The length $l(w)$ depends on the base $\Delta$ of the root system?

Question

Let $\Phi$ a root system and $W$ the Weyl group relative to basis $\Delta$. When $w \in W$ is written as $\sigma_{\alpha_1}\sigma_{\alpha_2} ... \sigma_{\alpha_t}$ ($\alpha_i \in \Delta$, $t$ minimal), we call the expression reduced, and write $l(w) = t$ this is the length of $w$, relative to $\Delta$.

The length $l(w)$ depends on the base $\Delta$ of the root system?

Answer 1

For sure it does. Let $\Delta := \lbrace \alpha_1, \alpha_2 \rbrace$ be a basis of a root system of type $A_2$. Then for example $\Delta' := \lbrace \beta_1 := \alpha_1+\alpha_2, \beta_2:= -\alpha_2 \rbrace$ is another basis.

The element $\sigma_{\alpha_1}\sigma_{\alpha_2}\sigma_{\alpha_1} = \sigma_{\beta_1}$ has length $3$ with respect to $\Delta$, but length $1$ with respect to $\Delta'$.