A topological construction for all two-row Springer varieties

TL;DR

This paper extends Khovanov's topological construction to all two-row Springer varieties, providing a new homology basis and skein-theoretic formulation for their Springer representations.

Contribution

It introduces a comprehensive topological construction for all two-row Springer varieties, enhancing understanding of their homology and representation theory.

Findings

Constructed a homology basis for two-row Springer varieties

Developed a skein-theoretic formulation of Springer representations

Extended Khovanov's construction beyond the original case

Abstract

Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of $(n/2, n/2)$ Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic properties of this construction we provide a particularly useful homology basis and construct the Springer representation using this basis. We also provide a skein-theoretic formulation of the representation in this case.