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1 results

  • Springer representations on the Khovanov Springer varieties

  • HEATHER M. RUSSELL, JULIANNA S. TYMOCZKO
  • Journal:
    Mathematical Proceedings of the Cambridge Philosophical Society / Volume 151 / Issue 1 / July 2011
    Published online by Cambridge University Press:
    09 March 2011, pp. 59-81
    Print publication:
    July 2011

  • Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H∗(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H∗(Xn) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of Sn corresponding to the partition (n/2, n/2).