ScalingStacks

Let $(W,S)$ be a finite Coxeter group. Let $V$ be the geometric representation of $W$ over $k={\mathbf{C}}$: it comes with a basis $\{e_{s}\}_{s\in S}$. Given $s\in S$, we denote by $\alpha_{s}$ the linear form on $V$ such that $s(x)-x=\alpha_{s}(x)e_{s}$ for all $x\in V$. The set $\{\alpha_{s}\}_{s\in S}$ is a basis of $V^{*}$. Let $P=P_{S}=P_{(W,S)}=k[V]$ (we will denote by $X_{S}$ or $X_{(W,S)}$ a given object constructed from $(W,S)$).

The braid group $B_{S}=B_{(W,S)}$ associated to $(W,S)$ is the group generated by $\{\sigma_{s}\}_{s\in S}$ with relations

$$\underbrace{\sigma_{s}\sigma_{t}\sigma_{s}\cdots}_{m_{st}\text{ terms}}=\underbrace{\sigma_{t}\sigma_{s}\sigma_{t}\cdots}_{m_{st}\text{ terms}}$$

for any $s,t\in S$ such that the order $m_{st}$ of $st$ is finite.

We denote by $l:B_{S}\to{\mathbf{Z}}$ the length function. It is the morphism of groups defined by $l(\sigma_{s})=1$ for $s\in S$.

Let us recall, following [Rou1], how to lift in a non-trivial way the action of $W$ on the derived category $D(P)$ to an action of $B_{S}$ on the homotopy category $\operatorname{Ho}\nolimits(P)$.

Let $s\in S$. We put

$$\theta_{s}=P\otimes_{P^{s}}P\text{ and }F_{s}=0\to\theta_{s}\langle 1\rangle\xrightarrow{m}P\langle 1\rangle\to 0.$$

The latter is a complex of graded $P^{{\mathrm{en}}}$-modules, where $P\langle 1\rangle$ is in cohomological degree $1$ and $m$ denotes the multiplication map. We put

$$F_{s}^{-1}=0\to P\langle-1\rangle\xrightarrow{a\mapsto a\alpha_{s}\otimes 1+a\otimes\alpha_{s}}\theta_{s}\to 0.$$

This is a complex of graded $P^{{\mathrm{en}}}$-modules, where $P\langle-1\rangle$ is in cohomological degree $-1$.

Let us recall a result of [Rou1, §9]. Given $i_{1},\ldots,i_{r}$, $j_{1},\ldots,j_{r^{\prime}}\in S$ and $\delta_{1},\ldots,\delta_{r}$, $\varepsilon_{1},\ldots,\varepsilon_{r^{\prime}}\in\{\pm 1\}$ such that $\sigma_{i_{1}}^{\delta_{1}}\cdots\sigma_{i_{r}}^{\delta_{r}}=\sigma_{j_{1}}^{\varepsilon_{1}}\cdots\sigma_{j_{r^{\prime}}}^{\varepsilon_{r^{\prime}}}$, there is a canonical isomorphism in $\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$

$$F_{i_{1}}^{\delta_{1}}\otimes_{P_{n}}\cdots\otimes_{P_{n}}F_{i_{r}}^{\delta_{r}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F_{j_{1}}^{\varepsilon_{1}}\otimes_{P_{n}}\cdots\otimes_{P_{n}}F_{j_{r^{\prime}}}^{\varepsilon_{r^{\prime}}}$$

and these isomorphisms form a transitive system of isomorphisms.

Given $b\in B_{S}$, we put

$$F_{b}=\lim_{\begin{subarray}{c}i_{1},\ldots,i_{r}\\ \varepsilon_{1},\ldots,\varepsilon_{r}\\ b=\sigma_{i_{1}}^{\varepsilon_{1}}\cdots\sigma_{i_{r}}^{\varepsilon_{r}}\end{subarray}}F_{i_{1}}^{\varepsilon_{1}}\otimes_{P}\cdots\otimes_{P}F_{i_{r}}^{\varepsilon_{r}}\in\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits).$$

The $2$-braid group ${\mathcal{B}}_{(W,S)}$ is the full monoidal subcategory of $\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$ with objects the $F_{b}$’s, with $b\in B_{S}$.

Let $I\subset S$ and let $W_{I}$ be the subgroup of $W$ generated by $I$. Let $V_{I}=\bigoplus_{s\in I}ke_{s}$ and $P_{I}=k[V_{I}]$. We have $V=V_{I}\oplus V^{I}$, hence $P=P_{I}\otimes k[V^{I}]$. We deduce also that $V^{*}=(V^{I})^{\perp}\oplus(V_{I})^{\perp}$, hence the composition of canonical maps $(V^{I})^{\perp}\hookrightarrow V^{*}\twoheadrightarrow(V_{I})^{*}$ is an isomorphism. We identify $(V^{I})^{\perp}=\bigoplus_{s\in I}k\alpha_{s}$ and $(V_{I})^{*}$ via this isomorphism.

The compositions of canonical maps $\bigcap_{s{\not\in}I}\ker\alpha_{s}\to V\to V/V^{I}$ and $V_{I}\to V\to V/V^{I}$ are isomorphisms: this provides an isomorphism $V_{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigcap_{s{\not\in}I}\ker\alpha_{s}$. We denote by $\rho_{I}:P\twoheadrightarrow P_{I}$ the morphism given by the composition $V_{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigcap_{s{\not\in}I}\ker\alpha_{s}\hookrightarrow V$.

We have a functor $\gamma_{I}:P_{I}^{\mathrm{en}}\operatorname{\!-Mod}\nolimits\to P^{\mathrm{en}}\operatorname{\!-Mod}\nolimits$ sending $M$ to $k[V^{I}]\otimes M$, where $k[V^{I}]$ is the regular $k[V^{I}]^{\mathrm{en}}$-module and $P^{\mathrm{en}}$ is decomposed as $P^{\mathrm{en}}=k[V^{I}]^{\mathrm{en}}\otimes P_{I}^{\mathrm{en}}$. We obtain a fully faithful monoidal functor

$${\mathcal{B}}_{W_{I}}\to{\mathcal{B}}_{W},\ F\mapsto\gamma_{I}(F)=k[V^{I}]\otimes F.$$