Subsection 3.2. (0PF4) · ScalingStacks

3.2. Extended affine symmetric groups

3.2.1. Finite case

Fix $n\geq 0$ . The symmetric group ${\mathfrak{S}}_{n}$ is a Coxeter group with generating set $\{(1,2),\ldots,(n-1,n)\}$ .

Its differential nil Hecke algebra $H_{n}$ is the $k$ -algebra generated by $T_{1},\ldots,T_{n-1}$ with relations

(3.2.1)

T_{i}^{2}=0,\ T_{i}T_{j}=T_{j}T_{i}\text{ if }|i-j|>1\text{ and }T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}

and with differential given by $d(T_{i})=1$ .

The algebra $H_{n}$ has a basis $(T_{w})_{w\in{\mathfrak{S}}_{n}}$ .

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 $\hat{{\mathfrak{S}}}_{0}=1$ .

3.2.3. Diagrammatic representation

The permutations of ${\mathbf{Z}}$ can be described as collections of strands in $[-1,1]\times{\mathbf{R}}$ going leftwards from integer points on the vertical line $x=1$ to integer points on the vertical line $x=-1$ . Thanks to their $n$ -periodicity, those permutations that are elements of $\hat{{\mathfrak{S}}}_{n}$ can also be encoded in a collection of strands drawn on a cylinder, going from right to left, by passing to the quotient of the vertical strip $[-1,1]\times{\mathbf{R}}$ by the vertical action by translation of $n{\mathbf{Z}}$ .

Here are some elements of $\hat{{\mathfrak{S}}}_{3}$ :

The multiplication $\sigma\sigma^{\prime}$ of $\sigma$ and $\sigma^{\prime}$ in $\hat{{\mathfrak{S}}}_{n}$ corresponds to the concatenation of the diagram of $\sigma$ put to the left of the diagram of $\sigma^{\prime}$ as in the following example:

The defining relations for $\hat{{\mathfrak{S}}}_{n}$ are depicted as follows

The elements of ${\mathfrak{S}}_{n}$ correspond to diagrams whose strands do not go in the back of the cylinder, hence can be drawn on a rectangle. For example, $s_{12}$ above can be represented as follows:

3.2.4. Length

Assume now again that $n\geq 1$ . We extend the length function on the Coxeter group $W_{n}$ to one on $W_{n}\rtimes\langle c\rangle$ by setting $\ell(wc^{d})=\ell(w)$ for $w\in W_{n}$ and $d\in{\mathbf{Z}}$ . Note that the action of $c$ on $W_{n}$ preserves lengths. Similarly, we extend the Chevalley-Bruhat order on $W_{n}\rtimes\langle c\rangle$ by setting $w^{\prime}c^{d^{\prime}}<wc^{d}$ if $w^{\prime}<w$ and $d^{\prime}=d$ and we consider the corresponding order on $\hat{{\mathfrak{S}}}_{n}$ . Note that the action of $c$ on $W_{n}$ preserves the order, hence $w^{\prime}c^{d^{\prime}}<wc^{d}$ if and only if $c^{d^{\prime}}w^{\prime}<c^{d}w$ .

0P4Y

Lemma 3.2.2. Let $\sigma^{\prime},\sigma^{\prime\prime}\in\hat{{\mathfrak{S}}}_{n}$ and $\sigma=\sigma^{\prime}\sigma^{\prime\prime}$ . Assume $\ell(\sigma)=\ell(\sigma^{\prime})+\ell(\sigma^{\prime\prime})$ . Let $a\in{\mathbf{Z}}/n$ such that $\ell(\sigma s_{a})<\ell(\sigma)$ and $\ell(\sigma^{\prime\prime}s_{a})>\ell(\sigma^{\prime\prime})$ .

Let $\alpha^{\prime\prime}=\sigma^{\prime\prime}s_{a}$ and $\alpha^{\prime}=\sigma^{\prime}\sigma^{\prime\prime}s_{a}\sigma^{\prime\prime-1}$ . We have $\sigma=\alpha^{\prime}\alpha^{\prime\prime}$ and $\ell(\sigma)=\ell(\alpha^{\prime})+\ell(\alpha^{\prime\prime})$ .

0P4Z

Proof. Multiplying if necessary $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ by a power of $c$ , we can assume $\sigma$ , $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are in $W_{n}$ .

