Lemma 8.2.7 (0PD4) · ScalingStacks

•

Lemma 8.2.7. $B_{n}$ and $D_{n}$ are stable under the action of $H_{n}^{\bullet}\wedge(H_{n}^{\bullet})^{\operatorname{opp}\nolimits}$ .
•

$E_{n}$ is stable under the action of $H_{n}^{\bullet}$ and $F_{n}$ is stable under the action of $(H_{n}^{\bullet})^{\operatorname{opp}\nolimits}$ .
•

$A$ and $C$ are stable under multiplication
•

Given $\alpha\in B$ and $\beta\in G$ , we have $\alpha\ast\beta\in B$ and $\beta\ast\alpha\in B$ .

Proof. Let $\sigma\in B_{n}$ and $r\in\{1,\ldots,n-1\}$ . Assume $\sigma T_{r}\neq 0$ .

If there is $1\leq j\leq i\leq n$ with $\sigma(-i)=n-j+1$ and $i\neq n+1-r$ , then $\sigma T_{r}\in B_{n}$ .

Assume now $\sigma(-i)\in T\sqcup(1,n-i)$ for all $i\neq n+1-r$ . We deduce that $L(\sigma_{|\{-(n+1-r),-(n-r)\}})\neq\emptyset$ , hence $\sigma T_{r}=0$ (cf Lemma 8.2.4), a contradiction.

Using Remark 8.2.6, we deduce that $T_{r}\sigma\in B_{n}$ .

The other assertions of the lemma are immediate. ∎