Nicolas Jacon
This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.
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We show the q-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $H_{n,q}$, if $(a_{λμ}^ν(n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $Γ_{λ,n}$ and $Γ_{μ,n}$, then each coefficient $a_{λμ}^ν(n,q)$ depends on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.
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 introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove tha…
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.
Inspired by the work of Rostam, we establish an explicit categorical equivalence between affine Yokonuma-Hecke algebras and quiver Hecke algebras associated to disjoint copies of quivers of (affine) type $A,$ generalizing Rouquier's categorical equivalence theorem.
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…
Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.
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 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.
We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type $G(p,p,n)$. This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003) 7-20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004) 446--490]. Furthermore, we use Naito--Sagaki's work [S. Naito & D. Sagaki, J. Algebra 251 (2002) 461--474] on Lakshmibai--Seshadri paths fixed by diagram automorphisms to derive explicit formu…
Let $W$ be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that $W$ is a wreath product of a symmetric group with a cyclic group of order $l$.
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 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…
In 1998 Broué, Malle and Rouquier conjectured that the Hecke algebra of a finite complex reflection group W is a free module over the algebra of parameters of rank |W|. We give an exposition of a proof of this conjecture in characteristic zero (and sufficiently large positive characteristic), due to I. Losev and I. Marin - G. Pfeiffer.
A conjecture of C. Bonnafé, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.
We construct irreducible representations of affine Khovanov-Lauda-Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the 'cuspidal modules' are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.
The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type $B$. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.
We consider the $ r = 0 $ case of the conjectures by Bonnafé, Geck, Iancu and Lam on cellular structures on the Hecke algebra of type $ B $. We show that this case induces the natural cell structure on the blob algebra $ b_n $ by restriction to one-line bipartitions.
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。
We calculate all decomposition matrices of the cyclotomic Hecke algebras of the rank 2 exceptional complex reflection groups in characteristic 0. We prove the existence of canonical basic sets in the sense of Geck-Rouquier and show that all modular irreducible representations can be lifted to the ordinary ones.
We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.
Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over $\Q[ξ]$ in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and Rouquier. In principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements. In this paper we find an explicit non-recursive expression for the coefficients appearing in these linear combinations. These coefficients are expressed in terms of readily computable numbers involving orders of symmetric groups and conjuga…
Between 1994 and 1998, the work of M. Broué, G. Malle, and R. Rouquier generalized in a natural way the definition of the Hecke algebra associated to a finite Coxeter group, for the case of an arbitrary complex reflection group. Attempting to also generalize the properties of the Coxeter case, they stated a number of conjectures concerning these Hecke algebras. One specific example of importance regarding those yet unsolved conjectures is the so-called BMR freeness conjecture. This conjecture is known to be true apart from 16 cases, that are almost all the exceptional groups of rank 2. These e…
Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Broué-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.
This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category ${\mathcal O}$, $W$-algebras, ...
In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions.
We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra $H$. These models enable concrete calculations inside $H$. For each of the eight groups, we compute an explicit integral basis of the center of $H$.
A new class of algebras have been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.
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.
This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012.
DG-Enhanced Hecke and KLR Algebras
本文构造了退化与非退化仿射Hecke代数的DG-增强版本，扩展了相关同构，证明其完备化后与KLR代数同构，并推导出其同调集中于零次且同构于分圆Hecke代数。
We review the state-of-the-art concerning the freeness conjecture stated in the 1990's by Broué, Malle and Rouquier for generic Hecke algebras associated to complex reflection groups, and in particular we expose in detail one of the main differences with the ordinary case, namely the lack of 0-Hecke algebras. We end the paper by proving a new case of this conjecture, the exceptional group called G26 in Shephard-Todd classification, namely the largest linear group of automorphisms of the Hessian configuration.
We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Broué, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group $G_{32}$, of order 155,520.
Soergel Calculus with patches
本文将Hecke范畴的图示表示适配到由Rouquier复数标准代表元构成的dg范畴中，借此恢复了辫群关系的范畴化与Rouquier公式等基本结果。
We verify the Broué-Malle-Rouquier (BMR) freeness for cyclotomic Hecke algebras associated with complex reflection groups $G_{17}$, $G_{18}$, $G_{19}$ in the Shephard-Todd classification. Together with results of Ariki, Ariki-Koike, Broué-Malle, Marin, Marin-Pfeiffer and Chavli, this settled affirmatively the BMR freeness conjecture. Our verification is inspired by Bergman's diamond lemma and requires 1GB memory and 5 days calculation on a PC.
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 give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra $\mathcal{H}$ whose category of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite-dimensional modules over an associated degenerate affine Hecke algebra. Moreover, unlike the traditional grading on degenerate affine Hecke algebras, this grading factors through central characters, and thus gives a grading to the irreducible representa…
Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We establish two generalizations of this result. On the one hand to the extension of the Hecke algebra associated to the normaliser of a reflection subgroup and on the other hand to the extension of the Hecke algebra by a lattice.
