ScalingStacks

We now define a groupoid of $n$-periodic bijections.

Given $I$ a subset of ${\mathbf{Z}}/n$ we denote by $\tilde{I}$ its inverse image in ${\mathbf{Z}}$.

Let ${\mathcal{S}}_{n}$ be the category with objects the subsets of ${\mathbf{Z}}/n$ and where $\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ is the set of $n$-periodic bijections $\sigma:\tilde{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{J}$. The group $n{\mathbf{Z}}$ acts by translation on $\operatorname{Hom}\nolimits$-sets. Note that $\hat{{\mathfrak{S}}}_{n}=\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}({\mathbf{Z}}/n)$.

Given $i,j\in\tilde{I}$ with $i-j{\not\in}n{\mathbf{Z}}$, the element $s_{ij}\in\hat{{\mathfrak{S}}}_{n}$ restricts to an $n$-periodic bijection $\tilde{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}$, which we also denote by $s_{ij}$.

Let $I$ be a subset of ${\mathbf{Z}}/n$. There is a unique increasing bijection $\beta_{I}:\{1,\ldots,|I|\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}\cap\{1,\ldots,n\}$. We extend it to an increasing bijection ${\mathbf{Z}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\tilde{I}$ by $\beta_{I}(r+d|I|)=\beta_{I}(r)+dn$ for $r\in\{1,\ldots,|I|\}$ and $d\in{\mathbf{Z}}$. There is an isomorphism of groups

Consider $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$. We define

and $\tilde{L}(\sigma)=\{(i,i^{\prime})\in L(\sigma)\ |\ 1\leq i\leq n\}$. The canonical map $\tilde{L}(\sigma)\to L(\sigma)/n{\mathbf{Z}}$ is bijective. We define $\ell(\sigma)=|\tilde{L}(\sigma)|$.

Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$. We have $\ell(\sigma)=0$ if and only if $\sigma$ is an increasing bijection.

Given $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,I)$ with $\ell(\tau)=0$, we have $L(\tau\circ\sigma)=L(\sigma)=(\tau\times\tau)\bigl(L(\sigma\circ\tau)\bigr)$, hence $\ell(\tau\circ\sigma)=\ell(\sigma\circ\tau)=\ell(\sigma)$.

Since $L(\tau\circ\sigma)=(\beta_{I}\times\beta_{I})(L(F_{I}^{-1}(\tau\circ\sigma)))$, we have $\ell(\sigma)=\ell(F_{I}^{-1}(\tau\circ\sigma))$. As a consequence, we deduce the following result from Lemma 3.2.3.

The next lemma relates length and number of intersections of paths on a cylinder.

Given $I,J\subset{\mathbf{Z}}/n$, we define $\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}^{\geq-r}}(I,J)=\{\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ l(\sigma)\leq r\}$ for $r\in{\mathbf{Z}}_{\geq 0}$. It follows from Lemma 6.2.1 that this defines a structure of ${\mathbf{Z}}_{\leq 0}$-filtered category on ${\mathcal{S}}_{n}$. We put ${\mathcal{H}}_{n}=\mathrm{gr}{\mathcal{S}}_{n}^{\bullet}$, a pointed ${\mathbf{Z}}_{\leq 0}$-graded category.

Note that a map $\sigma$ of length $0$ is invertible in ${\mathcal{H}}_{n}$. Note also that $F_{I}$ induces an isomorphism of graded pointed monoids $\hat{{\mathfrak{S}}}_{|I|}^{\operatorname{nil}\nolimits}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{{\mathcal{H}}_{n}}(I)$.

Let us consider the free abelian groups $R_{n}=\bigoplus_{a\in{\mathbf{Z}}/n}{\mathbf{Z}}\alpha_{a}$ and $L_{n}=\bigoplus_{a\in{\mathbf{Z}}/n}{\mathbf{Z}}\varepsilon_{a}$. We define a linear map $\rho:R_{n}\to L_{n}$ by $\rho(\alpha_{a})=\varepsilon_{a+1}-\varepsilon_{a}$ and a representation of the group $R_{n}$ on $L_{n}$ given by

Note that $\delta=\sum_{a\in{\mathbf{Z}}/n}\alpha_{a}\in\ker\rho$ and $\delta\cdot\varepsilon_{b}=2\varepsilon_{b}$ for all $b$.

