Automorphisms of GKM graphs and regular semisimple Hessenberg varieties

TL;DR

This paper characterizes the automorphism groups of regular semisimple Hessenberg varieties, revealing their structure as algebraic tori and relating their symmetries to symmetric groups, with implications for GKM manifolds.

Contribution

It provides a detailed description of the automorphism groups of Hessenberg varieties, connecting their reductive parts to algebraic tori and symmetric groups, and extends results to GKM manifolds.

Findings

The reductive part of the automorphism group is an algebraic torus of dimension n-1.

The quotient of the automorphism group by its connected component is related to symmetric groups.

Automorphism groups of projective GKM manifolds have finite automorphism group quotients.

Abstract

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the full flag variety $\mathrm{Fl}(\mathbb{C}^n)$ associated with a regular semisimple matrix $S$ of order $n$ and a function $h$ from $\{1,2,\dots,n\}$ to itself satisfying a certain condition. We show that when $\mathrm{Hess}(S,h)$ is connected and not the entire space $\mathrm{Fl}(\mathbb{C}^n)$, the reductive part of the identity component $\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ of the automorphism group $\mathrm{Aut}(\mathrm{Hess}(S,h))$ of $\mathrm{Hess}(S,h)$ is an algebraic torus of dimension $n-1$ and $\mathrm{Aut}(\mathrm{Hess}(S,h))/\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ is isomorphic to a subgroup of $\mathfrak{S}_n$ or $\mathfrak{S}_n\rtimes \{\pm 1\}$, where $\mathfrak{S}_n$ is the symmetric group of degree $n$. As a byproduct of our argument, we show that $\mathrm{Aut}(X)/\mathrm{Aut}^0(X)$ is a finite group for any projective GKM manifold $X$.