Simple Reflections on Simple Roots I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a simple system in $\Pi$. I know that if $\alpha \in \Delta$, then $s_{\alpha}(\Pi \setminus \{\alpha \}) = \Pi \setminus \{\alpha \}$.
My first question is, is it also true that $s_{\alpha}(\Delta \setminus \{\alpha\}) = \Delta \setminus \{\alpha \}$?
I feel like it should be true; maybe it has to do with the fact that positive systems contain unique simple systems.
My second question, which can be answer affirmatively if the above has an affirmative answer, is, if $\alpha \in \Delta$ and $\beta \in \Pi \setminus \Delta$, and $s_{\alpha}$ denotes ethe reflection of the hyperplane with normal vector $\alpha$, is it true that $ht(\beta) > ht(s_{\alpha}(\beta))$? I feel like it should be true, and I need it for something else I'm trying to prove, but I can't quite figure it out.
Allow me to recall what $ht(\beta)$ is, which denotes the height of $\beta$. First, recall that $\Delta$ is a basis for $\text{span } \Phi$ such that every $\beta \in \Phi$, $\beta$ is a linear combination of $\Delta$ with coefficients all of the same sign. Given $\beta \in \text{span } \Phi$, $\beta$ has a unique decomposition as $\beta = \sum_{\gamma \in \Delta} c_{\gamma} \gamma$, where $c_{\gamma} \in \Bbb{R}$. Then $ht(\beta) := \sum_{\gamma \in \Delta} c_{\gamma}$ is well-defined.
EDIT Not so sure the first question has a positive answer anymore. However, here are some thoughts on the second question. Let $\beta = \sum_{\gamma \in \Delta} c_{\gamma} \gamma$. By the reflection formula, we have
$$s_{\alpha}(\beta) = \beta - \frac{2 \langle \beta, \alpha \rangle}{||\alpha||^2} \alpha$$
$$= \sum_{\gamma} c_{\gamma} \gamma - b_{\alpha, \beta} \alpha$$
$$= \sum_{\ gamma \neq \alpha} c_{\gamma} \gamma + (c_{\alpha} - b_{\alpha, \beta}) \alpha$$
and therefore
$$ht(s_{\alpha}(\beta)) = \sum_{\ gamma \neq \alpha} c_{\gamma} + c_{\alpha} - b_{\alpha, \beta}$$
$$= ht(\beta) - b_{\alpha, \beta}$$
Now, if we knew or could argue that $b_{\alpha, \beta} > 0$, then we'd be done. Is this possible? In other words, if $\alpha \in \Delta$ and $\beta \in \Pi \setminus \Delta$, is it true that $\langle \alpha, \beta \rangle > 0$?
 A: Neither question has an affirmative answer. Think about the root system of type $A_3$: this consists of the vectors
$$\pm \alpha_1 \quad \pm \alpha_2 \quad \pm \alpha_3 \quad \pm(\alpha_1+\alpha_2) \quad \pm(\alpha_2+\alpha_3) \quad \pm(\alpha_1+\alpha_2+\alpha_3)$$
where $$\alpha_1=(1,-1,0,0) \quad \alpha_2=(0,1,-1,0) \quad \text{and} \quad \alpha_3=(0,0,1,-1).$$ You may take $\Pi=\{\alpha_1,\alpha_2,\alpha_3,\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_1+\alpha_2+\alpha_3\}$ and $\Delta=\{\alpha_1,\alpha_2,\alpha_3\}$. Now compute: letting $s_1$ be the reflection for $\alpha_1$, check that it interchanges the first and second coordinates of a vector in $\mathbf{R}^4$. The highest root is $$\phi=\alpha_1+\alpha_2+\alpha_3=(1,0,0,-1)$$ and we have
$$s_1(\phi)=(0,1,0,-1)=\alpha_2+\alpha_3.$$ Thus if you take $\beta=\alpha_2+\alpha_3$ the height of $s_1(\beta)$ is greater than that of $\beta$. The obvious generalization of this example shows that the answer to your second question is no for every irreducible root system except $A_1$, $A_2$ and $B_2$.