We define a bilinear map

Let $\Gamma_{n}^{\prime}=L_{n}\times R_{n}$. We define a group structure on $\Gamma_{n}^{\prime}$ by

Given $I\subset{\mathbf{Z}}/n$, we put $\varepsilon_{I}=\sum_{a\in I}\varepsilon_{a}\in L_{n}$. Given $i,j\in{\mathbf{Z}}$, we put

Note that $\alpha_{i,i+1}=\alpha_{i+n{\mathbf{Z}}}$, $\alpha_{i+n,j+n}=\alpha_{i,j}$ and $\alpha_{i,j}+\alpha_{j,k}=\alpha_{i,k}$ for all $i,j,k\in{\mathbf{Z}}$. Note also that $\delta=\alpha_{i,i+n}$ for all $i\in{\mathbf{Z}}$. Note finally that $\rho(\alpha_{i,j})=\varepsilon_{j+n{\mathbf{Z}}}-\varepsilon_{i+n{\mathbf{Z}}}$.

Consider $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$. We put

Note that $\rho(\llbracket\sigma\rrbracket)=\varepsilon_{J}-\varepsilon_{I}$ and $\llbracket\sigma^{\prime}\circ\sigma\rrbracket=\llbracket\sigma^{\prime}\rrbracket+\llbracket\sigma\rrbracket$ for any $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K)$.

We define

We put $\Gamma_{n}=\frac{1}{2}{\mathbf{Z}}\times\Gamma^{\prime}_{n}$. We endow $\Gamma_{n}$ with a structure of ${\mathbf{Z}}$-monoid by using the canonical embedding ${\mathbf{Z}}\hookrightarrow\frac{1}{2}{\mathbf{Z}}\hookrightarrow\Gamma_{n}$.

Given $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$, we put $\deg(\sigma)=(-\ell(\sigma),\operatorname{dm}\nolimits(\sigma))\in\Gamma_{n}$.

Let $D$ be a subset of $\{1,\ldots,n\}\times\{\pm 1\}$ that embeds in its projection on $\{1,\ldots,n\}$. We denote by $\Gamma_{D}$ the quotient of $\Gamma_{n}$ by the subgroup generated by $(0,\varepsilon_{i+n{\mathbf{Z}}})+(\frac{1}{2}\nu_{i},0)$, where $(i,\nu_{i})\in D$. We identify $\frac{1}{2}{\mathbf{Z}}$ with the image of $\frac{1}{2}{\mathbf{Z}}\times 0$ in $\Gamma_{D}$. We define a partial order on $\Gamma_{D}$ by $h\geq g$ if $hg^{-1}$ is in $\frac{1}{2}{\mathbf{Z}}_{\geq 0}$. We denote by $\deg_{D}(\sigma)$ the image of $\deg(\sigma)$ in $\Gamma_{D}$.

Given $E$ a subset of $\{1,\ldots,n\}$, we put $E^{+}=\{(i,1)\ |\ i\in E\}$.

By Lemmas 6.2.1 and 6.2.5, we obtain a $\Gamma_{D}$-filtration on ${\mathcal{S}}_{n}$ by defining

It follows from Lemma 6.2.5 that the pointed category ${\mathcal{H}}_{n}$ is isomorphic to the graded pointed category associated to the $\Gamma_{D}$-filtration of ${\mathcal{S}}_{n}$ (after forgetting the $\Gamma_{D}$-grading to ${\mathbf{Z}}$).

Note that if $D=\emptyset$, then $\Gamma_{D}=\Gamma_{n}$, $\deg_{D}=\deg$ and the ${\mathbf{Z}}_{\leq 0}$ grading on ${\mathcal{H}}_{n}$ given by the length can be recovered from the $\Gamma_{n}$-grading by using the quotient map $\Gamma_{n}\to\Gamma_{n}/\Gamma^{\prime}_{n}=\frac{1}{2}{\mathbf{Z}}$.

