Irreducible components of two-column $\Delta$-Springer fibers

TL;DR

This paper studies the geometric structure of $ abla$-Springer fibers, proving smoothness of all irreducible components and describing their cohomology and intersection properties in detail.

Contribution

It establishes the smoothness of all irreducible components of specific $ abla$-Springer fibers and provides explicit descriptions of their cohomology rings and intersection combinatorics.

Findings

All irreducible components of $Y_{n,n-1}$ are smooth.

Intersections of components are smooth Hessenberg varieties.

Provides combinatorial formulas for Poincaré polynomials.

Abstract

The $\Delta$-Springer fibers $Y_{n,\lambda,s}$, introduced by Levinson, Woo, and the second author, generalize Springer fibers for $\mathrm{GL}_n(\mathbb{C})$ and give a geometric interpretation of the of the Delta Conjecture from algebraic combinatorics (at $t=0$). We prove that all irreducible components of the $\Delta$-Springer fiber $Y_{n,n-1}=Y_{n,(1^{n-1}),n-1}$ are smooth. In fact, we prove that any intersection of irreducible components of $Y_{n,n-1}$ is a smooth Hessenberg variety which has the structure of an iterated Grassmannian fiber bundle. We then give a presentation of the singular cohomology ring of each irreducible component of $Y_{n,n-1}$ and a combinatorial formula for the Poincar\'e polynomial of an arbitrary union of intersections of irreducible components in terms of arm and leg statistics on Dyck paths.