Kun Wang, Haitao Ma, Zhu-Jun Zheng
In this paper we introduce the notion of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence associated with Hecke algebras is representation stable.
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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…
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…
The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.
The spin analogues of several classical concepts and results for Hecke algebras are established. A Frobenius type formula is obtained for irreducible characters of the Hecke-Clifford algebra. A precise characterization of the trace functions allows us to define the character table for the algebra. The algebra is endowed with a canonical symmetrizing trace form, with respect to which the spin generic degrees are formulated and shown to coincide with the spin fake degrees. We further provide a characterization of the trace functions and the symmetrizing trace form on the spin Hecke algebra which…
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.
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.
We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are establishe…
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
We generalize graded Hecke algebras to include a twisting two-cocycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
We prove that the double affine Hecke algebra of type A is Morita equivalent to the quantized affine Schur algebra.
Affine Hecke algebras and generalisations of quiver Hecke algebras for type B
本文定义并研究B型仿射Hecke代数的分圆商，通过构造A型同构族并选取特定同构，建立了其块的直和与B型分圆箭图Hecke代数推广之间的同构。
Based on a recent result of Mathas and the author, we prove that Uno's conjecture on representation types of Hecke algebras is true for all Hecke algebras of classical type.
Affine Hecke algebras of type D and generalisations of quiver Hecke algebras
本文定义并研究D型仿射Hecke代数的分圆商，通过将其与分圆箭图Hecke代数的推广建立同构，补全了经典型仿射Hecke代数与箭图Hecke代数之间的同构族。
This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.
We categorify the Hecke algebra with parameters 1 and v using a variation of the category of Soergel bimodules.
In this article, we define an algebraic version of the Knizhnik--Zamolodchikov functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo--Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.
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 begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.
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.
In this paper we construct an analogue of Iwahori-Hecke algebras of SL_2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL_2, and prove that the product is well-defined. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori-Matsumoto type relations.
Hecke algebras and edge contractions
本文建立了一个从边收缩后的Coxeter系统所关联的Hecke代数到原始Coxeter系统所关联的Hecke代数的嵌入映射。
We prove the modular branching rule of the cyclotomic Hecke algebras, which has remained open.
In the previous paper "Symmetric Crystals and Affine Hecke Algebras of Type B", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$.
We establish bounds for the representation dimension of skew group algebras and wreath products. Using this, we obtain bounds for the representation dimension of a block of a Hecke algebra of type A, in terms of the weight of the block. This includes certain blocks of group algebras of symmetric groups.
Let $k$ be a field and suppose $p, q\in k$. We prove that the two affine Hecke algebras $H_q$ and $H_p$ of type $A_n$ are isomorphic as $k$-algebras if and only if $p=q^{\pm 1}$.
We establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We show that the Hall-Littlewood sypersymmetric function, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case and the classical super case.
In this paper, we introduce the sign q-permutation representation of the Iwahori-Hecke algebra on the tensor space of the graded vector space. We establish Schur-Weyl reciprocity between the quantum general super Lie algebra and the Iwahori-Hecke algebra. Our result includes Schur-Weyl reciprocity of both the super case and the quantum case, and unifies those two important cases.
We classify the finite dimensional representations of the double affine Hecke algebra of type $C^{\vee}C_1$ in the case when $q$ is not a root of 1.
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.
We review some facts about the representation theory of the Hecke algebra. We adapt for the Hecke algebra case the approach of Okounkov and Vershik which was developed for the representation theory of symmetric groups. We justify an explicit construction of the idempotents in the Hecke algebra in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented.
We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner type) irreducible representations of the Hecke algebra.
In this paper we give a new proof for the classification of irreducible modules of an affine Hecke algebra of type $A_n$, which was obtained by G. E. Murphy in 1995.
We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vignéras.
We introduce the notion of affinizations and R-matrices for arbitrary quiver Hekcke algebras. We show that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define the notion of a duality datum and construct a tensor functor between graded module categories of two quiver Hecke algebras. We give several examples of such functors .
We develop and collect techniques for determining Hochschild cohomology of skew group algebras S(V)#G and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups G(r,p,n) to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of S(V)#G and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of G on V, and we show that there ex…
We show that the center of the Iwahori--Hecke algebra of the symmetric group $S_n$ carries a natural filtered algebra structure, and that the structure constants of the associated graded algebra are independent of $n$. A series of conjectures and open problems are also included.
We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain "generic" Hecke algebra, infinite-dimensional in general, coming from the universal enveloping algebra of gl_n (or sl_n). The endomorphism algebras and the generic algebras are cellular. We give several equivalent descriptions of these algebras, find a number of explicit bases, and describe indexing sets for their irreducible representations.
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.
We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and most of the algebras of types B and D, we also establish upper bounds. Moreover, we establish bounds for the representation dimension of group algebras of some symmetric groups.
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 introduce a new class (in two versions) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov's degenerate affine Hecke-Clifford algebra and to a new degenerate affine Hecke algebra introduced here. We formulate a Morita equivalence between the two versions of rational DaHa's. The trigonometric generalization of the above constructions is also formulated and its relation to the rational counterpart is established.
This paper determines the representation type of the Iwahori-Hecke algebras of type B when $q\ne\pm1$. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's.
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…
This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra containing subalgebra B with character e. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A,B,e) naturally acts in the B-cohomology and B-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor…
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.
We determine a basis of the cocenter (i.e., the zeroth Hochschild homology) of the degenerate affine Hecke-Clifford and spin Hecke algebras in classical types.
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…
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…
q-rook monoid algebras, Hecke algebras and Schur-Weyl duality
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 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…
This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application we determine a splitting field for the alternating Hecke algebras in the semisimple case.
This article introduces the duplex Hecke algebra, which is an infinite dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate duplex Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups. We will study the finite dimensional natural representation of the duplex Hecke algebra on tensor space and prove that the duplex Hecke algebra forms a duality with the Levi type quantum group.
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of t…
The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.
For simply-laced quivers, we consider the fixed-point subalgebra of the quiver Hecke algebra under the homogeneous sign map. This leads to a new family of algebras we call alternating quiver Hecke algebras. We give a basis theorem and a presentation by generators and relations which is strikingly similar to the KLR presentation for quiver Hecke algebras.
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dime…
We define an imbedding of the Hecke algebra module carried by the involutions in a Weyl group W (defined by the author and Vogan) into a completion of the Hecke algebra. An analogous result is proved for any Coxeter group. A variant of the construction gives an imbedding of the biregular representation of a Hecke algebra of W in a completion of the Hecke algebra of W^2.
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