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Automorphisms of GKM graphs and regular semisimple Hessenberg varieties
Algebraic Geometry · Submitted 26 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $n$.
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The abstract, sentence by sentence
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 $X$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.07 | 0.02 | asserted 1.00 |
| s1 | 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 $n$. | result 0.99 | 0.97 | 0.91 | asserted 1.00 |
| s2 | 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 $X$. | result 0.99 | 0.95 | 0.06 | asserted 1.00 |
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Original 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 $X$.
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