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YCor
YCor
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Dynkin Automorphismautomorphism of $SO_$\mathrm{SO}_{2n}$

Let $\sigma$ be a Dynkin automorphism of $G=SO_{2n}$$G=\mathrm{SO}_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried $\sigma:A \mapsto XAX$ where $$X= \begin{pmatrix} -1 & 0 & 0 & \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}. $$ Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?

Dynkin Automorphism of $SO_{2n}$

Let $\sigma$ be a Dynkin automorphism of $G=SO_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried $\sigma:A \mapsto XAX$ where $$X= \begin{pmatrix} -1 & 0 & 0 & \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}. $$ Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?

Dynkin automorphism of $\mathrm{SO}_{2n}$

Let $\sigma$ be a Dynkin automorphism of $G=\mathrm{SO}_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried $\sigma:A \mapsto XAX$ where $$X= \begin{pmatrix} -1 & 0 & 0 & \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}. $$ Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?

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Kurodachi
Kurodachi
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Dynkin Automorphism of $SO_{2n}$

Let $\sigma$ be a Dynkin automorphism of $G=SO_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried $\sigma:A \mapsto XAX$ where $$X= \begin{pmatrix} -1 & 0 & 0 & \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}. $$ Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?