521 research outputs found
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.Comment: 43 pages. In version 2, at the referee's suggestion, we slightly
expand some statements (Theorem 1.2 and Proposition 3.19) to include the
relevant affine varieties. This version is to appear in Advances in
Mathematic
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.Comment: 53 page
A survey of Heisenberg categorification via graphical calculus
In this expository paper we present an overview of various graphical
categorifications of the Heisenberg algebra and its Fock space representation.
We begin with a discussion of "weak" categorifications via modules for Hecke
algebras and "geometrizations" in terms of the cohomology of the Hilbert
scheme. We then turn our attention to more recent "strong" categorifications
involving planar diagrammatics and derived categories of coherent sheaves on
Hilbert schemes.Comment: 23 pages; v2: Some typos corrected and other minor improvements made;
v3: Some small errors corrected; v4: Code corrected to fix problem with
missing arrows on some diagram
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