Let $\sigma^{\prime}=s_{a_{1}}\cdots s_{a_{m}}$ and $\sigma^{\prime\prime}=s_{a_{m+1}}\cdots s_{a_{d}}$ be two reduced decompositions. The Exchange Lemma [Hu, Theorem 5.8] shows that there is $i$ such that $\sigma s_{a}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{d}}$ .

If $i>m$ , then $\sigma^{\prime\prime}s_{a}=s_{a_{m+1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{d}}$ and this contradicts $\ell(\sigma^{\prime\prime}s_{a})>\ell(\sigma^{\prime\prime})$ . So, $i\leq m$ . We have $\sigma s_{a}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{m}}\sigma^{\prime\prime}$ . We deduce that $\alpha^{\prime}=s_{a_{1}}\cdots s_{a_{i-1}}s_{a_{i+1}}\cdots s_{a_{m}}$ has length $m-1$ and the lemma follows. ∎

Given $\sigma\in\hat{{\mathfrak{S}}}_{n}$ , we put $L(\sigma)=\{(i,j)\in{\mathbf{Z}}\times{\mathbf{Z}}\ |\ i<j,\ \sigma(i)>\sigma(j)\}$ . This set has a diagonal action of $n{\mathbf{Z}}$ by translation. We put $\tilde{L}(\sigma)=\{(i,j)\in L(\sigma)\ |\ 1\leq i\leq n\}$ . The canonical map $\tilde{L}(\sigma)\to L(\sigma)/n{\mathbf{Z}}$ is bijective.

The next lemma is a variation on classical results (cf [Sh, Lemma 4.2.2], [BjBr, Proposition 8.3.6] and [BjBr, §2.2]).

0P50

Lemma 3.2.3. Let $\sigma\in\hat{{\mathfrak{S}}}_{n}$ . We have $L(\sigma)=L(c^{d}\sigma)$ for all $d\in{\mathbf{Z}}$ and

\ell(\sigma)=|\tilde{L}(\sigma)|=\sum_{0\leq i<j<n}\bigl|{\lfloor\frac{\sigma(j)-\sigma(i)}{n}\rfloor}\bigr|.

If $(i,j)\in L(\sigma)$ , then $\sigma s_{ij}<\sigma$ .

Assume $\sigma=c^{d}w$ and $w=s_{a_{1}}\cdots s_{a_{l}}$ is a reduced decomposition of $w\in W_{n}$ . Given $1\leq r\leq l$ , let $i_{r}\in\{1,\ldots,n\}$ with $i_{r}+n{\mathbf{Z}}=a_{r}$ .

The set $\{\bigl(s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}),s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)\bigr)\}_{1\leq r\leq l}$ is a subset of $L(\sigma)$ . This induces a bijection

\{\bigl((s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}),s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)\bigr)\}_{1\leq r\leq l}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L(\sigma)/n{\mathbf{Z}}.

0P51

Proof. Consider a pair $(i,j)\in L(\sigma)$ with $1\leq i\leq n$ and such that $(i,j^{\prime}){\not\in}L(\sigma)$ and $(j^{\prime},j){\not\in}L(\sigma)$ for $i<j^{\prime}<j$ . Given $j^{\prime}$ with $i<j^{\prime}<j$ , we have $\sigma(i)<\sigma(j^{\prime})<\sigma(j)$ , a contradiction. It follows that $j=i+1$ . We have

L(\sigma)=\bigl(\{(i,i+1)\}+n{\mathbf{Z}}\bigr)\coprod(s_{i,i+1},s_{i,i+1})(L(\sigma s_{i,i+1})).

We deduce by induction on $|\tilde{L}(\sigma)|$ that $\ell(\sigma)\leq|\tilde{L}(\sigma)|$ .

