Automorphisms of GKM graphs and regular semisimple Hessenberg varieties

2024/05/26 by Donghoon Jang, Jang, Donghoon, Kuroki, Shintarô +6
Mathematics · #14N15 (Secondary) #57S25 (Primary) 57S12 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2405.16399

openalex publication_date 2024/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A regular semisimple Hessenberg variety Hess(S,h) is a smooth subvariety of the full flag variety Fl(ℂⁿ) associated with a regular semisimple matrix S of order n and a function h from \1,2,…,n\ to itself satisfying a certain condition. We show that when Hess(S,h) is connected and not the entire space Fl(ℂⁿ), the reductive part of the identity component Aut⁰(Hess(S,h)) of the automorphism group Aut(Hess(S,h)) of Hess(S,h) is an algebraic torus of dimension n-1 and Aut(Hess(S,h))/Aut⁰(Hess(S,h)) is isomorphic to a subgroup of \mathfrakS_n or \mathfrakS_n\rtimes \± 1\, where \mathfrakS_n is the symmetric group of degree n. As a byproduct of our argument, we show that Aut(X)/Aut⁰(X) is a finite group for any projective GKM manifold X.

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