We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's "special" bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.
We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …
Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq h$, the Coxeter number of $G$. We observe an easy `recursion formula' for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a `duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with $p$-regular …
Soergel bimodule category B is a categorification of the Hecke algebra of a Coxeter system (W,S). We find a presentation of B (as a tensor category) by generators and relations when W is a right-angled Coxeter group.
We study the projective objects in an exact category naturally associated to a Coxeter system. We discuss an analog of the Kazhdan-Lusztig conjecture and show how it follows from a "genericity" conjecture and how the latter follows from a "Hard Lefschetz" conjecture.
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we…
It is shown that there is an order isomorphism $ϕ'$ from the poset $V$ of $B\times B$-orbits on the wonderful compactification of a semi-simple adjoint group $G$ with Weyl group $W$ to an interval in reverse Chevalley-Bruhat order on a non-canonically associated Coxeter group $\hat{W}$ (in general neither finite nor affine). Moreover, $ϕ'$ preserves the corresponding Kazhdan-Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan-Lusztig polynomials for the analogues of $V$ for general Coxeter groups $W$ is completed by reducing it by a similar order isomorphism to known r…
In this paper, we study the Coxeter transformation of the derived categories of coherent sheaves on smooth complete varieties. We first obtain that if the rank of the Grothendieck group is finite, say $m$, then its characteristic polynomials is $(λ+(-1)^{n})^m$, where $n$ is dimension of the variety. We then study the Jordan canonical forms of the Coxeter transformations for rational surfaces, smooth complete toric varieties with ample canonical or anticanonical bundles, and prove that their Jordan canonical forms can determine and be determined by the Betti numbers of these varieties. As an a…
We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the equivalences between $d-$cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1…
The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander--Reiten sequences and study the Auslander--Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the assoc…
We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the "truncated simple reflections" on the set of almost positive roots $Φ_{\ge -1}$ associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simply-laced or non-simply-laced). This confirms th…
Let g be a symmetrizable Kac-Moody algebra and U_h(g) its quantized enveloping algebra. The quantum Weyl group operators of U_h(g) and the universal R-matrices of its Levi subalgebras endow U_h(g) with a natural quasi-Coxeter quasitriangular quasibialgebra structure which underlies the action of the braid group of g and Artin's braid groups on the tensor product of integrable, category O modules. We show that this structure can be transferred to the universal enveloping algebra Ug[[h]]. The proof relies on a modification of the Etingof-Kazhdan quantization functor, and yields an isomorphism be…
We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac-Moody algebra U_h(g), together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter struc…
The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La catégorie B de Soergel d'un système de Coxeter (W,S) est une catégorie de bimodules sur une algèbre de polynômes sur laquelle W agit. C'est une catégorification de l'algèbre de Hecke de (W,S). Dans cet article…
We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T' form a basis of the Grothendieck group of T and that, if T and T' are related by a mutation, then the indices with respect to T and T' are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras …
We determine for which Coxeter types the associated small quotient of the $2$-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive $2$-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive $2$-representations. For the small quotients of general Coxeter types, we give a description for the cell $2$-representations.
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when it is $\Hom$-finite. By definition, the cluster categories of \cite{Bua} are a special case. In this paper we show that a large class of 2-Calabi-Yau triangulated categories, including those associated with elements in Coxeter groups from \cite{Bua2}, are triangle equivalent to generalized cluster categories. This was already shown for some special elements in \cite{Ami3}.
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 investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of …
In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{Λ_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the 2-Auslander algebras introduced in \cite{I1}. Generalizing work in \cite{GLS1} we show that they are quasihereditary, even strongly quasihereditary in the sense of \cite{R}. We also describe the cluster tilting object giving rise to the Ringel dual, and prove that there is a duality between $\Sub{Λ_w}$ and the category $\mathcal{F}(Δ)$ of good modules over the quasihereditary al…
We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign (l,0,2) and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group.
We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over th…
We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver $…
Let A be a dg category, F:A->A a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/F the associated dg orbit category. For every A1-homotopy invariant (e.g. homotopy K-theory, K-theory with coefficients, etale K-theory and periodic cyclic homology), we construct a distinguished triangle expressing E(A/F) as the cone of the endomorphism E(F)-Id of E(A). In the particular case where F is the identity dg functor, this triangle splits and gives rise to the fundamental theorem. As a first application, we compute the A1-homotopy invariants of cluster (dg) categories, …
For a tuple of square complex-valued $N\times N$ matrices $A_1,\dots,A_n$ the determinant of their linear combination $x_1A_1+\cdots +x_nA_n$, which is called \textit{a pencil}, is a homogeneous polynomial of degree $N$ in $\C[x_1,...x_n]$. Zero-set of this polynomial is an algebraic set in the projective space $\C\Po^{n-1}$. This set is called the determinantal hypersurface or determinantal manifold of the tuple $(A_1,...,A_n)$. It was shown in Cuckovic, Stessin, Tchernev (2021) that if $G$ is a non-special Coxeter group of type $A,B$, or $D$, $ρ_1$ and $ρ_2$ are two linear representations of…