We prove the statements on $\{\bigl(s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}),s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)\bigr)\}_{1\leq r\leq l}$ by induction on $\ell(\sigma)$ . By induction, the statements hold for $\sigma s_{a_{l},a_{l}+1}$ . In particular, $\ell(\sigma s_{a_{l},a_{l}+1})=|\tilde{L}(\sigma s_{a_{l},a_{l}+1})|$ . It follows that $\ell(\sigma)=\ell(\sigma s_{a_{l},a_{l}+1})+1>|\tilde{L}(\sigma s_{a_{l},a_{l}+1})|$ . Assume $(i_{l},i_{l}+1){\not\in}L(\sigma)$ . It follows that $L(\sigma s_{a_{l},a_{l}+1})=s_{a_{l},a_{l}+1}(L(\sigma))\coprod\bigl(\{(i_{l},i_{l}+1)\}+n{\mathbf{Z}}\bigr)$ , hence $|\tilde{L}(\sigma)|<|\tilde{L}(\sigma s_{a_{l},a_{l}+1})|=\ell(\sigma s_{a_{l},a_{l}+1})=\ell(\sigma)-1$ , a contradiction. It follows that $(i_{l},i_{l}+1)\in L(\sigma)$ , hence

L(\sigma)=s_{a_{l},a_{l}+1}(L(\sigma s_{i_{l},i_{l}+1}))\coprod\bigl(\{(i_{l},i_{l}+1)\}+n{\mathbf{Z}}\bigr).

The last statement of the lemma follows now by induction.

Consider now $(i,j)\in L(\sigma)$ . Up to translating $(i,j)$ diagonally by $n{\mathbf{Z}}$ , we can assume there is $r$ such that $i=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r})$ and $j=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)$ . So $\sigma s_{i,j}=c^{d}s_{a_{1}}\cdots s_{a_{r-1}}s_{a_{r+1}}\cdots s_{a_{l}}$ , hence $\sigma s_{i,j}<\sigma$ . The lemma follows. ∎

0P52

Lemma 3.2.4. Given $\sigma,\sigma^{\prime}\in\hat{{\mathfrak{S}}}_{n}$ , we have $\sigma^{\prime}<\sigma$ and $\ell(\sigma^{\prime})=\ell(\sigma)-1$ if and only if there is $(j_{1},j_{2})\in L(\sigma)$ such that $\sigma^{\prime}=\sigma s_{j_{1},j_{2}}$ and

•

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

given $i\in{\mathbf{Z}}$ with $j_{1}<i<j_{2}$ , we have $\sigma(j_{1})<\sigma(i)$ or $\sigma(i)<\sigma(j_{2})$ .

0P53

Proof. Consider $(j_{1},j_{2})\in L(\sigma)$ and let $s=s_{j_{1},j_{2}}$ . Consider integers $i<j$ with $i-j{\not\in}n{\mathbf{Z}}$ .

If $s(i)<s(j)$ , then $(i,j)\in L(\sigma)$ if and only if $s(i,j)=(s(i),s(j))\in L(\sigma s)$ .

Assume now $s(i)>s(j)$ . We have three possibilities:

$\bullet\$ $i-j_{1}\in n{\mathbf{Z}}$ , $j-j_{2}{\not\in}n{\mathbf{Z}}$ : we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma(i)>\sigma(j)>\sigma s(i)$ (and then $(i,j){\not\in}L(\sigma s)$ )

$\bullet\$ $i-j_{1}{\not\in}n{\mathbf{Z}}$ , $j-j_{2}\in n{\mathbf{Z}}$ : we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma s(j)>\sigma(i)>\sigma(j)$ (and then $(i,j){\not\in}L(\sigma s)$ ).

$\bullet\$ $i=j_{1}+nr$ , $j=j_{2}+nr^{\prime}$ with $r,r^{\prime}\in{\mathbf{Z}}$ : we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma(j_{1})-\sigma(j_{2})>n(r^{\prime}-r)>\sigma(j_{2})-\sigma(j_{1})$ (and then $(i,j){\not\in}L(\sigma s)$ ).

We deduce there is an injective map $a:L(\sigma s)\to L(\sigma)$ given by

a((i,j))=\begin{cases}(i,j)&\text{ if }s(i)>s(j)\\ s(i,j)&\text{ otherwise}\end{cases}

and

L(\sigma)=a(L(\sigma s))\sqcup\coprod_{|r|<\min\bigl(\frac{j_{2}-j_{1}}{n},\frac{\sigma(j_{1})-\sigma(j_{2})}{n}\bigr)}\bigl((j_{1}+nr,j_{2})+n{\mathbf{Z}}\bigr)\sqcup\\ \coprod_{\begin{subarray}{c}j_{1}<i<j_{2}\\ \sigma(j_{1})>\sigma(i)>\sigma(j_{2})\end{subarray}}\Bigl(\bigl((j_{1},i)+n{\mathbf{Z}}\bigl)\sqcup\bigl((i,j_{2})+n{\mathbf{Z}}\bigr)\Bigr).

