Anthony Licata

Heisenberg categorification and Hilbert schemes

Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G whose Grothendieck group is isomorphic to an integral form of the Heisenberg algebra. We construct an action of H^G on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions of C^2/G.

Derived equivalences for cotangent bundles of Grassmannians via categorical SL2 actions

Abstract We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang–Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.

Categorical geometric skew Howe duality

AbstractWe categorify the R-matrix isomorphism between tensor products of minuscule representations of

COHERENT SHEAVES AND CATEGORICAL sl2 ACTIONS

U_{q}({\mathfrak{sl}}_{n})

Coherent sheaves on quiver varieties and categorification

by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of

Hecke algebras, finite general linear groups, and Heisenberg categorification

U_{q}({\mathfrak{sl}}_{2})

Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

which are related to representations of

Diagram automorphisms of quiver varieties

U_{q}({\mathfrak{sl}}_{n})

Braid group actions via categorified Heisenberg complexes

by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.

A graphical calculus for the Jack inner product on symmetric functions

We introduce the concept of a geometric categorical sl2 action and relate it to that of a strong categorical sl2 action. The latter is a special kind of 2-representation in the sense of Rouquier. The main result is that a geometric categorical sl2 action induces a strong categorical sl2 action. This allows one to apply the theory of strong sl2 actions to various geometric situations. Our main example is the construction of a geometric categorical sl2 action on the derived category of coherent sheaves on cotangent bundles of Grassmannians.