Mathematical Proceedings of the Cambridge Philosophical Society
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).
Russell, Heather M. and Tymoczko, Julianna S., "Springer Representations on the Khovanov Springer Varieties" (2011). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.