Note that $a(L(\sigma s))\sqcup((j_{1},j_{2})+n{\mathbf{Z}})\subset L(\sigma)$ .

Let us now prove the lemma. We have $\sigma=c^{d}w$ and $\sigma^{\prime}=c^{d^{\prime}}w^{\prime}\in\hat{{\mathfrak{S}}}_{n}$ for some $w,w^{\prime}\in W_{n}$ . Assume $\sigma^{\prime}<\sigma$ and $\ell(\sigma^{\prime})=\ell(\sigma)-1$ . We have $d=d^{\prime}$ , $w^{\prime}<w$ and $\ell(w^{\prime})=\ell(w)-1$ . It follows that there is a reduced decomposition $w=s_{a_{1}}\cdots s_{a_{l}}$ and $r\in\{1,\ldots,l\}$ such that $w^{\prime}=s_{a_{1}}\cdots s_{a_{r-1}}s_{a_{r+1}}\cdots s_{a_{l}}$ . Let $j_{1}=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r})$ and $j_{2}=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)$ . We have $(j_{1},j_{2})\in L(\sigma)$ and $\sigma^{\prime}=\sigma s_{j_{1},j_{2}}$ (Lemma 3.2.3).

The discussion above shows that $\{i\in{\mathbf{Z}}\ |\ j_{1}<i<j_{2},\ \sigma(j_{1})>\sigma(i)>\sigma(j_{2})\}=\emptyset$ and $\min\bigl(\frac{j_{2}-j_{1}}{n},\frac{\sigma(j_{1})-\sigma(j_{2})}{n}\bigr)<1$ . The lemma follows. ∎

0P54

Example 3.2.5. The elements of $\tilde{L}(\sigma)$ are in bijection with intersection points between strands of a “good diagram” representing $\sigma$ . Here, we define a strand diagram to be good if no more than two strands intersect at a given point and if the diagram minimizes the total number of intersection points. Similarly, the elements of $L(\sigma)$ correspond to intersections in an unfolded good strand diagram.

These descriptions can be deduced from Lemma 6.2.3 below, that shows those statements hold for pairs of strands. Now, the intersection point set for a good diagram is the disjoint union over intersection sets between pairs of strands, and a good diagram minimizes the intersection number among good diagrams if and only of each pair of strands minimizes its intersection number.

For example:

3.2.5. Extended affine Hecke algebra

We let $c$ act on the differential graded algebra $H^{\mathrm{nil}}(W_{n})$ by $c(T_{a})=T_{a+1}$ . Let $\hat{H}_{n}=H^{\mathrm{nil}}(W_{n})\rtimes\langle c\rangle$ . For $n\geq 2$ , it is the differential graded ${\mathbf{F}}_{2}$ -algebra generated by $\{T_{a}\}_{a\in{\mathbf{Z}}/n}$ and $c^{\pm 1}$ with relations

T_{a}^{2}=0,\ cT_{a}=T_{a+1}c,\ T_{a}T_{b}=T_{b}T_{a}\text{ if }a\neq b\pm 1

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

and differential $d(T_{a})=1$ , $d(c)=0$ . The element $c$ has degree $0$ , while $T_{a}$ has degree $-1$ . Note that $\hat{H}_{1}={\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{1}]={\mathbf{F}}_{2}\langle c\rangle$ , a differential graded algebra in degree $0$ with $d=0$ .

Let $w\in W_{n}$ , $d\in{\mathbf{Z}}$ and $w^{\prime}=wc^{d}$ . We put $T_{w^{\prime}}=T_{w}c^{d}$ . We also put $T_{\sigma}=T_{w}c^{d}$ for $\sigma=wc^{d}$ . The set $\{T_{\sigma}\}_{\sigma\in\hat{{\mathfrak{S}}}_{n}}$ is a basis of $\hat{H}_{n}$ .

