In a given coxeter group $(W,S)$, a reflection is an element of $W$ that can be written with a symmetric word in the generators $S$.

In multiple sources, I found the following formula $$ \mathrm{dp}(\alpha) = \frac{1}{2}(l(t_\alpha) + 1) $$ where $\alpha$ is a positive root, $t_\alpha$ the corresponding reflection and the depth $\mathrm{dp}(\alpha)$ is the length of a shortest word $w$ such that $w\cdot \alpha$ is a negative root.

Assuming that any reflection has length achievable by a symmetric word, this formula is rather easy to check, but I couldn't find a proof for just this fact.

In X Fu's thesis, Lemma 1.3.19, the formula is proven but I'm looking for more elementary proof of this fact:

Question: Is the length of a reflection in a Coxeter group achievable by a symmetric word?

In a given coxeter group $(W,S)$, a reflection is an element of $W$ that can be written with a symmetric word in the generators $S$.

In multiple sources, I found the following formula: $$ \mathrm{dp}(\alpha) = \frac{1}{2}(l(t_\alpha) + 1) $$ where $\alpha$ is a positive root, $t_\alpha$ the corresponding reflection and the depth $\mathrm{dp}(\alpha)$ is the length of a shortest word $w$ such that $w\cdot \alpha$ is a negative root.

Assuming that any reflection has length achievable by a symmetric word, this formula is rather easy to check, but I couldn't find a proof for just this fact.

In X Fu's thesis, Lemma 1.3.19, the formula is proven but I'm looking for a more elementary proof of this fact:

Question: Is the length of a reflection in a Coxeter group achievable by a symmetric word?