# Universal graph Schubert varieties

## TL;DR

This paper introduces a unified geometric framework for graph Schubert varieties and their variants, connecting them with back-stable Schubert polynomials and providing new formulas and interpretations.

## Contribution

It establishes that the classes of these loci are represented by back-stable Schubert polynomials and extends the framework to symmetric and skew-symmetric cases with Pfaffian formulas.

## Key findings

- Classes of graph Schubert varieties are represented by back-stable Schubert polynomials.
- Derived new Pfaffian formulas for symmetric and skew-symmetric cases.
- Provided geometric interpretations for involution Stanley symmetric functions.

## Abstract

We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks. Identifying an invertible linear map with its graph viewed as a point in a Grassmannian, we show that the closures of these loci have cohomology classes represented by the back-stable Schubert polynomials of Lam, Lee, and Shimozono. As a special case, we recover the result of Knutson, Lam, and Speyer that Stanley symmetric functions represent the classes of graph Schubert varieties. We consider similar loci where $f$ is restricted to be symmetric or skew-symmetric. Their classes are now given by back-stable versions of the polynomials introduced by Wyser and Yong to represent classes of orbit closures for the orthogonal and symplectic groups acting on the type A flag variety. Using degeneracy locus formulas of Kazarian and of Anderson and Fulton, we obtain new Pfaffian formulas for these polynomials in the vexillary case. We also give a geometric interpretation of the involution Stanley symmetric functions of Hamaker, Marberg, and the author: they represent classes of involution graph Schubert varieties in isotropic Grassmannians.