Subsection 3.2.2. (0PF6) · ScalingStacks

3.2.2. Definition

Let $n\geq 1$ . We denote by $\hat{{\mathfrak{S}}}_{n}$ the extended affine symmetric group: this is the subgroup of the group of permutations of ${\mathbf{Z}}$ with elements those bijections $\sigma:{\mathbf{Z}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{Z}}$ such that $\sigma(n+r)=n+\sigma(r)$ for all $r\in{\mathbf{Z}}$ .

Given $i,j\in{\mathbf{Z}}$ with $i-j{\not\in}n{\mathbf{Z}}$ , we denote by $s_{ij}$ the element of $\hat{{\mathfrak{S}}}_{n}$ defined by

s_{ij}(r)=\begin{cases}j-i+r&\text{ if }r=i\pmod{n}\\ i-j+r&\text{ if }r=j\pmod{n}\\ r&\text{otherwise.}\end{cases}

Note that $s_{i+n,j+n}=s_{i,j}$ , $s_{ij}=s_{ji}$ and $s_{ij}^{2}=1$ .

The symmetric group ${\mathfrak{S}}_{n}$ identifies with the subgroup of $\hat{{\mathfrak{S}}}_{n}$ of permutations $\sigma$ such that $\sigma(\{1,\ldots,n\})=\{1,\ldots,n\}$ . We have a surjective morphism $\hat{{\mathfrak{S}}}_{n}\to{\mathfrak{S}}_{n}$ sending $\sigma$ to the induced permutation of ${\mathbf{Z}}/n$ . We identify its kernel with ${\mathbf{Z}}^{n}$ via the injective morphism

{\mathbf{Z}}^{n}\to\hat{{\mathfrak{S}}}_{n},\ (\lambda_{1},\ldots,\lambda_{n})\mapsto(\{1,\ldots,n\}\ni i\mapsto i+n\lambda_{i}).

We have $\hat{{\mathfrak{S}}}_{n}={\mathbf{Z}}^{n}\rtimes{\mathfrak{S}}_{n}$ .

Assume $n\geq 2$ . Let $W_{n}$ be the Coxeter group of type $\hat{A}_{n-1}$ : it is generated by $\{s_{a}\}_{a\in{\mathbf{Z}}/n}$ with relations

s_{a}^{2}=1,\ s_{a}s_{b}=s_{b}s_{a}\text{ if }a\neq b\pm 1

s_{a}s_{a+1}s_{a}=s_{a+1}s_{a}s_{a+1}\ (\text{ for }n>2).

Consider the semi-direct product $W_{n}\rtimes\langle c\rangle$ of $W_{n}$ by an infinite cyclic group generated by an element $c$ , with relation $cs_{a}c^{-1}=s_{a+1}$ .

0P4W

Lemma 3.2.1. There is an isomorphism of groups

W_{n}\rtimes\langle c\rangle\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\hat{{\mathfrak{S}}}_{n},\ c\mapsto(j\mapsto j+1),\ s_{i+n{\mathbf{Z}}}\mapsto s_{i,i+1}\text{ for }i\in\{1,\ldots,n\}.

0P4X

Proof. Denote by $f$ the map of the lemma. By [Lus, §3.6] (cf also [BjBr, Proposition 8.3.3]), the restriction of $f$ to $W_{n}$ induces an isomorphism with the subgroup of $\hat{{\mathfrak{S}}}_{n}$ of elements $\sigma$ such that $\sum_{i=1}^{n}(\sigma(i)-i)=0$ . It is immediate to check that $f$ extends to a morphism of groups $W_{n}\rtimes\langle c\rangle\to\hat{{\mathfrak{S}}}_{n}$ .

Consider $\sigma\in\hat{{\mathfrak{S}}}_{n}$ and let $N=\sum_{i=1}^{n}(\sigma(i)-i)$ . Note that $n|N$ . Put $\sigma^{\prime}=\sigma f(c)^{-N/n}$ . We have $\sigma^{\prime}\in f(W_{n})$ , so $f$ is surjective. Let $\sigma=f(wc^{d})$ . We have $\sum_{i=1}^{n}(\sigma(i)-i)=nd$ . So, if $\sigma=1$ , then $d=0$ , hence $w\in\ker(f)\cap W_{n}=1$ . This shows that $f$ is injective. ∎

We will identify $W_{n}\rtimes\langle c\rangle$ and $\hat{{\mathfrak{S}}}_{n}$ via the isomorphism of Lemma 3.2.1.

We put $W_{1}=1$ , so that $\hat{{\mathfrak{S}}}_{1}\simeq\langle c\rangle=W_{1}\rtimes\langle c\rangle$ . We also put $\hat{{\mathfrak{S}}}_{0}=1$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2