Two-row Delta Springer varieties

TL;DR

This paper explores the geometry and topology of two-row Delta Springer varieties, providing explicit descriptions of their irreducible components and extending symmetric group actions to degenerate affine Hecke algebra actions, connecting to symmetric functions and tensor spaces.

Contribution

It offers a combinatorial description of two-row Delta Springer varieties and extends their symmetry actions to a degenerate affine Hecke algebra, linking geometry to algebraic representations.

Findings

Explicit description of irreducible components

Comparison with ordinary and exotic Springer fibers

Extension of symmetric group action to degenerate affine Hecke algebra

Abstract

We study the geometry and topology of $\Delta$-Springer varieties associated with two-row partitions. These varieties were introduced in recent work by Griffin-Levinson-Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund-Remmel-Wilson. We provide an explicit and combinatorial description of the irreducible components of the two-row $\Delta$-Springer variety and compare it to the ordinary two-row Springer fiber as well as Kato's exotic Springer fiber corresponding to a one-row bipartition. In addition to that, we extend the action of the symmetric group on the homology of the two-row $\Delta$-Springer variety to an action of a degenerate affine Hecke algebra and relate this action to a $\mathfrak{gl}_{2}$-tensor space.