ScalingStacks

Proof. Note that $\tau$ is an increasing bijection since $\ell(\tau)=0$. We have $D(\tau\circ\sigma)=D(\sigma)$ and given $(i_{1},i_{2})\in D(\sigma)$, we have $(\tau\circ\sigma)^{i_{1},i_{2}}=\tau\circ\sigma^{i_{1},i_{2}}$. This shows the first equivalence. The second equivalence follows from the fact that $D(\sigma\circ\tau)=(\tau^{-1}\times\tau^{-1})(D(\sigma))$ and given $(i_{1},i_{2})\in D(\sigma)$, we have $(\sigma\circ\tau)^{\tau^{-1}(i_{1}),\tau^{-1}(i_{2})}=\sigma^{i_{1},i_{2}}\circ\tau$. ∎