Proof of the modular branching rule for cyclotomic Hecke algebras

Lectures on cyclotomic Hecke algebras

These are lecture notes prepared for London Mathematical Society Symposium "Quantum Groups" held in Durham 19-29 July 1999. We give a survey on cyclotomic Hecke algebras. These algebras have been studied by Dipper, James, Malle, Mathas and the author. In lecture one, we start with definitions. We then go on to Specht module theory of cyclotomic Hecke algebras constructed by Dipper, James and Mathas. In lecture two, we explain Ginzburg and Lusztig's theory on affine Hecke algebras and its application to cyclotomic Hecke algebras due to the author. As Kazhdan-Lusztig bases of Hecke algebras desc…

Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras

We prove that cyclotomic Yokonuma--Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.

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…

On representations of cyclotomic Hecke algebras

An approach, based on Jucys--Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys--Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without use of the representation theory.

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…

Semisimple representations of alternating cyclotomic Hecke algebras

We define alternating cyclotomic Hecke algebras in higher levels as subalgebras of cyclotomic Hecke algebras under an analogue of Goldman's hash involution. We compute the rank of these algebras and construct a full set of irreducible representations in the semisimple case, generalising Mitsuhashi's results.

Cyclotomic double affine Hecke algebras and affine parabolic category O, I

Using the orbifold KZ connection we construct a functor from an affine parabolic category O of type A to the category O of a cyclotomic rational double affine Hecke algebra. We give several results concerning this functor.

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.

On the centre of the cyclotomic Hecke algebras of $G(m,1,2)$

We compute the centre of the cyclotomic Hecke algebra attached to $G(m,1,2)$ and show that if $q \neq 1$ it is equal to the image of the centre of the affine Hecke algebra $\Ha$. We also briefly discuss what is known about the relation between the centre of an arbitrary cyclotomic Hecke algebra and the centre of the affine Hecke algebra of type $A$.

Blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n)

We classify blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n) by using the "residue equivalence" for multi-partitions.

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…

循环Hecke代数的标准对称形式与Murphy基

Standard symmetrizing forms and Murphy bases of the cyclotomic Hecke algebras

本文研究了循环Hecke代数类型$G(\ell,1,n)$的标准对称形式和Murphy基，计算了其在不同类型的Hecke代数上的显式值，并推导了基元素间过渡矩阵的系数方程。

Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

A categorification of a cyclotomic Hecke algebra

We propose a categorification of the cyclotomic Hecke algebra in terms of the equivariant K-theory of the framed matrix factorizations. The construction generalizes the earlier construction of the authors for a categorification of the finite Hecke algebra of type A. We also explain why our construction provides a faithful realization of the Hecke algebras and discuss a geometric realization of the Jucys-Murphy subalgebra.

Positive Jantzen sum formulas for cyclotomic Hecke algebras

We prove a ``positive'' Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules $E^ν_{f,e}$, which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of $E^ν_{f,e}$ in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see …

On simple modules of cyclotomic quiver Hecke algebras of type A

We give a classification of the graded simple modules of cyclotomic quiver Hecke algebras of type A using the diagram calculus of the diagrammatic Cherednik algebra. We also obtain a non-trivial lower bound for the dimension of the simple modules which is independent of the field characteristic.

Schur elements and basic sets for cyclotomic Hecke algebras

We study the Schur elements and the a-function for cyclotomic Hecke algebras. As a consequence, we show the existence of canonical basic sets, as defined by Geck-Rouquier, for certain complex reflection groups. This includes the case of finite Weyl groups for all choices of parameters (in characteristic 0).

The number of simple modules for the Hecke algebras of type G(r,p,n) (with an appendix by Xiaoyi Cui)

We derive a parameterization of simple modules for the cyclotomic Hecke algebras of type $G(r,p,n)$ over field of any (coprime to $p$) characteristic. We give explicit formulas for the number of simple modules over these cyclotomic Hecke algebras.

Graded decomposition numbers for cyclotomic Hecke algebras

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.

退化分圆 Hecke 代数的单模之间的同构

Isomorphisms between simple modules of degenerate cyclotomic Hecke algebras

本文研究退化分圆 Hecke 代数单模间的关系，构造了由不同胞腔基定义的单模之间的显式同构，并得到退化情形下的广义 Mullineux 对合。

关于环状箭图Hecke超代数的RoCK块的维数

On dimensions of RoCK blocks of cyclotomic quiver Hecke superalgebras

本文显式计算环状箭图Hecke超代数的RoCK块幂等截断的维数，等价于计算扭曲仿射Kac-Moody李代数基本表示中Shapovalov形式的值。

任意Weyl群的分圆非退化Hecke代数

The cyclotomic non-degenerate Hecke algebras of arbitrary Weyl groups

该研究为任意Weyl群的非退化仿射Hecke代数构造KLR型生成元，建立了分圆Hecke代数块直和与KLR型代数分圆商的同构，明确了两类模分解的对应关系。

Centers of degenerate cyclotomic Hecke algebras and parabolic category O

We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type B_d(l) consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then deduce that the center of an integral block of parabolic category O for the Lie algebra gl_n(C) is generated by the center of its universal enveloping algebra.