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Remark 3.2.6. Define a filtration on ${\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{n}]$ with $({\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{n}])^{\geq-i}$ the subspace spanned by group elements $w\in\hat{{\mathfrak{S}}}_{n}$ with $\ell(w)\leq i$ . The associated graded algebra is $\hat{H}_{n}$ .

We put $\hat{H}_{0}={\mathbf{F}}_{2}$ .

0P56

Remark 3.2.7. The group $\hat{{\mathfrak{S}}}_{n}$ is more classically described as a semi-direct product ${\mathbf{Z}}^{n}\rtimes{\mathfrak{S}}_{n}$ (cf §3.2.2) coming from its description as the extended affine Weyl group of $\operatorname{GL}\nolimits_{n}$ . The nil affine Hecke algebra of $\operatorname{GL}\nolimits_{n}$ associated with this description (cf e.g. [Rou2, §2.2.2]) is not isomorphic to $\hat{H}_{n}$ . When considering invertible (instead of $0$ ) parameters, the two algebras are isomorphic.

0P57

Example 3.2.8. An element $T_{\sigma}$ of $\hat{H}_{n}$ will be representated by a good strand diagram for $\sigma$ . The multiplication of $T_{\sigma}$ and $T_{\sigma^{\prime}}$ is obtained by concatenating the diagrams of $\sigma$ and $\sigma^{\prime}$ (as in the multiplication of $\sigma$ and $\sigma^{\prime}$ ). If the corresponding diagram is good, then $T_{\sigma}T_{\sigma^{\prime}}=T_{\sigma^{\prime\prime}}$ , where $\sigma^{\prime\prime}$ is represented by the concatenated diagram. Otherwise, $T_{\sigma}T_{\sigma^{\prime}}=0$ . For example:

3.2.6. Positive versions

Let $\hat{{\mathfrak{S}}}_{n}^{+}$ be the submonoid of $\hat{{\mathfrak{S}}}_{n}$ of permutations $\sigma$ such that $\sigma({\mathbf{Z}}_{>0})\subset{\mathbf{Z}}_{>0}$ . Note that $\hat{{\mathfrak{S}}}_{n}^{+}$ is stable under left and right multiplication by ${\mathfrak{S}}_{n}$ .

There is a decomposition $\hat{{\mathfrak{S}}}_{n}^{+}=({\mathbf{Z}}_{\geq 0})^{n}\rtimes{\mathfrak{S}}_{n}$ .

We have $s_{r-1}s_{r-2}\cdots s_{1}cs_{n-1}s_{n-2}\cdots s_{r}=(\underbrace{0,\ldots,0,1,0,\ldots,0}_{\mathrm{pos.}r})\in({\mathbf{Z}}_{\geq 0})^{n}$ for $r\in\{1,\ldots,n\}$ , hence $v$ restricts to an isomorphism from the submonoid of $W_{n}\rtimes\langle c\rangle$ generated by $s_{1},\ldots,s_{n-1},c$ to $\hat{{\mathfrak{S}}}_{n}^{+}$ .

Let $\hat{H}_{n}^{+}=\bigoplus_{w\in\hat{{\mathfrak{S}}}_{n}^{+}}{\mathbf{F}}_{2}T_{w}$ , an ${\mathbf{F}}_{2}$ -subspace of $\hat{H}_{n}$ containing $H_{n}$ .

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Proposition 3.2.9. $\hat{H}_{n}^{+}$ is a differential graded subalgebra of $\hat{H}_{n}$ .

The algebra $\hat{H}_{n}^{+}$ has a presentation with generators $T_{1},\ldots,T_{n-1},c$ and relations

T_{i}^{2}=0,\ T_{i}T_{j}=T_{j}T_{i}\text{ if }|i-j|>1,\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}(\text{ if }n>2\ )

cT_{i}=T_{i+1}c\text{ for }1\leq i<n-1\text{ and }c^{2}T_{n-1}=T_{1}c^{2}.

The remainder of §3.2.6 will be devoted to the proof of Proposition 3.2.9.

Let $A_{n}$ be the $k$ -algebra with generators $t_{1},\ldots,t_{n-1},b$ and relations

t_{i}^{2}=0,\ t_{i}t_{j}=t_{j}t_{i}\text{ if }|i-j|>1,\ t_{i}t_{i+1}t_{i}=t_{i+1}t_{i}t_{i+1}(\text{ if }n>2\ )

bt_{i}=t_{i+1}b\text{ for }1\leq i<n-1\text{ and }b^{2}t_{n-1}=t_{1}b^{2}.

Given $i\in\{1,\ldots,n\}$ , we put $\beta_{i}=bt_{n-1}\cdots t_{i}$ . Given $I\subset\{1,\ldots,n\}$ non-empty with elements $1\leq i_{1}<\cdots<i_{r}\leq n$ , we put $\gamma_{I}=\beta_{i_{1}+r-1}\beta_{i_{2}+r-2}\cdots\beta_{i_{r}}$ . Note that $\gamma_{\{1,\ldots,n\}}=b^{n}$ .

There is a morphism of algebras $H_{n}\to A_{n},\ T_{i}\mapsto t_{i}$ and we denote by $t_{w}$ the image of $T_{w}$ for $w\in{\mathfrak{S}}_{n}$ .

0P59

Example 3.2.10. The elements of $\hat{{\mathfrak{S}}}_{n}^{+}$ correspond to strand diagrams where the strands wind positively around the cylinder. The relation $c^{2}T_{n-1}=T_{1}c^{2}$ is illustrated below:

We describe some elements $w\in\hat{{\mathfrak{S}}}_{7}^{+}$ and the image of $T_{w}$ in $A_{7}$ :

The element $(0,0,0,0,1,0,0)\in({\mathbf{Z}}_{\geq 0})^{7}$ corresponds to the following element of ${\mathfrak{S}}_{7}^{+}$ :

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Lemma 3.2.11. The set $\{t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}}\}$ with $w\in{\mathfrak{S}}_{n}$ , $m\geq 0$ and $I_{1}\subset\{1,\ldots,n\}$ , $I_{r}\subset\{1,\ldots,|I_{r-1}|\}$ for $1<r\leq m$ generates $A_{n}$ as a $k$ -vector space.

0P5B

Proof. Let $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,n-1\}$ . We have

\beta_{i}t_{j}=\begin{cases}t_{j+1}\beta_{i}&\text{ if }j<i-1\\ \beta_{i-1}&\text{ if }j=i-1\\ 0&\text{ if }j=i\\ t_{j}\beta_{i}&\text{ if }j>i.\end{cases}

Consider $I\subset\{1,\ldots,n\}$ non-empty with elements $1\leq i_{1}<\cdots<i_{r}\leq n$ . We put $i_{0}=0$ and $i_{r+1}=n+1$ .

Consider $j\in\{1,\ldots,n-1\}$ . Fix $k\in\{0,\ldots,r\}$ such that $i_{k}\leq j<i_{k+1}$ . Let us show that

(3.2.2)

\gamma_{I}t_{j}=\begin{cases}t_{j+r-k}\gamma_{I}&\text{ if }i_{k}<j<i_{k+1}-1\\ 0&\text{ if }i_{k}=j<i_{k+1}-1\\ \gamma_{\{i_{1}<\cdots<i_{k}<i_{k+1}-1<i_{k+2}<\cdots<i_{r}\}}&\text{ if }i_{k}<j=i_{k+1}-1\\ t_{k}\gamma_{I}&\text{ if }i_{k}=j=i_{k+1}-1.\end{cases}

We have

\gamma_{I}t_{j}=\beta_{i_{1}+r-1}\cdots\beta_{i_{k+1}+r-k-1}t_{j+r-k-1}\beta_{i_{k+2}+r-k-2}\cdots\beta_{i_{r}}.

If $j<i_{k+1}-1$ , then $\beta_{i_{k+1}+r-k-1}t_{j+r-k-1}=t_{j+r-k}\beta_{i_{k+1}+r-k-1}$ and we deduce the first two equalities in (3.2.2). Assume now $j=i_{k+1}-1$ . We have $\beta_{i_{k+1}+r-k-1}t_{j+r-k-1}=\beta_{i_{k+1}+r-k-2}$ and the third equality in (3.2.2) follows. The last equality from the fact that given $i\in\{1,\ldots,n-1\}$ , we have

	$\displaystyle\beta_{i+1}\beta_{i}$	$\displaystyle=b^{2}t_{n-2}\cdots t_{i}t_{n-1}\cdots t_{i}=b^{2}t_{n-1}\cdots t_{i}t_{n-1}\cdots t_{i+1}=t_{1}b^{2}t_{n-2}\cdots t_{i}t_{n-1}\cdots t_{i+1}$
		$\displaystyle=t_{1}\beta_{i+1}^{2}.$

We deduce that $\gamma_{I}t_{j}=u\gamma_{I^{\prime}}$ for some $I^{\prime}\subset\{1,\ldots,n\}$ with $|I^{\prime}|=|I|$ and $\max(I^{\prime})\leq\max(I)$ and $u\in\{0,1,t_{1},\ldots,t_{n-1}\}$ .

