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 describe decomposition numbers of quantum algebras at roots of unity, canonical bases of quantum affine algebras on integrable modules describe decomposition numbers of cyclotomic Hecke algebras at roots of unity. In particular, it turns out that crystal structure in the sense of Kashiwara exists behind the representation theory of cyclotomic Hecke algebras. This was first observed by Lascoux, Leclerc and Thibon. Its interpretation in terms of socle series was proposed by Rouquier. It is further developed in Vazirani's thesis. The special case of Hecke algebras of type A is extensively studied, and its relation to Lusztig conjecture is now very clear. We explain Varagnolo and Vasserot's work and several algorithms proposed so far. Its application to the decomposition numbers of finite general linear groups due to Dipper and James is also explained. The parametrization of simple modules of cyclotomic Hecke algebras over arbitrary fields in terms of crystal graphs of quantum affine algebras is given by Mathas and the author. In the third lecture, we show that this crystal graph description fits well in the Specht module theory. It uses work of Takemura and Uglov, Varagnolo and Vasserot.
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.
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).
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.
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.
We give a proof of a conjecture that Kleshchev multipartitions are those partitions which parametrize non-zero simple modules obtained as factor modules of Specht modules by their own radicals.
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.
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…
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 …
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.
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.
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.
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.
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…
In this paper we consider all possible generalizations of the B-type Hecke algebras, namely the cyclotomic and what we call 'generalized', and we construct Markov traces on each of them, so as to obtain all possible different levels of homfly-pt analogues in the solid torus related to the (Hecke) algebras of B-type.
The graded Specht module $S^λ$ for a cyclotomic Hecke algebra comes with a distinguished generating vector $z^λ\in S^λ$, which can be thought of as a "highest weight vector of weight $λ$". This paper describes the {\em defining relations} for the Specht module $S^λ$ as a graded module generated by $z^λ$. The first three relations say precisely what it means for $z^λ$ to be a highest weight vector of weight $λ$. The remaining relations are homogeneous analogues of the classical {\em Garnir relations}. The homogeneous Garnir relations, which are {\em simpler} than the classical ones, are associa…
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…
Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on a use of bases of these algebras which generalize a normal form for elements of the complex reflection groups $G(m,1,n)$, $m=1,2,\dots,\infty$, constructed by a recursive use of the Coxeter--Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented.
We clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the RSK construction for quiver Hecke algebras of type A, that was recently imported from the setting of representations of p-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules, that categorifies the Berenstein-Zelevinsky strings framework on quantum groups, and generalizes a graded variant of the classical Bernstein-Zelevinsky derivatives from the p-adic setting. Graded cyclotomic decomposition numbers are shown to be a special s…
We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are classified over an algebraically closed field of any characteristic p. The modular branching rules for these algebras are obtained, and when p does not divide the order of G, they are further identified with crystal graphs of integrable modules for quantum affine algebras. The key is to establish an equivalence between a module category of the (cyclotomic) wreath Hec…
The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with $(\eps,q)$-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type $G(s,1,m)$, where $1\le s\le r$ and $1\le m\le n$. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero. In proving these results, we develop a Spech…
These are the notes from a series of lectures the author gave at Harvard University in the Fall of 1994. The goal of these lectures is to give a self-contained exposition of recent result of Cherednik (\cite{C6}), who proved Macdonald's inner product conjectures (see \cite{M2}) for arbitrary root systems, using the algebraic structure he called ``double affine Hecke algebra''.
In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the "Hecke family"; on the other, we find generalizations of KLR algebras in finite and affine type A, the "KLR family." We show that these algebras have compatible isomorphisms generalizing those between Hecke and KLR algebras given by Brundan and Kleshchev. This allows us to organize a long list of algebras and categories into a single system, including (affine/cyclotomic) Hecke algebras, (affine/cyclo…
This chapter is based on a series of lectures that I gave at the National University of Singapore in April 2013. The notes survey the representation theory of the cyclotomic Hecke algebras of type A with an emphasis on understanding the KLR grading and the connections between the "classical" ungraded representation theory and the rapidly emerging graded theory. They are fairly self-contained and they try to give a leisurely introduction to these algebras, with many examples and calculations that don't appear elsewhere. We make extensive use of the interactions between the ungraded and graded r…
In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in $S^3$ carries a natural action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on 3 parameters $q, t_1, t_2$. As a consequence, for a knot $K$ satisfying this conjecture, we defined a three-variable polynomial invariant $J^K_n(q,t_1,t_2)$ generalizing the classical colored Jones polynomials $J^K_n(q)$. In this paper, we give explicit formulas and provide a quantum group interpretation for the generalized Jones polynomials $J^K_n(q,t_1,t_2)$. Our …
We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$ recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over $\C$ parametrized by a dominant weight of level $l$ for the root system of type $A_\infty$. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category $\mathcal{O}$ for the general linear…
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…
In this paper we categorify the Heisenberg action on the Fock space via the category O of cyclotomic rational double affine Hecke algebras. This permits us to relate the filtration by the support on the Grothendieck group of O to a representation theoretic grading defined using the Heisenberg action. This implies a recent conjecture of Etingof.
Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $p$-restricted partitions. We prove an analogue of this result in the very general setting of "Schur pairs". As an application we show that the socle of a Weyl module of a cyclotomic $q$-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type $A$. Finally, we prove a cyclotomic analogue of the Carter-Lusz…
----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman…
In this paper, we investigate the supercategories consisting of supermodules over quiver Hecke superalgebras and cyclotomic quiver Hecke superalgebras. We prove that these supercategories provide a supercategorification of a certain family of quantum superalgebras and their integrable highest weight modules. We show that, by taking a specialization, we obtain a supercategorification of quantum Kac-Moody superalgebras and their integrable highest weight modules.
In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the ($\imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Sch…
We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of…
Let $U$ be a quantized enveloping algebra. We consider the adjoint action of an $\mathfrak{sl}_2$-subalgebra of $U$ on a subalgebra of $U^+$ that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with $\mathfrak{sl}_2$ on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the cat…
This paper provides a combinatorial dictionary between three sets of objects: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and the irreducible modules of the affine Hecke algebra $H_n$ (for generic $q$). In particular, we compute the action of the crystal operator $\tilde{e}_i$ (a refinement of socle of Restriction) on an irreducible module both in terms of its parameterization by multisegments and by multipartitions. In other words, we give explicit crystal graph isomorphisms. A byproduct is the determination of which multisegments parameterize modules of the {\it cyclotomic…
We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively .
This paper has been withdrawn because of a gap in the proof of Lemma 3.10. The main reults in this paper have now been proved, and extended in the following papers: S. Ariki and A. Mathas, The number of simple modules of the Hecke algebras of type G(r,1,n) S. Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type G(m,1,n) and Kleshchev multipartitions
To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine wreath algebras. We study the structure theory of these new algebras and their natural cyclotomic quotients.
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
The paper an elementary introduction for non-specialists to the theory of quasi-invariants of Coxeter groups. The main object of study is the variety X_m of quasi-invariants for a finite Coxeter group, which arose in a work of O.Chalykh and A.Veselov about 10 years ago, as the spectral variety of the quantum Calogero-Moser system. Despite being singular, this variety has very nice properties (Cohen-Macaulay, Gorenstein, simplicity of the ring of differential operators, explicitly given Hilbert series). It is interesting that although the definition of X_m is completely elementary, to understan…
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a Quantum group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a p-adic group, are considered. We give a uniform geometric construction of these algebras in terms of homology of an appropriate …