分圆Hecke代数上的迹型与分圆Schur代数的余中心

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

本文在A型分圆Hecke代数上定义了统一的迹型，并构造了双基及分圆Schur代数余中心的显式基，证明其维数与基域、Hecke参数及分圆参数无关。

关于分圆 Hecke-Clifford 代数的表示理论

On representation theory of cyclotomic Hecke-Clifford algebras

本文在特征非2的代数闭域上，于特定参数条件下显式构造了非退化与退化分圆 Hecke-Clifford 超代数的单模，并通过维数比较给出其半单性的充分条件，进而证明一般情形下这些超代数是半单的。

The idempotents in cyclotomic Hecke algebras and periodic property of the Jucys-Murphy elements

This paper proves a periodic property of Jucys-Murphy elements of the degenerate and non-degenerate cy- clotomic Hecke algebras of type A. We do this by first giving a new closed formula for the KLR idempotents e(i) which, it tuns out, is very efficient computationally.

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.

Restricting Specht modules of cyclotomic Hecke algebras

This paper proves that the restriction of a Specht module for a (degenerate or non-degenerate) cyclotomic Hecke algebra, or KLR algebra, of type A has a Specht filtration.

与分圆Hecke代数相关的简单$B_4$表示

Simple $B_4$ representations associated to cyclotomic Hecke algebras

本文在生成元可对角化的非半单情形下，确定了复反射群$G_{25}$对应的分圆Hecke代数结构，分类了辫群$B_4$满足三次多项式的简单表示，并应用于$G_2$型辫子张量范畴的分类。

A note on degenerate cyclotomic Yokonuma-Hecke algebras

In this note, we first prove that the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}$ is cellular by constructing an explicit cellular basis. We then develop the fusion procedure for $Y_{r,n}^{d}$, that is, a complete set of pairwise orthogonal primitive idempotents for $Y_{r,n}^{d}$ is defined by consecutive evaluations of a certain rational function.

Cyclotomic quiver Hecke algebras corresponding to minuscule representations

In the paper, we give an explicit basis of the cyclotomic quiver Hecke algebra corresponding to a minuscule representation of finite type.

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

We associate a monoidal category $\mathcal{H}^λ$ to each dominant integral weight $λ$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $λ$. We show that, in the $\mathfrak{sl}_\infty$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of $\mathcal{H}^λ$, where $d$ is the level of $λ$. The categories $\mathcal{H}^λ$ can be viewed as a graphical calculus describing induction and restriction functors between categor…

On the semisimplicity of the cyclotomic quiver Hecke algebra of type C

We provide criteria for the cyclotomic quiver Hecke algebras of type C to be semisimple. In the semisimple case, we construct the irreducible modules.

Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions

We relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.

Blocks of cyclotomic Hecke algebras

This paper classifies the blocks of the cyclotomic Hecke algebras of type G(r,1,n) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of `Jantzen equivalence'. We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999.…

Rouquier blocks of the cyclotomic Hecke algebras of G(de,e,r)

The "Rouquier blocks" of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the "families of characters", defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series, G(de,e,r), thus completing their calculation for all complex reflection groups.

Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r,p,n) and the corresponding cyclotomic Hecke algebras H_{r,p,n}, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of sim…

On the cyclotomic Hecke algebras of complex reflection groups

Following the definition of Rouquier for the "families of characters" of a Weyl group and its generalization to the case of complex reflection groups, already appeared in the works of Broue-Kim and Malle-Rouquier, we show that these "families" depend on a numerical datum of the group, its "essential hyperplanes". Using this characterization, we provide the algorithm and the results of the determination of the Rouquier blocks of the cyclotomic Hecke algebras of all exceptional complex reflection groups.

Slim cyclotomic q-Schur algebras

We construct a new basis for a slim cyclotomic $q$-Schur algebra $\cysSr$ via symmetric polynomials in Jucys--Murphy operators of the cyclotomic Hecke algebra $\cysHr$. We show that this basis, labelled by matrices, is not the double coset basis when $\cysHr$ is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at $q=1$. As further applications, we then discuss the cyclotomic Schur--Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects.

循环Hecke-Clifford超代数的半正常基

Seminormal bases of cyclotomic Hecke-Clifford algebras

本文研究了循环Hecke-Clifford超代数的半正常基构造，通过标准生成元作用和原始幂等元集的显式构造，揭示了其在简单模中的结构特性。

循环Hecke代数块的交集

Intersections of blocks of cyclotomic Hecke algebras

本文证明了关于循环Hecke代数块交集的猜想，并推广到更广泛的反射群类别。

An isomorphism theorem for degenerate cyclotomic Yokonuma-Hecke algebras and applications

Inspired by the work [PA], we establish an explicit algebra isomorphism between the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ and a direct sum of matrix algebras over tensor products of degenerate cyclotomic Hecke algebras of type $A$. We then develop several applications of this result, including a new proof of the modular representation theory of $Y_{r,n}^{d}(q)$, a semisimplicity criterion for it and cellularity of it. Moreover, we prove that $Y_{r,n}^{d}(q)$ is a symmetric algebra and determine the associated Schur elements by using the isomorphism theorem for it.

