Jonathan Brundan, Alexander Kleshchev
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.
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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.
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…
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.
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.
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.
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 $A$.
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.
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…
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。
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.
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.
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.
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).
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。
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。
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。
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.
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。
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。
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.
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.
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…
We provide criteria for the cyclotomic quiver Hecke algebras of type C to be semisimple. In the semisimple case, we construct the irreducible modules.
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.
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.
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…
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.
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。
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.
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。
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。
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。
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.
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…
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.
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.
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。
We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type $A$. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of typ…
Following the generalization of the notion of families of characters, defined by Lusztig for Weyl groups, to the case of complex reflection groups, thanks to the definition given by Rouquier, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the "families" of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups.
The families of characters, defined by Lusztig for Weyl groups, play an important role in the representation theory of finite reductive groups. The definition of Rouquier for the families of characters in terms of blocks of the Hecke algebra has made possible the generalization of this notion to the case of complex reflection groups. In this article, we study the Rouquier blocks of the cyclotomic Hecke algebras and we show that they depend on numerical data determined by the generic Hecke algebra. Moreover, we provide the algorithm and the results of their determination for all exceptional com…
We first give a direct proof of a basis theorem for the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q).$ Our approach follows Kleshchev's, which does not use the representation theory of $Y_{r,n}^{d}(q),$ and so it is very different from that of [ChP2]. We also present two applications. Then we prove that the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ is cellular by constructing an explicit cellular basis, and show that the Jucys-Murphy elements for $Y_{r,n}^{d}(q)$ are JM-elements in the abstract sense. In the appendix, we shall develop the fusion procedure for $Y_{r,n}^{d}(q).$
We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a "super-strong linkage principle" which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalise the notion of homomorphisms between Weyl/Specht modules which are "generically" placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebr…
Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $β$ in the positive root lattice with height $n$ and any integral dominant weight $Λ$, one can associate a quiver Hecke algebras $R_β(K)$ and its cyclotomic quotient $R_β^Λ(K)$ over $K$. It has been conjectured that the natural map from $R_β(K)$ to $R_β^Λ(K)$ maps the center of $R_β(K)$ surjectively onto the center of $R_β^Λ(K)$. A similar conjecture claims that the center of the affine Hecke algebra of type $A$ maps surjectively onto the center of its cyclotomic quotient---the cyclotomic Hecke alge…
We prove the modular branching rule of the cyclotomic Hecke algebras, which has remained open.
Defect in cyclotomic Hecke algebras
本文证明了亏格在无限系列复反射群及分圆Yokonuma-Hecke代数中是块不变量，并揭示了其与Fayers权概念及钩长推广的对应关系。
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.
An inductive approach to the representation theory of cyclotomic Hecke algebras, inspired by Okounkov and Vershik, is developed. We study the common spectrum of the Jucys-Murphy elements using representations of the simplest affine Hecke algebra. Representations are constructed with the help of a new associative algebra whose underlying vector space is the tensor product of the cyclotomic Hecke algebra with the free associative algebra generated by standard m-tableaux.
In the paper, we give an explicit basis of the cyclotomic quiver Hecke algebra corresponding to a minuscule representation of finite type.
We prove a super Frobenius formula for the characters of the cyclotomic Hecke algebras by applying the super Schur-Weyl reciprocity between the quantum superalgebras and cyclotomic Hecke algebras, which is a super analogue of the Frobenius formula in (T. Shoji, J. Algebra 226: 2000, 818--856) and a cyclotomic analogue of the super Frobeinus formula in (H. Mitsuhashi, Linear Multilinear Algebra 58: 2010, 941--955).
In this paper we derive a closed formula for the $(\mathbb{Z}\times\mathbb{Z}_2)$-graded dimension of the cyclotomic quiver Hecke superalgebra $R^Λ(β)$ associated to an {\it arbitrary} Cartan superdatum $(A,P,Π,Π^\vee)$, polynomials $(Q_{i,j}({\rm x}_1,{\rm x}_2))_{i,j\in I}$, $β\in Q_n^+$ and $Λ\in P^+$. As applications, we obtain a necessary and sufficient condition for which $e(ν)\neq 0$ in $R^Λ(β)$. We construct an explicit monomial basis for the bi-weight space $e(\widetildeν)R^Λ(β)e(\widetildeν)$, where $\widetildeν$ is a certain specific $n$-tuple defined in (1.4). In particular, this g…
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.…
We study the cocenter of the cyclotomic quiver Hecke algebra $R^Λ_α$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, $Λ\in P^+$ and $α\in Q_n^+$. We introduce a notion called "piecewise dominant sequence" and use it to construct some explicit homogeneous elements which span the maximal degree component of the cocenter of $R^Λ_α$. We show that the minimal degree components of the cocenter of $R^Λ_α$ is spanned by the image of some KLR idempotent $e(ν)$, where each $ν\in I^α$ is piecewise dominant. As an application, we show that the weight space $L(Λ)_{Λ-α}…
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.
This paper studies the seminormal bases $\{f_{\mathfrak{s}\mathfrak{t}}\}$ and the dual seminormal bases $\{g_{\mathfrak{s}\mathfrak{t}}\}$ of the non-degenerate and the degenerate cyclotomic Hecke algebras ${H}_{\ell,n}$ of type $G(\ell,1,n)$. We present some explicit formulae for the constants $α_{\mathfrak{s}\mathfrak{t}}:=g_{\mathfrak{s}\mathfrak{t}}/f_{\mathfrak{s}\mathfrak{t}}\in K^\times$ in terms of the $γ$-coefficients $\{γ_{\mathfrak{u}}, γ'_{\mathfrak{u}}\}$ of $H_{\ell,n}$. In particular, we answer a question of Mathas on the rationality of square roots of some quotients of product…
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。
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.
Representation type of cyclotomic quiver Hecke algebras of type $C^{(1)}_{\ell}$
针对仿射类型C，通过构造支配极大权连通箭图，完全确定了任意层级分圆KLR代数的表示型，并给出了部分表示有限与驯顺型代数的Morita等价类及分次分解矩阵。
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.
On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras
本文引入了超对称化超代数的 Schur 元素，证明了 cyclotomic Hecke-Clifford 代数在特定条件下具有超对称性或对称性，并计算了半单情况下的 Schur 元素，同时定义了新的对称化形式。
A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.
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