Springer representations on the Khovanov Springer varieties

TL;DR

This paper demonstrates that Khovanov's Springer varieties admit a natural symmetric group action on their homology, resulting in an irreducible Springer representation, with explicit combinatorial and algebraic descriptions.

Contribution

It constructs a geometric realization of Springer representations on Khovanov varieties and proves their irreducibility with explicit basis identification.

Findings

The symmetric group acts naturally on the homology of Khovanov Springer varieties.

The Springer representation on these varieties is irreducible in each degree.

Explicit identification of the Kazhdan-Lusztig basis for the relevant irreducible representation.

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 $H_*(X_n)$. 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 $H_*(X_n)$ is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of $S_n$ corresponding to the partition $(n/2,n/2)$.