Fusion procedure for Degenerate cyclotomic Hecke algebras

The primitive idempotents of the generic degenerate cycloctomic Hecke algebras are derived by consecutive evaluations of a certain rational function. This rational function depends only on the Specht modules and the normalization factors are the weights of the Brundan-Kleshchev trace.

关于 cyclotomic 有向图 Hecke-Clifford 超代数的广义分层细胞基

On the generalized graded cellular bases for cyclotomic quiver Hecke-Clifford superalgebras

本文构造了 cyclotomic 有向图 Hecke-Clifford 超代数的半单变形，推导了四种类型的 bi-重量空间维度公式，并引入了广义分层细胞超代数的概念，证明了四种类型的 cyclotomic 有向图 Hecke-Clifford 超代数属于广义分层细胞代数。

仿射类型C的高阶环形有理格子代数的表示类型

Representation type of higher level cyclotomic quiver Hecke algebras in affine type C

本文研究了仿射类型C的环形有理格子代数的表示类型，通过单元格性的莫里塔不变性，在特征不等于2的情况下明确描述了其基本代数。

Matrix units and Schur elements for the degenerate cyclotomic Hecke algebras

The paper uses the cellular basis of the (semi-simple) degenerate cyclotomic Hecke algebras to investigate these algebras exhaustively. As a consequence, we describe explicitly the "Young's seminormal form" and a orthogonal bases for Specht modules and determine explicitly the closed formula for the natural bilinear form on Specht modules and Schur elements for the degenerate cyclotomic Hekce algebras.

环状quiver Hecke代数 $A_\ell^{(1)}$ 型的表示类型

Representation type of cyclotomic quiver Hecke algebras of type $A_\ell^{(1)}$

本文构造了仿射A型支配极大权的连通quiver，并利用它完全确定了任意阶环状KLR代数的表示类型，从而完整分类了环状Hecke代数块的表示类型。

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.

Morita equivalences for cyclotomic Hecke algebras of type B and D

We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the representation theory of affine Hecke algebras of type B and D reduces to the study of their cyclotomic quotients with eigenvalues in a single orbit under multiplication by $q^2$ and inversion. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B an…

Categorifications and cyclotomic rational double affine Hecke algebras

Varagnolo and Vasserot conjectured an equivalence between the category O for CRDAHA's and a subcategory of an affine parabolic category O of type A. We prove this conjecture. As applications, we prove a conjecture of Rouquier on the dimension of simple modules of CRDAHA's and a conjecture of Chuang-Miyachi on the Koszul duality for the category O of CRDAHA's.

Finite dimensional Hecke algebras

These are notes prepared for ICRA workshop at Torun, Poland, August 2007. In the first part, we explain results on canonical basic sets by Geck and Jacon and propose a categorification framework which is suitable for our example of Hecke algebras. In the second part, we review basics of Kashiwara crystal and explain the Fock space theory of cyclotomic Hecke algebras and its applications. In the third part, we explain Rouquier's theory of quasihereditary covers of cyclotomic Hecke algebras. We add detailed explanation of the proofs here. The third part is based on my intensive course given at N…

Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)

We prove a Morita reduction theorem for the cyclotomic Hecke algebras H_{r,p,n}({q,Q})$ of type G(r,p,n). As a consequence, we show that computing the decomposition numbers of H_{r,p,n}(Q) reduces to computing the p'-splittable decomposition numbers of the cyclotomic Hecke algebras H_{r',p',n'}(Q'), where $1\le r'\le r$, $1\le n'\le n$, $ p'\mid p$ and where the parameters Q' are contained in a single $(ε,q)$-orbit and $ε$ is a primitive p'th root of unity.

On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2

Brundan and Kleshchev introduced graded decomposition numbers for representations of cyclotomic Hecke algebras of type $A$, which include group algebras of symmetric groups. Graded decomposition numbers are certain Laurent polynomials, whose values at 1 are the usual decomposition numbers. We show that in quantum characteristic 2 every such polynomial has non-zero coefficients either only in odd or only in even degrees. As a consequence, we find the first examples of graded decomposition numbers of symmetric groups with non-zero coefficients in some negative degrees.

关于类型$G(r,1,n)$的循环Hecke代数的余核

On the cocenter of the cyclotomic Hecke algebra of type $G(r,1,n)$

本文基于Geck和Pfeiffer的工作，构建了类型$G(r,1,n)$的循环Hecke代数$\mathscr{H}_{n,K}$的整数基，并证明其余核和中心的维度与基域特征、Hecke参数和循环参数无关。