Lemma 3.2.2 (0P4Y) · ScalingStacks

0P4Y

Lemma 3.2.2. Let $\sigma^{\prime},\sigma^{\prime\prime}\in\hat{{\mathfrak{S}}}_{n}$ and $\sigma=\sigma^{\prime}\sigma^{\prime\prime}$ . Assume $\ell(\sigma)=\ell(\sigma^{\prime})+\ell(\sigma^{\prime\prime})$ . Let $a\in{\mathbf{Z}}/n$ such that $\ell(\sigma s_{a})<\ell(\sigma)$ and $\ell(\sigma^{\prime\prime}s_{a})>\ell(\sigma^{\prime\prime})$ .

Let $\alpha^{\prime\prime}=\sigma^{\prime\prime}s_{a}$ and $\alpha^{\prime}=\sigma^{\prime}\sigma^{\prime\prime}s_{a}\sigma^{\prime\prime-1}$ . We have $\sigma=\alpha^{\prime}\alpha^{\prime\prime}$ and $\ell(\sigma)=\ell(\alpha^{\prime})+\ell(\alpha^{\prime\prime})$ .

0P4Z

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. ∎