Lemma 6.2.10 (0P7W) · ScalingStacks

0P7W

Lemma 6.2.10. Consider $\sigma^{\prime\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ and $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K)$ and let $\sigma=\sigma^{\prime}\sigma^{\prime\prime}$ . Assume $\ell(\sigma)=\ell(\sigma^{\prime})+\ell(\sigma^{\prime\prime})$ .

Let $(i_{1},i_{2})\in D(\sigma)\setminus(D(\sigma)\cap D(\sigma^{\prime\prime}))$ . Let $\alpha^{\prime\prime}=\sigma^{\prime\prime}s_{i_{1},i_{2}}$ and $\alpha^{\prime\prime}=(\sigma^{\prime})^{\sigma^{\prime\prime}(i_{1}),\sigma^{\prime\prime}(i_{2})}$ . We have $\sigma=\alpha^{\prime}\alpha^{\prime\prime}$ and $\ell(\sigma)=\ell(\alpha^{\prime})+\ell(\alpha^{\prime\prime})$ .

0P7X

Proof. Assume first $I=J=K$ . The lemma follows in that case from Lemmas 3.2.4 and 3.2.2.

Consider now the general case. There are increasing bijections $\tau:J\to I$ and $\tau^{\prime}:K\to J$ . We have $D(\sigma)=\tau^{-1}(D(\tau^{\prime}\sigma\tau))$ and $D(\sigma^{\prime\prime})=\tau^{-1}(D(\sigma^{\prime\prime}\tau))$ (proof of Lemma 5.4.7). The lemma follows now from the previous case applied to the decomposition $\tau^{\prime}\sigma\tau=(\tau^{\prime}\sigma^{\prime})(\sigma^{\prime\prime}\tau)$ . ∎