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LSpice
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Characterize an element of $SL_n$\operatorname{SL}_n(\mathbb Z)$

I'm trying to generalize a theorem on $SL_n(\mathbb Z)$$\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= \begin{pmatrix} 1 & 0 & \cdots & m\\ & 1 & & \vdots\\ & & 1 & 0\\ & & & 1\\ \end{pmatrix} $$$$e_{1,n}(m)= \begin{pmatrix} 1 & 0 & \cdots & m\\ & 1 & & \vdots\\ & & 1 & 0\\ & & & 1\\ \end{pmatrix}, $$
together with a general element from the Bruhat big cell.
I

I was wanderingwondering if there is a way to charectizecharacterize $e_{1,n}(1)$ in the general Chevalley groups over $\mathbb Z$?

Characterize an element of $SL_n(\mathbb Z)$

I'm trying to generalize a theorem on $SL_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= \begin{pmatrix} 1 & 0 & \cdots & m\\ & 1 & & \vdots\\ & & 1 & 0\\ & & & 1\\ \end{pmatrix} $$
together with a general element from the Bruhat big cell.
I was wandering if there is a way to charectize $e_{1,n}(1)$ in the general Chevalley groups over $\mathbb Z$?

Characterize an element of $\operatorname{SL}_n(\mathbb Z)$

I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= \begin{pmatrix} 1 & 0 & \cdots & m\\ & 1 & & \vdots\\ & & 1 & 0\\ & & & 1\\ \end{pmatrix}, $$
together with a general element from the Bruhat big cell.

I was wondering if there is a way to characterize $e_{1,n}(1)$ in the general Chevalley groups over $\mathbb Z$?

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Ami
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Characterize an element of $SL_n(\mathbb Z)$

I'm trying to generalize a theorem on $SL_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= \begin{pmatrix} 1 & 0 & \cdots & m\\ & 1 & & \vdots\\ & & 1 & 0\\ & & & 1\\ \end{pmatrix} $$
together with a general element from the Bruhat big cell.
I was wandering if there is a way to charectize $e_{1,n}(1)$ in the general Chevalley groups over $\mathbb Z$?