64 pages; numerous TikZ figures, PDF is preferable to DVI. v4: Revision in response to referee's report. Several proofs rewritten, examples and pictures added
发表信息：
Forum of Mathematics, Sigma , Volume 5 , 2017 , e27
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 representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category $\mathcal{O}$ in this case, with an explicit basis.
This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.
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.
H. Miyachi and W. Turner have independently proved that Broue's Abelian Defect Group Conjecture holds for certain unipotent blocks of the finite general linear group, the so-called Rouquier blocks. This together with A. Marcus and J. Chuang and R. Rouquier proves that the conjecture holds for all blocks of such groups. We prove that other finite classical groups also possess unipotent Rouquier blocks at linear primes.
Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise naturally in geometric representation theory and in modular representation theory of symmetric groups. A framework for studying categorical representations was introduced by Rouquier and Khovanov-Lauda. Their definitions are algebraic/diagrammatic, but are connected to the topology of quiver varieties by the work of Rouquier and Varagnolo-Vasserot. In this …
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…
We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztig's idempotent and integral quantum group $\…
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which generalizes James Conjecture. We also describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur-Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We …
We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we prove a conjecture describing the global dimension of these algebras.
In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(Λ)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ gives a categorification of $V(Λ)$.
We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O. We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier's deformati…
We prove the validity of the freeness conjecture of Broué, Malle and Rouquier for the generic Hecke algebras associated to the exceptional complex reflection groups of rank 2 belonging to the tetrahedral and octahedral families, and we give a description of the basis similar to the classical case of the finite Coxeter groups.
Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.
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.
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.
We initiate the study of Gaitsgory's central sheaves using complexes of Soergel bimodules, in extended affine type A. We conjecture that the complex associated to the standard representation can be flattened to a central complex in finite type A, which is the complex sought after in a conjecture of Gorsky-Negut-Rasmussen.
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 consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in case the biclosed set is an inversion set of an e…
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 define a quotient of the category of finitely generated modules over the cyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a module category structure over a direct sum of certain cyclotomic Khovanov-Lauda-Rouquier algebras of type An-1, this way categorifying the branching rules for the inclusion of sl(n) in sl(n+1). Using this we give a new, elementary proof of Khovanov-Lauda cyclotomic conjecture. We show that continuing recursively gives the Gelfand-Tsetlin basis for type An. As an application we prove a conjecture of Mackaay, Stosic and Vaz concerning categorical W…
In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra. One of our primary results is that these categories coincide when both are defined. We also inves…
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…
In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke categor…
We study equivalences for category O_p of the rational Cherednik algebras H_p of type G_l(n) = μ_l^n\rtimes S_n: a highest weight equivalence between O_p and O_{σ(p)} for σ\in S_l and an action of S_l on a non-empty Zariski open set of parameters p; a derived equivalence between O_p and O_{p'} whenever p and p' have integral difference; a highest weight equivalence between O_p and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.
Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for $W$ depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.
The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser fami…
We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types A_n and D_n, then we apply the folding method to D_{n+1} and A_{2n-1} to obtain types B_n and C_n. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of c-vectors for a general cluster algebra.
A Diagrammatic Category for the Representation Theory of U_q(sl_n)
本文部分回答了Greg Kuperberg在q-alg/9712003中提出的一个问题，以及Justin Roberts在"Problems on invariants of knots and 3-manifold"，math. GT/0406190中提出的问题12.18："能否通过生成元和关系描述量子群U_q（sl_n）的表示范畴？"对于每n个\geq 0，我定义一个三价图的张量范畴，模合痕，并从这个范畴构造一个函子到量子群U_q（sl_n）的表示范畴（的一个满子范畴）。人们希望通过提供生成器来完整地描述这个函子的内核。由此得到的图解范畴的商则是等价于U_q（sl_n）的表示范畴的范畴。我在这方面取得了重大进展，描述了内核的某些生成器，以及对进一步元素的一些障碍。这仍然是一个猜想，这些关系产生的内核。本文的结果推广了q-alg/9712003、MR1659228、math. QA/0310143和math. GT/0506403的结果。这个论证实质上是通过构造一个由U_q（sl_{n-1}）包含在U_q（sl_n}）中而得到的健忘函子的图解形式。我们知道这个函子是忠实的，所以一个图在n的核中，如果它在图遗忘函子下的像在n-1的核中。这允许我们执行归纳计算，既建立内核的元素族，又找到障碍物。
A conjecture by the second author, proven by Bonnafé-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra. In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank one property with respect to the Cartan matrix of $A$. Examples where the result is ap…
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…
Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras …
We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.
We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel in arXiv:math/0006056. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action o…
We construct a finite dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$, acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of $\mathcal{A}(B)$, viewed as mapping class group of the punctured disc -- hence our exposition can be seen as a type $B$ analogue of Khovanov-Seidel's work in arXiv:math/0006056v2. Moreover, we show that certain category of bimodule…
We give a diagrammatic presentation in terms of generators mod relations of the representation category of $U_q(\mathfrak{sl}_n)$. More precisely, we produce all the relations among $\rm{SL}_n$-webs, thus describing the full subcategory tensor-generated by fundamental representations $\bigwedge^k \mathbb{C}^n$ (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim [arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an application of q…
The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov-Finkelberg developed a derived version of this equivalence which relates the derived category of $ G^\vee$-equivariant constructible sheaves on $ Gr $ with the category of $G$-equivariant ${\mathcal O}(\mathfrak g)$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $ U_q \mathfrak g $. We d…
We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$ of quantum $\mathfrak{sp}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\mathbf{Web}(\mathfrak{sp}_6) \rightarrow \mathbf{FundRep}(U_q(\m…