More than 10 years ago, Dipper, James and Murphy developped the theory of Specht modules for Hecke algebras of type $B\_n$. More recently, using Lusztig's a-function, Geck and Rouquier showed how to obtain parametrisations of the irreducible representations of Hecke algebras (of any finite type) in terms of so-called canonical basic sets. For certain values of the parameters in type $B\_n$, combinatorial descriptions of these basic sets were found by Jacon, based on work of Ariki and Foda-Leclerc-Okado-Thibon-Welsh. Here, we consider the canonical basic sets for all the remaining choices of th…
We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. This is done by considering the examples of rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to \cite[Problem 6]{Rouquier} stating that the restricted rational Cherednik algebra at the value $t=0$ is symmetric.
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 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 consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight $d$ of a symmetric group $\mathfrak S_n$ defined over a field $F$ of characteristic $e$ is Morita equivalent to the principal block of the wreath product $\mathfrak S_e \wr \mathfrak S_d$. This generalises a theorem of Chuang and Kessar that applies to RoCK blocks with abelian defect groups. Our proof relies crucially on an isomorphism between $F\mathfrak S_n$ and a cyclotomic Khovanov-Lauda-Rouq…
We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier an…
In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these $q$-Schur algebras as the duals of the $d$th graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the $q$-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representatio…
A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis is indexed by certain equivalence classes of the Coxeter group whose explicit description is rather complicated. Even their number is not obvious. We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. It follows that the maximal compositions of $n$ index the basis of the cent…
Let $H_k(W,q)$ be the Iwahori--Hecke algebra associated with a finite Weyl group $W$, where $k$ is a field and $0 \neq q \in k$. Assume that the characteristic of $k$ is not ``bad'' for $W$ and let $e$ be the smallest $i \geq 2$ such that $1+q+q^2+... +q^{i-1}=0$. We show that the number of simple $H_k(H,q)$-modules is ``generic'', i.e., it only depends on $e$. The proof uses some computations in the {\sf CHEVIE} package of {\sf GAP} and known results due to Dipper--James, Ariki--Mathas, Rouquier and the author.
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…
Using a characterization of a generalized $τ$-invariant for intermediate parameter Hecke algebras in type $B_n$, we verify a conjectural description of Kazhdan-Lusztig cells in this setting due to C. Bonnafé, L. Iancu, M. Geck, and T. Lam.
Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras
本文通过在 Borel-Moore 同调中引入 Grassmannian-Steinberg quiver flag 簇，将 Naisse-Vaz 扩展 KLR 代数几何实现为卷积代数，从而对 Verma 模进行几何范畴化，并为代数基定理提供了几何解释。
We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category $\mathcal{O}$; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the re…
Let $S_λ$ denote the Specht module defined by Dipper and James for the Iwahori-Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. When $e=2$ we determine the decomposability of all Specht modules corresponding to hook partitions $(a,1^b)$. We do so by utilising the Brundan-Kleshchev isomorphism between $\mathscr{H}$ and a Khovanov-Lauda-Rouquier algebra and working with the relevant KLR algebra, using the set-up of Kleshchev-Mathas-Ram. When $n$ is even, we easily arrive at the conclusion that $S_λ$ is indecomposable. When $n$ is odd, we find an endomorphism of $S_λ$ and us…
A celebrated result of Farahat and Higman constructs an algebra $\mathrm{FH}$ which "interpolates" the centres $Z(\mathbb{Z}S_n)$ of group algebras of the symmetric groups $S_n$. We extend these results from symmetric group algebras to type $A$ Iwahori-Hecke algebras, $H_n(q)$. In particular, we explain how to construct an algebra $\mathrm{FH}_q$ "interpolating" the centres $Z(H_n(q))$. We prove that $\mathrm{FH}_q$ is isomorphic to $\mathcal{R}[q,q^{-1}] \otimes_{\mathbb{Z}} Λ$ (where $\mathcal{R}$ is the ring of integer-valued polynomials, and $Λ$ is the ring of symmetric functions). The iso…
Skew cellularity of the Hecke algebras of type $G(\ell,p,n)$
本文引入分次斜胞代数并发展其 Clifford 理论，证明 $G(\ell,p,n)$ 型 Hecke 代数为分次斜胞代数，推广 Geck 的胞性结果，并应用于分解矩阵、单模分类及正特征调整矩阵。
Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of finite type A,D,E. When $J$ is the vector representation of $A^{(1)}_{n-1}$, we recover the affine Schur…
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 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 attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group …
Asymptotic Schur algebras and cellularity of q-Schur algebras
本文证明了有限型 q-Schur 代数的胞腔性，推广了 Geck 的定理，并研究了渐近 Schur 代数的特殊模与 Schur 代数的左胞腔表示，推广了 Lusztig 的相关工作。
We prove the freeness conjecture of Broue, Malle and Rouquier for the Hecke algebras associated to the primitive complex 2-reflection groups with a single conjugacy class of reflections.
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