I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root string is at most 4:

Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$

where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}

The part of the proof I don't understand is when he writes that \begin{equation} W_{\alpha}(\beta + q\alpha) = \beta - p\alpha \quad (1) \end{equation} meaning that the reflection (in the hyperplane orthogonal to $\alpha$) reverses a root string. I read somewhere else that this is geometrically an obvious fact but I can't see how. I get that by reflecting through $\alpha$ we just add a positive or negative multiple of $\alpha$ to any root but I really cannot see why $(1)$ is true.

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root string is at most 4:

Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$

where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}

The part of the proof I don't understand is when they write that \begin{equation} W_{\alpha}(\beta + q\alpha) = \beta - p\alpha \quad (1) \end{equation} meaning that the reflection (in the hyperplane orthogonal to $\alpha$) reverses a root string. I read somewhere else that this is geometrically an obvious fact but I can't see how. I get that by reflecting through $\alpha$ we just add a positive or negative multiple of $\alpha$ to any root but I really cannot see why $(1)$ is true.

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root is at most 4:

Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$

where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}

The part of the proof I don't understand is when he writes that \begin{equation} W_{\alpha}(\beta + q\alpha) = \beta - p\alpha \quad (1) \end{equation} meaning that the reflection (in the hyperplane orthogonal to $\alpha$) reverses a root string. I read somewhere else that this is geometrically an obvious fact but I can't see how. I get that by reflecting through $\alpha$ we just add a positive or negative multiple of $\alpha$ to any root but I really cannot see why $(1)$ is true.

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root string is at most 4:

Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$

where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}

The part of the proof I don't understand is when he writes that \begin{equation} W_{\alpha}(\beta + q\alpha) = \beta - p\alpha \quad (1) \end{equation} meaning that the reflection (in the hyperplane orthogonal to $\alpha$) reverses a root string. I read somewhere else that this is geometrically an obvious fact but I can't see how. I get that by reflecting through $\alpha$ we just add a positive or negative multiple of $\alpha$ to any root but I really cannot see why $(1)$ is true.