ScalingStacks

Proof. The length property of the bijection $\beta$ above shows that the map of the lemma is an isomorphism of $(H_{r}\otimes H_{n})$-modules. Since

$$d(T_{w_{E}})=\sum_{i\in E,\ j\in\{1,\ldots,r+n\}\setminus E,\ j<i}T_{w_{E}s_{i,j}},$$

it follows that the map of the lemma intertwines $d_{V}$ and the differential of $L^{+}(r,n)$. The lemma follows. ∎