# Euler characteristics in the quantum $K$-theory of flag varieties

## TL;DR

This paper establishes a fundamental link between Euler characteristics and quantum $K$-theory of flag varieties, revealing that the sum of structure constants in Schubert class products equals one, with implications for rational curves.

## Contribution

It proves a new formula connecting sheaf Euler characteristics with degrees of rational curves in quantum $K$-theory of flag varieties, and characterizes minimal degrees via projections.

## Key findings

- Euler characteristic of product equals q^d
- Sum of structure constants in Schubert products equals 1
- Description of minimal degree d via projections

## Abstract

We prove that the sheaf Euler characteristic of the product of a Schubert class and an opposite Schubert class in the quantum $K$-theory ring of a (generalized) flag variety $G/P$ is equal to $q^d$, where $d$ is the smallest degree of a rational curve joining the two Schubert varieties. This implies that the sum of the structure constants of any product of Schubert classes is equal to 1. Along the way, we provide a description of the smallest degree $d$ in terms of its projections to flag varieties defined by maximal parabolic subgroups.