We generalize the geometric construction of quiver Hecke algebras from Varagnolo and Vasserot to a setup with arbitrary connected reductive groups. This corresponds to replacing quiver representations by generalized quiver representations introduced by Derksen and Weyman. The class of algebras which we construct contains (affine) nil Hecke algebras, skew group rings of Weyl groups with polynomial rings and quiver Hecke algebras. We describe an explicit faithful representation in a polynomial ring and we calculate the generators and relations for these algebras.
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.
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 introduce a family of quiver Hecke algebras which give a categorification of quantum Borcherds algebra associated to an arbitrary Borcherds-Cartan datum.
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 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…
We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of type $C_{\ell}^{(1)}$.
We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type A. We also give some related results on the center of quiver Hecke algebras and cohomology of quiver varieties.
We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $A^{(1)}_{\ell}$. Note that each finite quiver Hecke algebra lies in one parameter family, and the original Erdmann-Nakano theorem studied the finite quiver Hecke algebra at a special parameter value. We study the general case in our paper. Our result shows in particular that their representation type does not depend on the parameter. Moreover, when the parameter value is nonzero, we show that finite quiver Hecke algebras of tame representation type are biserial algebras.
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 give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $D^{(2)}_{\ell+1}$, which tells their representation type. If $R^{Λ_0}(β)$ is not of wild representation type, we may compute its stable Auslander-Reiten quiver.
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…
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.
We define and study RoCK blocks for double covers of symmetric groups. We prove that RoCK blocks of double covers are Morita equivalent to standard `local' blocks. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué's abelian defect group conjecture for symmetric groups. Indeed we prove Broué's conjecture for the RoCK blocks defined in this article. Our methods involve translation into the quiver Hecke superalgebras setting using the Kang-Kashiwara-Tsuchioka isomorphism and categorification m…
We construct characteristic-free bases and BGG resolutions of unitary simple modules of quiver Hecke algebras and Cherednik algebras. We hence solve and vastly generalise Berkesch-Griffeth-Sam's conjecture, calculate the Castelnuovo-Mumford regularity of ideals of of certain highly symmetric linear subspace arrangements, and compute Dirac homology of Cherednik algebras.
On the generalized graded cellular bases for cyclotomic quiver Hecke-Clifford superalgebras
School of Mathematics and Statistics Beijing Institute of Technology（北京理工大学数学与统计学院）；Academy of Mathematics and Systems Science Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Max-Planck-Institut für Mathematik（马克斯·普朗克数学研究所）
In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection $ π_{λ, μ} : V_{\mathbb{A}}(λ)^\vee \otimes_{\mathbb{A}} V_\mathbb{A}(μ)^\vee \twoheadrightarrow V_\mathbb{A}(λ+ μ)^\vee $ sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic $0$, we obtain a positivity condition on some coefficients associated with the proj…
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.
Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\infty}\gmod$ to $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.
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 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…
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}…
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.
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 recast the classical notion of standard tableaux in an alcove-geometric setting and extend these classical ideas to all reduced paths in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias--Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the …
Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cit…
We construct all the irreducible representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras $osp(1|2n),$ and show that their highest weights are given by the dominant words. We use the dominant Lyndon words to construct the cuspidal modules and show that the irreducible representations are the simple heads of standard representations constructed by induction from the cuspidal modules.
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.
We study cyclotomic quiver Hecke algebras $R^{Λ_0}(β)$ in type $A^{(2)}_{2\ell}$, where $Λ_0$ is the fundamental weight. The algebras are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras associated with the symmetric group, from the viewpoint of the Fock space theory developed by the first author and his collaborators. We give a formula for the dimension of the algebra, and a simple criterion to tell the representation type. The criterion is a natural generalization of Erdmann and Nakano's for the Iwahori-Hecke algebras. Except for the examples coming from cyclotomic Hecke alg…
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…
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…
We relate the representations of the rational Cherednik algebras associated with the complex reflection group G(m,1,n) to sheaves on Nakajima quiver varieties associated with extended Dynkin gaphs via a Z-algebra construction. As the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. We interpret the ordering on category O geometrically using this relationship; we also relate the geometry to the a-function for Hecke algebras with unequal parameters.
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 provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $\mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$. We construct the localization $\widetilde{\mathscr{C}_w}$ of $\mathscr{C}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization $\widetilde{\mathscr{C}_w}$ is left rigid and we expect that it is rigid.
We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in the quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon for the basis of the quantum unipotent coordinate ring $\mathcal{A}_q(\mathfrak{n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying GLS-clusters are Laurent families by using the PBW-decomposition vector of a simple module $X$ and categorical interpre…
We construct and investigate Specht modules $\mathcal{S}^λ$ for cyclotomic quiver Hecke algebras in type $C^{(1)}_\ell$ and $C_\infty$, which are labelled by multipartitions $λ$. It is shown that in type $C_\infty$, the Specht module $\mathcal{S}^λ$ has a homogeneous basis indexed by standard tableaux of shape $λ$, which yields a graded character formula and good properties with the exact functors $E_i^Λ$ and $F_i^Λ$. For type $C^{(1)}_\ell$, we propose a conjecture.
We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.
We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez-Leclerc's category $\mathcal{C}_Q$. This was previously proved in untwisted $ADE$ types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to…
This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit "integral" closed formula for the Gram determinants of the Specht modules in terms of the combinatorics which utilizes the KLR gradings. We then use seminormal forms to give a deformation of the KLR algebras of type $A$. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As…
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…
We show that for a finite-type Lie algebra $\mathfrak{g}$, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton-Okounkov bodies associated to the quantum coordinate rings $\Aqnw$. When $\mathfrak{g}$ is simply-laced, we use Kang-Kashiwara-Kim-Oh's monoidal categorification to investigate the cluster theory of these bodies. In particular, our construction yields a simplex $\ds$ for every seed $\s$ of $\Aqnw$. We exhibit various properties of these simplices by characterizing their rational points, normal fans, and volumes. As an applicat…
We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that we compute explicitly.
