Minimal bandwidth $\mathbb{C}^*$-actions on generalized Grassmannians

TL;DR

This paper investigates $ ext{C}^*$-actions on rational homogeneous spaces, identifying those with minimal bandwidth and linking this to fundamental weights and Dynkin diagrams, with applications to the Cayley plane's Chow ring.

Contribution

It characterizes minimal bandwidth $ ext{C}^*$-actions on generalized Grassmannians and relates them to Dynkin diagram coefficients, providing new insights into their geometric structure.

Findings

Minimal bandwidth corresponds to the smallest fundamental weight coefficient.

Identifies which rational homogeneous spaces admit minimal bandwidth $ ext{C}^*$-actions.

Analyzes the Chow ring of the Cayley plane $ ext{E}_6(6)$.

Abstract

The bandwidth of a $\mathbb{C}^*$-action of a polarized pair $(X,L)$ is a natural measure of its complexity. In this paper we study $\mathbb{C}^*$-actions on rational homogeneous spaces, determining which provide minimal bandwidth. We prove that the minimal bandwidth is linked to the smallest coefficient of the fundamental weight, in a base of simple roots, which describes the variety as a marked Dynkin diagram. As a direct application of the results we study the Chow ring of the Cayley plane $\mathrm{E}_6(6)$.