Springer representations on the Khovanov Springer varieties

Abstract

Springer varieties are studied because their cohomology carries a natural action of the symmetric group $S_n$ and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties $X_n$ as subvarieties of the product of spheres $(S^2{)}^n$. We show that if $X_n$ is embedded antipodally in $(S^2{)}^n$ then the natural $S_n$-action on $(S^2{)}^n$ induces an $S_n$-representation on the image of $H_{\ast }(X_n)$. 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 $H_{\ast }(X_n)$ is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of $S_n$ corresponding to the partition $(n/2,n/2)$.

References in corpus (2)

• Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
• A remark on the topology of (n,n) Springer varieties

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