Positive roots and elements in a Coxeter group.

Question

In the paper, a set $L$ associated to an element $w$ in a Coxeter group $W$ is defined as follows. Let $w=s_{i_1} \cdots s_{i_m}$ be a reduced expression. Define $L=\{\beta_1, \ldots, \beta_m\}$, where \begin{align} & \beta_1 = \alpha_{i_1}, \\ & \beta_k = s_{i_1} \cdots s_{i_{k-1}}(\alpha_{i_k}), k=2,3,\ldots,m. \end{align} The set $L$ is denoted by $R_w$. If $w$ is reduced, $\beta_1, \ldots, \beta_m$ are different from each other. I think that the following is true. If $w=s_{i_1} \cdots s_{i_m}$ is not reduced, then there are two elements $\alpha, -\alpha$ in $\beta_1, \ldots, \beta_m$ whose sum is $0$. Is this true? Are there some references about this?

In the case of type $A_2$, for $w= s_1 s_2 s_1 s_2$ which is not reduced, we have \begin{align} \beta_1 = \alpha_1, \beta_2 = \alpha_1+\alpha_2, \beta_3 = \alpha_2, \beta_4=-\alpha_1. \end{align} We have $\alpha_1+(-\alpha_1)=0$. Thank you very much.

Answer 1

This is a standard observation in Lie theory: if you let the Weyl group act on the root system on the left, $\beta_k$ is the unique root such that $s_{i_k}\cdots s_{i_1}\beta_k$ is negative and $s_{i_{k-1}}\cdots s_{i_1}\beta_k$ is positive. That is, for a positive root $\gamma$, the number of times it appears as $\pm \beta_k$ is the number of times that $s_{i_k}\cdots s_{i_1}\gamma$ changes sign as $k$ goes from 0 to $m$, and it's $\beta_k$ if the change is positive to negative, and $-\beta_k$ if the change is negative to positive.

An expression is unreduced if and only if there is a positive root that changes sign twice, so the places where this sign change happens gives the cancelling $\beta$'s.