580 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
Hecke algebras, finite general linear groups, and Heisenberg categorification
We define a category of planar diagrams whose Grothendieck group contains an
integral version of the infinite rank Heisenberg algebra, thus yielding a
categorification of this algebra. Our category, which is a q-deformation of one
defined by Khovanov, acts naturally on the categories of modules for Hecke
algebras of type A and finite general linear groups. In this way, we obtain a
categorification of the bosonic Fock space. We also develop the theory of
parabolic induction and restriction functors for finite groups and prove
general results on biadjointness and cyclicity in this setting.Comment: 46 pages, many figures; v2: some formulas corrected and additional
explanations added; v3: minor typos corrected and section numbering changed
to match published version; v4: fixed problem of missing arrows on strands in
v
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
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
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