turns out to be a cellular algebra. In the case where the modified Ariki-Koike algebra $H^{\flat}$ is defined, $\bar{S^0}(Λ)$ coincides with the cyclotomic q-Schur algebra associated to $H^{\flat}$. In this paper, we discuss a relationship among the decomposition numbers of $S(Λ)$, $S^0(Λ)$ and $\bar{S^0}(Λ)$. In particular, we…

Modular representations of cyclotomic Hecke algebras of type G(r,p,n)

We give a classification of the simple modules for the cyclotomic Hecke algebras over $\mathbb{C}$ in the modular case. We use the unitriangular shape of the decomposition matrices of Ariki-Koike algebras and Clifford theory.

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

稳定的 Khovanov-Lauda-Rouquier 代数与块的局部模型

Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks

本文通过引入 KLRW 代数的稳定商，为 Ariki--Koike 代数的所有块（包括 $S_n$ 的 RoCK 块）提供了一种新的 Morita 等价局部模型。

The cyclotomic BMW algebra associated with the two string type B braid group

The cyclotomic Birman-Murakami-Wenzl (or BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki-Koike, in the same way as the BMW algebras are extensions of the Hecke algebras of type A. In this paper we focus on the case n=2, producing a basis of B_2^k and constructing its left regular representation.

Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space

In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula.

仿射范畴表示的RoCK块

RoCK blocks for affine categorical representations

本研究针对仿射范畴化中的块等价问题，提出用有限超平面配置的房分类Scopes等价类，重点研究最大的RoCK等价类，验证其与Lyle相关工作的契合性，并提供了测试和查找Ariki-Koike代数RoCK块的Sage代码。

分圆q-网范畴的分解矩阵

Decomposition matrices of cyclotomic $q$-web categories

该研究构造了介于分圆Hecke代数与分圆q-Schur代数之间的分圆q-网自同态代数的胞腔结构，证明其模范畴的Grothendieck群同构于量子仿射$\boldsymbol{\text{gl}}_p$的可积最高权模，推广了相关经典成果。

Markov traces on affine and cyclotomic Yokonuma-Hecke algebras

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that t…

On the freeness of the cyclotomic BMW algebras: admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras

The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Häring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory. In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over ground rings with parameters satisfying so-called "admissibility conditions". These conditions are necessary in order for these results to hold and originally arise from the representation theory of B_2^k, which is analysed by the authors in a previous paper. Furthermore, we o…

On the Mackey formulas for cyclotomic Hecke algebras and categories O of rational Cherednik algebras

In this paper, we shall establish the Mackey formulas in the following two set ups: (i) on the tensor induction and restriction functors on the modules over cyclotomic Hecke algebras (Ariki-Koike algebras) and their standard subalgebras of parabolic subgroups. (ii) on the Bezrukavnikov-Etingof induction and restriction functors among categories O of rational Cherednik algebras for the complex reflection group of type G(r, 1, n) and their parabolic subgroups.

Modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these alg…

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.

Cyclotomic quiver Hecke algebras and Hecke algebra of G(r,p,n)

Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike al…

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…