Nicolas Jacon
We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using Lusztig $a$-values.
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We compute the generic degrees of the Ariki--Koike algebras by first constructing a basis of matrix units in the semisimple case. As a consequence, we also obtain an explicit isomorphism from any semisimple Ariki--Koike algebra to the group algebra of the corresponding complex reflection group.
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…
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 $H_{n_k,r_k}$.
This article is a comprehensive review of the representation theory of the Ariki-Koike algebras and the cyclotomic Schur algebras.
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.
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…
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.
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.
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 $H_{m,n}(q,\mathbf{Q})$.
We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.
We prove that every Ariki-Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki-Koike algebras which have q-connected parameter sets. A similar result is proved for the cyclotomic q-Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki-Koike algebras defined over fields of characteristic zero are now known in principle.
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.
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.
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。
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.
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.
The aim of this note is to study a generalization of theorems by James and Fayers on the modular representations of the symmetric group and its Hecke algebra to the case of the complex reflection groups of type $G(l,1,n)$ and the associated Ariki-Koike algebra.
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.
We study a natural generalization of the notion of cores for l-partitions attached with a multi-charge s $\in$ Z^l : the (e, s)-cores. We rely them both to the combinatorics and the notion of weight defined by Fayers. Next we study applications in the context of the block theory for Ariki-Koike algebras.
This note is concerned with a natural generalization of the Mullineux involution for Ariki-Koike algebras. Using a result of Fayers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our algorithm notably does not involve the determination of paths in affine crystals.
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。
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。
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.
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。
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。
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.
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.
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。
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.
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。
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。
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。
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…
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。
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。
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.
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…
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。
This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.
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…
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。
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。
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 …
We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field C as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.
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…
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.
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。
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.
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.
We give two different approaches to classifying the simple modules of $0$-Yokonuma-Hecke algebras $Y_{r,n}(0)$ over an algebraically closed field of characteristic $p$ such that $p$ does not divide $r.$ Using the isomorphism between the $0$-Yokonuma-Hecke algebra and $0$-Ariki-Koike-Shoji algebra, we in fact give another way to obtain the simple modules of the latter, which was previously studied by Hivert, Novelli and Thibon (Adv. Math. $\bf{205}$ (2006) 504-548). In the appendix, we give the classification of simple modules of the nil Yokonuma-Hecke algebra.
We determine the structure of category $\cO$ for the rational Cherednik algebra of $G(m,1,n)$ in the case where the $\KZ$ functor satisfies a condition called \emph{separating simples}. As a consequence, we show that the property of having exactly $N-1$ simple modules, where $N$ is the number of simple modules of $G(m,1,n)$, determines the Ariki-Koike algebra up to isomorphism.
The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Sch{ü}tzenberger. We show that this procedure is a part of a more general construction holding in the Kac-Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which notably give interesting analogues of the Young lattice and are expected to parametrize distinguishe…
On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-modules
本文提出了一种新的方法，用于计算不可约最高权 $U_q (\mathfrak{gl}_{\infty})$-模的 canonical basis，并将其应用于 Calogero-Moser 细胞特征标和 Ariki-Koike 代数的分解矩阵。
The level l Fock space admits canonical bases G_e and G_\infty. They correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in N[v]. Restriction to the highest weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki-Koike algebras.
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…
In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type $B$ are described by using the lower global basis of symmetric crystals of $V_θ(λ)$. In this paper, we prove the existence of crystal bases and global bases of $V_θ(0)$ for any symmetric quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin diagram invo…
Generalised hook lengths and Schur elements for Hecke algebras
本研究比较分拆钩长的两种推广，应用于Ariki-Koike代数模表示论，证明单模Schur元可被其广义核的Schur元整除，且A型Hecke代数下该值等于核与商的Schur元乘积。
Quantum wreath products and Schur--Weyl duality II
本文通过引入花环模，利用抛物诱导和多分拆标记方案，统一了Ariki-Koike代数、Hu代数、仿射Hecke代数及其pro-p Iwahori变体中的多种模族，并解决了Ginzburg-Guay-Opdam-Rouquier问题，从而具体实现了D型有理Cherednik代数的范畴O。
Let G be a symplectic or orthogonal complex Lie group with Lie algebra g. As a G-module, the decomposition of the symmetric algebra S(g) into its irreducible components can be explicitely obtained by using identities due to Littlewood. We show that the multiplicities appearing in the decomposition of the k-th graded component of S(g) do not depend on the rank n of g providing n is sufficiently large. Thanks to a classical result by Kostant, we establish a similar result for the k-th graded component of the space H(g) of G-harmonic polynomials. These stabilization properties are equivalent to t…
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