The Springer Geometry of Specht Polynomials and Schubert cycle positivity for two row Springer fiber components

Abstract

We show that the type A Springer representation is realized geometrically in the homology of the complete flag variety by Specht polynomials. For any partition, we identify the classical Specht polynomial generators of the Specht module with the classes of a family of disjoint Levi--Richardson varieties, and this family degenerates to the corresponding Springer fiber. This factors Springer's Schubert positivity problem for Springer fiber components through a chain of positive expansions, from Specht polynomials through the Joseph polynomials to the Schubert cycles. For two-row partitions we make each of these expansions combinatorially explicit, giving manifestly nonnegative Schubert cycle expansions of both the Levi-Richardson cycles and the Springer fiber components. This resolves Springer's question for two-row fibers and proves two conjectures of Precup and Sabando-Alvarez, and identifies the Springer basis with the web basis for two-row Specht modules. As an application, we deduce the Schubert cycle expansions of the components of the Poisson degeneracy locus of the flag variety.