ScalingStacks

We describe now $L^{+}(r,n)$ as a twisted free $(H_{r}\otimes H_{n})$-module.

Consider $E\subset\{1,\ldots,r+n\}$ with $|E|=r$. Let $w_{E}\in{\mathfrak{S}}_{r+n}$ be the permutation such that $w_{E}(E)=\{1,\ldots,r\}$ and the restrictions of $w_{E}$ to $E$ and to $\{1,\ldots,r+n\}\setminus E$ are increasing. If $E=\{i_{1}<\cdots<i_{r}\}$, then we have a reduced decomposition

and

There is a bijection

where $f_{r}(v^{\prime})\in{\mathfrak{S}}_{n+r}$ is given by $f_{r}(v^{\prime})(i)=i$ for $i\leq r$ and $f_{r}(v^{\prime})(r+i)=r+v^{\prime}(i)$ for $1\leq i\leq n$. We have $\ell(\beta(v,v^{\prime},E))=\ell(v)+\ell(v^{\prime})+\ell(w_{E})$.

Given $(a,i_{b})\in\tilde{L}(w_{E})$, we define $v(E,a,b)\in{\mathfrak{S}}_{r}$ and $v^{\prime}(E,a,b)\in{\mathfrak{S}}_{n}$ as follows. Let $b^{\prime}\in\{1,\ldots,r\}$ be minimal such that $a<i_{b^{\prime}}$. We define $v(E,a,b)$ to be the cycle $(b,b-1,\ldots,b^{\prime})$ and $v^{\prime}(E,a,b)$ to be the cycle $(a-b^{\prime}+1,a-b^{\prime}+2,\ldots,i_{b}-b)$. We have

and $\ell(w_{E})-\ell(w_{(E\cup\{a\})\setminus\{i_{b}\}})=i_{b}-a$.

Given $m\geq 1$, we define a free differential $(H_{r}\otimes H_{n})$-module

Given $m^{\prime}<m$, we define $f_{m^{\prime},m}:V_{m}\to V_{m^{\prime}}$ as the morphism of $(H_{r}\otimes H_{n})$-modules given by

We will show below (Lemma 6.1.3) that $d(f_{m^{\prime},m})=\sum_{m>m^{\prime\prime}>m^{\prime}}f_{m^{\prime}m^{\prime\prime}}\circ f_{m^{\prime\prime}m}$. We denote by $V$ the differential $(H_{r}\otimes H_{n})$-module obtained as the corresponding twisted object $[\bigoplus V_{m},(f_{m^{\prime}m})]$ (cf §2.1.3). We have $V=\bigoplus_{m}V_{m}$ as a $(H_{r}\otimes H_{n})$-module and $d_{V}=\sum_{m}d_{V_{m}}+\sum_{m,m^{\prime}}f_{m^{\prime},m}$.

There is a dual version of Lemma 6.1.3. In particular, there is a decomposition of right $(H_{r}\otimes H_{n})$-modules