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Springer representations on the Khovanov Springer varieties

Published online by Cambridge University Press:  09 March 2011

HEATHER M. RUSSELL  and
JULIANNA S. TYMOCZKO
HEATHER M. RUSSELL
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail: hrussell@math.lsu.edu
JULIANNA S. TYMOCZKO
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A. e-mail: tymoczko@math.uiowa.edu
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Abstract

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).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 1 , July 2011 , pp. 59 - 81
Copyright
Copyright © Cambridge Philosophical Society 2011

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