ScalingStacks

Proof. Let $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,n-1\}$. We have

Consider $I\subset\{1,\ldots,n\}$ non-empty with elements $1\leq i_{1}<\cdots<i_{r}\leq n$. We put $i_{0}=0$ and $i_{r+1}=n+1$.

Consider $j\in\{1,\ldots,n-1\}$. Fix $k\in\{0,\ldots,r\}$ such that $i_{k}\leq j<i_{k+1}$. Let us show that

We have

If $j<i_{k+1}-1$, then $\beta_{i_{k+1}+r-k-1}t_{j+r-k-1}=t_{j+r-k}\beta_{i_{k+1}+r-k-1}$ and we deduce the first two equalities in (3.2.2). Assume now $j=i_{k+1}-1$. We have $\beta_{i_{k+1}+r-k-1}t_{j+r-k-1}=\beta_{i_{k+1}+r-k-2}$ and the third equality in (3.2.2) follows. The last equality from the fact that given $i\in\{1,\ldots,n-1\}$, we have

We deduce that $\gamma_{I}t_{j}=u\gamma_{I^{\prime}}$ for some $I^{\prime}\subset\{1,\ldots,n\}$ with $|I^{\prime}|=|I|$ and $\max(I^{\prime})\leq\max(I)$ and $u\in\{0,1,t_{1},\ldots,t_{n-1}\}$.

Fix $s\in\{1,\ldots,n\}$ with $s\geq\max(I)$. We have

Consider $I_{1},\ldots,I_{m}$ as in the lemma. Let $k$ be minimal such that $1{\not\in}I_{k}$. We put $k=m+1$ if there is no such $k$. Define $u=\gamma_{\{|I_{m}|\}}$ if $k=m+1$ and $u=1$ otherwise. Put $I_{0}=\{1,\ldots,n\}$. Recall that $b=\beta_{n}$. We have

where $I^{\prime}_{r}=\{i-1|i\in I_{r}\setminus\{1\}\}\cup\{|I_{r-1}|\}$ for $1\leq r<k$, $I^{\prime}_{k}=\{i-1|i\in I_{k}\}\cup\{|I_{k-1}|\}$ and $I^{\prime}_{r}=I_{r}$ for $r>k$.

We deduce that the set $B=\{t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}}\}$ of the lemma is stable under right multiplication by $t_{j}$ for $j\in\{1,\ldots,n-1\}$ and by $b$. Since $B$ contains $1$, it follows that $B$ is a generating family for $A_{n}$ as an ${\mathbf{F}}_{2}$-vector space. ∎

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

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

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. ∎