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Let $G$ be a pinned split reductive group. There exists a Chevalley system:

For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding root subgroup, and for each element $w$ of the Weyl group (of the root system) there are lifts $\dot{w}$, such that $\dot{w} x_b(c) \dot{w}^{-1}$ = $x_{w(b)} (\pm c)$.

Are there any explicit statements about these signs? Do they have any plain coherence properties?

(There are related questions on MO regarding signs in the commutator formula for root subgroups and signs in the Chevalley basis for Lie algebras; I wondered whether perhaps something else is known here.)

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One usually defines $w_\alpha(\varepsilon) = x_\alpha(\varepsilon) \cdot x_{-\alpha}(-\varepsilon^{-1}) \cdot x_\alpha(\varepsilon)$, which is the image of $$\begin{pmatrix} 0 & \varepsilon \\ -\varepsilon^{-1} & 0 \end{pmatrix}$$ under the mapping of $\operatorname{SL}_2\to G$ defined by $x_\alpha$ and $x_{-\alpha}$. Then there are relations (sometimes called Steinberg relations) on $x_\alpha$, $w_\alpha$ and $h_\alpha(\varepsilon)=w_\alpha(\varepsilon)w_\alpha(1)^{-1}$, among them $$w_\alpha(\varepsilon) \cdot x_\beta(\xi) \cdot w_\alpha(\varepsilon)^{-1} = x_{w_\alpha\beta}\left(\eta_{\alpha\beta} \cdot \varepsilon^{-\langle\beta,\alpha\rangle} \cdot \xi\right).$$ Here the numbers $\eta_{\alpha\beta}=\pm1$ satisfy the following properties: $$\eta_{\alpha\beta}=\eta_{\alpha,-\beta}, \qquad \eta_{\alpha\alpha} = -1, \qquad \eta_{\alpha\beta}\eta_{\alpha,w_\alpha\beta}=(-1)^{A_{\alpha\beta}},$$ where $A_{\alpha\beta} = 2(\alpha,\beta)/(\alpha,\alpha)$ are the Cartan integers. Moreover,

  • $\eta_{\alpha\beta}=1$ if $\alpha\pm\beta\neq0$ and $\alpha\pm\beta\notin\Phi$ (in this case $x_\beta$ commutes with $x_\alpha$ and $x_{-\alpha}$);
  • $\eta_{\alpha\beta} = -\eta_{\beta\alpha}$ if $\langle \alpha,\beta\rangle=\langle\beta,\alpha\rangle=-1$ (the case when $\alpha,\beta$ are the fundamental roots for an $\mathsf{A}_2$ subsystem);
  • $\eta_{\alpha\beta}=-1$ if $\langle \alpha,\beta\rangle=0$ and $\alpha\pm\beta\in\Phi$ (the case when $\alpha,\beta$ are two orthogonal short roots of a $\mathsf{C}_2$ subsystem).

When you look at a lift of an arbitrary $w\in W(\Phi)$, the result depends on a particular choice of lifts for $w_\alpha$, which, in turn, depends on the choice of $x_\alpha$.

The numbers $\eta_{\alpha\beta}$ can be expressed in terms of the structure constants $N_{\alpha\beta ij}$ (the expression is usually very simple, but in a few cases involves also the lengths of a certain root chain). It is more complicated in $\mathsf{G}_2$, and in my experience in this case it is much easier to just fix some signs in an explicitly chosen matrix representation and work with that.

The standard reference for all this is "Simple groups of Lie type" by R. W. Carter (Proposition 6.4.3). Since this book is not easy to get legally, you can also take a look at Chevalley groups over commutative rings: I. Elementary calculations by N. Vavilov and E. Plotkin (Section 13).

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  • $\begingroup$ Thank you! I'm confused right now - in this section it also says "if they are orthogonal, then $\eta_{\alpha \beta} = 1$". Doesn't that contradict the final formula you wrote (which is written in the same section)? $\endgroup$
    – Elle
    May 2, 2021 at 17:38
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    $\begingroup$ @Elle This is an inaccuracy, they definnitely mean the case when $\alpha$ and $\beta$ are of the same length. The relation for orthogonal short roots $\alpha$ and $\beta$ in $\mathsf{C}_2$ can be checked directly in $\operatorname{Sp}_4$ by setting, say, $x_\alpha(\xi)=I+\xi e_{1,2}-\xi e_{3,4}$, $x_\beta(\xi)=I+\xi e_{1,3}+\xi e_{2,3}$ and $x_{-\beta}(\xi)=x_\beta(\xi)^\top$. $\endgroup$
    – Andrei Smolensky
    May 2, 2021 at 18:53
  • $\begingroup$ Sorry I don't follow, the statements contradict each other regardless of root length? $\endgroup$
    – Elle
    May 2, 2021 at 19:50
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    $\begingroup$ Right, I was talking about the long roots. For a pair of orthogonal roots the sign of $\eta_{\alpha,\beta}$ depends on whether $\alpha\pm\beta$ is a root. The sum of two orthogonal roots is a root only if they form a pair of short roots in $\mathsf{C}_2$. The phrasing "if they are orthogonal" is clearly a mistake, because they then consider the case $|\alpha|=|\beta|$ and $\alpha+\beta\in\Phi\neq\mathsf{G}_2$, which includes the case $\mathsf{C}_2$ (compare also with Table II on p.96). I would guess that they meant "if they are strictly orthogonal". $\endgroup$
    – Andrei Smolensky
    May 2, 2021 at 20:14
  • $\begingroup$ Okay, thanks. I'm also confused about the second formula you wrote; let's consider type $A_2$ and write $x_1$ for $x_{\alpha_1}(\xi)$, etc. Then the simple reflection $w_1$ is constructed out of $x_1$ and $x_{-1}$. According to your formula, the sign for $x_{12}$ in $w_1 x_2 w_{1}^{-1}$ is now determined. But it seems to me that there is still a free choice for sign in the Chevalley base construction of $x_{12}$? $\endgroup$
    – Elle
    May 3, 2021 at 16:21

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