Webs and smooth components of two column Springer fibers

TL;DR

This paper explores the relationship between webs and the geometry of two column Springer fibers, providing a characterization of smooth components and their invariance properties, thus extending previous work on two row cases.

Contribution

It introduces a new correspondence between webs and smooth components of two column Springer fibers, expanding the understanding of their geometric and combinatorial structure.

Findings

Characterization of smooth components of two column Springer fibers

Description of the geometry of these smooth components

Invariance of Poincaré polynomial under dihedral action

Abstract

Webs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of $\mathfrak{sl}_k$, and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung's 1997 thesis gave the first evidence of a connection between $\mathfrak{sl}_2$ webs and Springer fibers, showing that webs naturally index and describe the components of certain "two row" Springer fibers. However, this case is known to be far from generic. This paper deepens this connection with a similar correspondence in the substantially more complicated "two column" case. In particular, and building on works of Fresse, Melnikov, and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components. We also show that the Poincar\'e polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs.