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Mikhail Borovoi
Mikhail Borovoi
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In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ bedenote the unique (up to conjugacy) elementary abelian $2$-subgroupsubgroup of maximal rank containedelements of order dividing 2 in the diagonal maximal torustorus; it is generated by the followingimages of the three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}$, $\begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix}$, $\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct$$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix},\quad \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\quad \begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}. $$ Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$$N_{G}(E)/C_{G}(E) \cong S_4\,$, which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculationcalculations. Is this observation true for any $n$? Is there any reference relevant?

In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}$, $\begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix}$, $\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant?

In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three matrices $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix},\quad \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\quad \begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}. $$ Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4\,$, which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculations. Is this observation true for any $n$? Is there any reference relevant?

`\operatorname`; deleted "Thank you!"
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LSpice
LSpice
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In the algebraic group $G = PGL_4(\mathbb{C})$$G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}$, $\begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix}$, $\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $PGL_{n}(\mathbb{C})$$\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!

In the algebraic group $G = PGL_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $PGL_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!

In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}$, $\begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix}$, $\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant?

added 31 characters in body
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user488802
user488802
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In the algebraic group $G = PGL_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $PGL_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!

In the algebraic group $G = PGL_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $PGL_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!

In the algebraic group $G = PGL_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank contained in the maximal torus generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}$. Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4$ which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $PGL_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!

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user488802
user488802
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