ScalingStacks

Proof. Let $\sigma\in B_{n}$ and $r\in\{1,\ldots,n-1\}$. Assume $\sigma T_{r}\neq 0$.

If there is $1\leq j\leq i\leq n$ with $\sigma(-i)=n-j+1$ and $i\neq n+1-r$, then $\sigma T_{r}\in B_{n}$.

Assume now $\sigma(-i)\in T\sqcup(1,n-i)$ for all $i\neq n+1-r$. We deduce that $L(\sigma_{|\{-(n+1-r),-(n-r)\}})\neq\emptyset$, hence $\sigma T_{r}=0$ (cf Lemma 8.2.4), a contradiction.

Using Remark 8.2.6, we deduce that $T_{r}\sigma\in B_{n}$.

The other assertions of the lemma are immediate. ∎