ScalingStacks

Proof of Proposition 3.2.9. Let $H$ be the subalgebra of $\hat{H}_{n}$ generated by $T_{1},\ldots,T_{n-1},c$. This is a differential graded subalgebra of $\hat{H}_{n}$. Given $w\in\hat{{\mathfrak{S}}}_{n}$, let $|w|=\sum_{i=1}^{n}w(i)$. Let $w\in\hat{{\mathfrak{S}}}_{n}^{+}$, $w\neq 1$. We show by induction on $\ell(w)+|w|$ that $T_{w}\in H$.

Assume $\ell(ws_{i})<\ell(w)$ for some $i\in\{1,\ldots,n-1\}$. We have $ws_{i}\in\hat{{\mathfrak{S}}}_{n}^{+}$ and $|ws_{i}|=|w|$, hence by induction $T_{ws_{i}}\in H$. We deduce that $T_{w}=T_{ws_{i}}T_{i}\in H$.

Otherwise, we have $0<w(1)<\cdots<w(n)$, hence $w(n)>n$ since $w\neq 1$. It follows that $wc^{-1}\in\hat{{\mathfrak{S}}}_{n}^{+}$ and $|wc^{-1}|<|w|$, hence $T_{wc^{-1}}\in H$ by induction. So $T_{w}=T_{wc^{-1}}T_{c}\in H$.

We have shown that $\hat{H}_{n}^{+}\subset H$. Since $\hat{H}_{n}^{+}$ is stable under right multiplication by $T_{c}$ and by $T_{i}$ for $i\in\{1,\ldots,n-1\}$, it follows that $H=\hat{H}_{n}^{+}$.

There is a surjective morphism of algebras $\rho:A_{n}\to\hat{H}_{n}^{+},\ t_{i}\mapsto T_{i},\ b\mapsto c$. Given $I=\{i_{1}<\cdots<i_{r}\}$ a non-empty subset of $\{1,\ldots,n\}$, we put

$$c_{I}=(cs_{n-1}\cdots s_{i_{1}+r-1})(cs_{n-1}\cdots s_{i_{2}+r-2})\cdots(cs_{n-1}\cdots s_{i_{r}})\in\hat{{\mathfrak{S}}}_{n}.$$

We have $c_{I}(i_{l})=n+l$ for $1\leq l\leq r$ and $c_{I}(j)=j+r-k$ if $i_{k}<j<i_{k+1}$ (where we put $i_{0}=0$ and $i_{r+1}=n+1$).

Let $E$ be the set of families $(I_{1},\ldots,I_{m})$ where $m\geq 0$, $I_{1}\subset\{1,\ldots,n\}$ and $I_{r}\subset\{1,\ldots,|I_{r-1}|\}$ for $1<r\leq m$.

Given $w\in{\mathfrak{S}}_{n}$ and $(I_{1},\ldots,I_{m})\in E$, we have $\rho(t_{w}\gamma_{I_{1}}\cdots\gamma_{I_{m}})=T_{w}T_{c_{I_{1}}}\cdots T_{c_{I_{m}}}$ and that element is either $T_{wc_{I_{1}}\cdots c_{I_{m}}}$ or $0$.

We define a map $\phi:({\mathbf{Z}}_{\geq 0})^{n}\to E$. Let $a\in({\mathbf{Z}}_{\geq 0})^{n}$. Let $m=\max\{a(i)\}_{1\leq i\leq n}$. We put $I_{1}=a^{-1}({\mathbf{Z}}_{\geq 1})$ and we define inductively $I_{r}$ for $2\leq r\leq m$ by $I_{r}=c_{I_{r-1}}\cdots c_{I_{1}}(a^{-1}({\mathbf{Z}}_{\geq r}))$. We put $\phi(a)=(I_{1},\ldots,I_{m})$. We have

$$c_{I_{m}}\cdots c_{I_{1}}(i)=na(i)+|a^{-1}({\mathbf{Z}}_{>a(i)})|+(\text{position of }i\text{ in }a^{-1}(a(i))).$$

We define a map $\psi:E\to({\mathbf{Z}}_{\geq 0})^{n}$. Let $(I_{1},\ldots,I_{m})\in E$. We define $a\in({\mathbf{Z}}_{\geq 0})^{n}$ by $a(i)=\lfloor\frac{c_{I_{m}}\cdots c_{I_{1}}(i)-1}{n}\rfloor$ and we put $\psi(I_{1},\ldots,I_{m})=a$. The maps $\psi$ and $\phi$ are inverse bijections. We deduce that the map $E\to({\mathfrak{S}}_{n}\setminus\hat{{\mathfrak{S}}}_{n}^{+})$ sending $(I_{1},\ldots,I_{m})$ to the class of $c_{I_{m}}\cdots c_{I_{1}}$ is bijective. It follows that the map ${\mathfrak{S}}_{n}\times E\to\hat{{\mathfrak{S}}}_{n}^{+},\ (w,(I_{1},\ldots,I_{m}))\mapsto wc_{I_{m}}\cdots c_{I_{1}}$ is bijective.

If $\rho(t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}})=T_{wc_{I_{m}}\cdots c_{I_{1}}}=0$ for some $w\in{\mathfrak{S}}_{n}$ and $(I_{1},\ldots,I_{m})\in E$, then the bijectivty of the map above shows that the image of $\rho$ is the span of a proper subset of a basis of $\hat{H}_{n}^{+}$, contradicting the surjectivity of $\rho$.

This shows that the elements $\rho(t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}})$ are distinct basis elements of $\hat{H}_{n}^{+}$, hence $\rho$ is an isomorphism. ∎