ScalingStacks

Proof. Multiplying if necessary $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ by a power of $c$, we can assume $\sigma$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are in $W_{n}$.

Let $\sigma^{\prime}=s_{a_{1}}\cdots s_{a_{m}}$ and $\sigma^{\prime\prime}=s_{a_{m+1}}\cdots s_{a_{d}}$ be two reduced decompositions. The Exchange Lemma [Hu, Theorem 5.8] shows that there is $i$ such that $\sigma s_{a}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{d}}$.

If $i>m$, then $\sigma^{\prime\prime}s_{a}=s_{a_{m+1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{d}}$ and this contradicts $\ell(\sigma^{\prime\prime}s_{a})>\ell(\sigma^{\prime\prime})$. So, $i\leq m$. We have $\sigma s_{a}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{m}}\sigma^{\prime\prime}$. We deduce that $\alpha^{\prime}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{m}}$ has length $m-1$ and the lemma follows. ∎