$\Delta$-Springer varieties and Hall-Littlewood polynomials

TL;DR

This paper establishes a positive Hall-Littlewood expansion for the cohomology of $ ext{Delta}$-Springer varieties, linking algebraic geometry, combinatorics, and representation theory, and providing geometric insights into the Delta Conjecture.

Contribution

It introduces a geometric approach to compute the Frobenius characteristic of $ ext{Delta}$-Springer varieties, connecting their cohomology to Hall-Littlewood polynomials and the Delta Conjecture.

Findings

Proved a positive Hall-Littlewood expansion formula for $ ext{Delta}$-Springer varieties.

Interpreted Frobenius characteristic via point counting over finite fields.

Provided geometric interpretation of a key formula in the Delta Conjecture at $t=0$.

Abstract

The $\Delta$-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo, and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall-Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta$-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb{F}_q$ and partitioning the $\Delta$-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades, and Shimozono for the Hall-Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.