Combinatorial realizations of crystals via torus actions on quiver varieties

Let V(λ) be a highest-weight representation of a symmetric Kac–Moody algebra, and let B(λ) be its crystal. There is a geometric realization of B(λ) using Nakajima’s quiver varieties. In many particular cases one can also realize B(λ) by elementary combinatorial methods. Here we study a general method of extracting combinatorial realizations from the geometric picture: we use Morse theory to index the irreducible components by connected components of the subvariety of fixed points for a certain torus action. We then discuss the case of $\widehat{\mathfrak{sl}}_{n}$ , where the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial realizations for each highest-weight crystal. In the case of B(Λ ₀) we recover a family of realizations which was recently constructed by Fayers. This gives a more conceptual proof of Fayers’ result as well as a generalization to higher level crystals. We also discuss a relationship with Nakajima’s monomial crystal.     

Translation quiver varieties

We introduce a framework of translation quiver varieties which includes Nakajima quiver varieties as well as their graded and cyclic versions. An important feature of translation quiver varieties is that the sets of their fixed points under toric actions can be again realized as translation quiver varieties. This allows one to simplify quiver varieties in several steps. We prove that translation quiver varieties are smooth, pure and have Tate motivic classes. We also describe an algorithm to compute those motivic classes.     

Crystal bases and quiver varieties

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.     

Desingularizations of quiver Grassmannians via graded quiver varieties

Inspired by recent work of Cerulli, Feigin and Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.     

Smoothing of quiver varieties

We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.     

Diagram automorphisms of quiver varieties

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.     

Coherent sheaves on quiver varieties and categorification

We construct geometric categorical $\mathfrak g $ actions on the derived category of coherent sheaves on Nakajima quiver varieties. These actions categorify Nakajima’s construction of Kac–Moody algebra representations on the K-theory of quiver varieties. We define an induced affine braid group action on these derived categories.     

Free group functors

We construct group functors whose Lie algebras are free.     

Sequences of reflection functors and the preprojective component of a valued quiver

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated with an indecomposable symmetrizable generalized Cartan matrix.     

Schur functors for the symplectic group

The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6).     

Additive group actions on Danielewski varieties and the cancellation problem

Given complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X  ×  and Y  ×  induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Sér. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the additive group on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

Reflection functors and symplectic reflection algebras for wreath products

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.     

Crystals, quiver varieties, and coboundary categories for Kac-Moody algebras

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.     

The braid group for a quiver with superpotential

We survey various generalizations of braid groups for quivers with superpotential and focus on the cluster braid groups, which are introduced in a joint work with King(2018). Our motivations come from the study of cluster algebras, Calabi-Yau categories and Bridgeland stability conditions.     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Abelian group actions on algebraic varieties with one fixed point

Dedicated to the memory of Ed Floyd with admiration and friendship