Glasgow Mathematical Journal

Table of Contents - May 2012 - Volume 54, Issue 02  

Research Article

A GRAPHICAL CALCULUS FOR 2-BLOCK SPALTENSTEIN VARIETIES

GISA SCHÄFERa1

a1 University of Bonn, Mathematikzentrum, Endenicher Allee 60, 53115 Bonn, Germany e-mail: gschaefe@uni-bonn.de

Abstract

We generalise statements known about Springer fibres associated to nilpotents with two Jordan blocks to Spaltenstein varieties. We study the geometry of generalised irreducible components (i.e. Bialynicki-Birula cells) and their pairwise intersections. In particular, we develop a graphical calculus that encodes their structure as iterated fibre bundles with ℂℙ1 as base spaces, and compute their cohomology. At the end, we present a connection with coloured cobordisms generalising the construction of Khovanov (M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000), 359–426) and Stroppel (C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Mathematica 145(4) (2009), 954–992).

(Received March 30 2011)

(Revised October 02 2011)

(Accepted October 16 2011)

2010 Mathematics Subject Classification

  • Primary: 14M15;
  • Secondary: 17B10;
  • 18D10;
  • 16S38