Oriented Schubert calculus in Chow-Witt rings of Grassmannians

TL;DR

This paper develops an oriented version of Schubert calculus using Chow-Witt rings of Grassmannians, providing detailed algebraic descriptions and applications involving quadratic form encodings of solution multiplicities.

Contribution

It introduces an oriented analogue of classical Schubert calculus with complete diagrammatic descriptions in Chow-Witt rings and twisted Witt groups, advancing arithmetic refinements.

Findings

Complete diagrammatic descriptions of ring structures

Arithmetic refinements with quadratic form encodings

Chow-Witt version of signed counts of subspaces

Abstract

We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and twisted Witt groups. In the resulting arithmetic refinements of Schubert calculus, the multiplicity of a solution subspace is a quadratic form encoding additional orientation information. We also discuss a couple of applications, such as a Chow-Witt version of the signed count of balanced subspaces of Feh\'er and Matszangosz.