This quotient map provides a ${\mathbf{Z}}$-grading on the $\Gamma_{n}$-graded pointed category associated to the $\Gamma_{n}$-filtration of ${\mathcal{S}}_{n}$. This ${\mathbf{Z}}$-graded pointed category is isomorphic to ${\mathcal{H}}_{n}$.

Assume $D=[1,n]^{+}$. Composing with the quotient map $\Gamma_{n}\twoheadrightarrow\Gamma_{D}$, the embedding $\frac{1}{2}{\mathbf{Z}}\hookrightarrow\Gamma_{n},\ r\mapsto(r,0)$ and the quotient map $\Gamma_{n}\twoheadrightarrow R_{n},\ (r,(l,\alpha))\mapsto\alpha$ induce an embedding of $\frac{1}{2}{\mathbf{Z}}$ as a central subgroup of $\Gamma_{D}$ with quotient map $\Gamma_{D}\twoheadrightarrow R_{n}$. So, $\Gamma_{[1,n]^{+}}$ identifies with the set $\frac{1}{2}{\mathbf{Z}}\times R_{n}$, with multiplication given by $(r,\alpha)\cdot(r^{\prime},\alpha^{\prime})=(r+r^{\prime}+\langle\langle\alpha,\alpha^{\prime}\rangle\rangle,\alpha+\alpha^{\prime})$. When $n\geq 3$, the group $\Gamma_{[1,n]^{+}}$ has a presentation with generators $z=(\frac{1}{2},0)$, $g_{a}=(0,\alpha_{a})$, $a\in{\mathbf{Z}}/n$ and relations

We define a morphism of groups

It follows from Lemma 6.2.7 that the $\Gamma_{n}$-grading on ${\mathcal{H}}_{n}$ comes from a grading by the kernel of the composition $\Gamma_{n}\xrightarrow{{\mathrm{can}}}\Gamma_{[1,n]^{+}}\xrightarrow{\epsilon}{\mathbf{Z}}/2$.

Given $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$, let $D(\sigma)$ be the set of pairs $(i_{1},i_{2})\in L(\sigma)$ such that

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  $i_{2}-i_{1}<n$ or $\sigma(i_{1})-\sigma(i_{2})<n$ and

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  given $i\in\tilde{I}$ with $i_{1}<i<i_{2}$, we have $\sigma(i_{1})<\sigma(i)$ or $\sigma(i)<\sigma(i_{2})$.

We put $\tilde{D}(\sigma)=D(\sigma)\cap\tilde{L}(\sigma)$. The diagonal action of $n{\mathbf{Z}}$ on $L(\sigma)$ preserves $D(\sigma)$ and we have a canonical bijection $\tilde{D}(\sigma)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}D(\sigma)/n{\mathbf{Z}}$.

Given $(i_{1},i_{2})\in L(\sigma)$, we put $\sigma^{i_{1},i_{2}}:=\sigma\circ s_{i_{1},i_{2}}$.

We define a partial order on $\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ as the transitive closure of $\sigma^{\prime}<\sigma$ if $\sigma^{\prime}=\sigma^{i_{1},i_{2}}$ for some $(i_{1},i_{2})\in D(\sigma)$.

When $I=J={\mathbf{Z}}/n$, this coincides with the extended Chevalley-Bruhat order on $\hat{{\mathfrak{S}}}_{n}$ by Lemma 3.2.4 and given $(i_{1},i_{2})\in L(\sigma)$, we have $\sigma^{i_{1},i_{2}}<\sigma$ (Lemma 3.2.3). The next lemma shows that this holds for general maps in ${\mathcal{S}}_{n}$.

Consider $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}}(I,J)$ non-zero. We put

Fix a positive integer $n^{\prime}\leq n$ and an increasing injection $\alpha:\{1,\ldots,n^{\prime}\}\hookrightarrow\{1,\ldots,n\}$. We extend $\alpha$ to an increasing injection ${\mathbf{Z}}\to{\mathbf{Z}}$ by $\alpha(r+dn^{\prime})=\alpha(r)+dn$ for $r\in\{1,\ldots,n^{\prime}\}$ and $d\in{\mathbf{Z}}$.