Fix $s\in\{1,\ldots,n\}$ with $s\geq\max(I)$ . We have

\gamma_{I}\beta_{s}=\begin{cases}\beta_{r}\gamma_{\{i_{2}-1,\ldots,i_{r}-1,s\}}&\text{ if }1\in I\\ \gamma_{(I-1)\cup\{s\}}&\text{ otherwise.}\end{cases}

Consider $I_{1},\ldots,I_{m}$ as in the lemma. Let $k$ be minimal such that $1{\not\in}I_{k}$ . We put $k=m+1$ if there is no such $k$ . Define $u=\gamma_{\{|I_{m}|\}}$ if $k=m+1$ and $u=1$ otherwise. Put $I_{0}=\{1,\ldots,n\}$ . Recall that $b=\beta_{n}$ . We have

\gamma_{I_{m}}\cdots\gamma_{I_{1}}b=u\gamma_{I^{\prime}_{m}}\cdots\gamma_{I^{\prime}_{1}}

where $I^{\prime}_{r}=\{i-1|i\in I_{r}\setminus\{1\}\}\cup\{|I_{r-1}|\}$ for $1\leq r<k$ , $I^{\prime}_{k}=\{i-1|i\in I_{k}\}\cup\{|I_{k-1}|\}$ and $I^{\prime}_{r}=I_{r}$ for $r>k$ .

We deduce that the set $B=\{t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}}\}$ of the lemma is stable under right multiplication by $t_{j}$ for $j\in\{1,\ldots,n-1\}$ and by $b$ . Since $B$ contains $1$ , it follows that $B$ is a generating family for $A_{n}$ as an ${\mathbf{F}}_{2}$ -vector space. ∎

0P5C

Remark 3.2.12. An example of the description of $\gamma_{I}t_{j}$ in the proof of Lemma 3.2.11 is given below:

0P5D

Proof of Proposition 3.2.9. Let $H$ be the subalgebra of $\hat{H}_{n}$ generated by $T_{1},\ldots,T_{n-1},c$ . This is a differential graded subalgebra of $\hat{H}_{n}$ . Given $w\in\hat{{\mathfrak{S}}}_{n}$ , let $|w|=\sum_{i=1}^{n}w(i)$ . Let $w\in\hat{{\mathfrak{S}}}_{n}^{+}$ , $w\neq 1$ . We show by induction on $\ell(w)+|w|$ that $T_{w}\in H$ .

Assume $\ell(ws_{i})<\ell(w)$ for some $i\in\{1,\ldots,n-1\}$ . We have $ws_{i}\in\hat{{\mathfrak{S}}}_{n}^{+}$ and $|ws_{i}|=|w|$ , hence by induction $T_{ws_{i}}\in H$ . We deduce that $T_{w}=T_{ws_{i}}T_{i}\in H$ .

Otherwise, we have $0<w(1)<\cdots<w(n)$ , hence $w(n)>n$ since $w\neq 1$ . It follows that $wc^{-1}\in\hat{{\mathfrak{S}}}_{n}^{+}$ and $|wc^{-1}|<|w|$ , hence $T_{wc^{-1}}\in H$ by induction. So $T_{w}=T_{wc^{-1}}T_{c}\in H$ .

We have shown that $\hat{H}_{n}^{+}\subset H$ . Since $\hat{H}_{n}^{+}$ is stable under right multiplication by $T_{c}$ and by $T_{i}$ for $i\in\{1,\ldots,n-1\}$ , it follows that $H=\hat{H}_{n}^{+}$ .

There is a surjective morphism of algebras $\rho:A_{n}\to\hat{H}_{n}^{+},\ t_{i}\mapsto T_{i},\ b\mapsto c$ . Given $I=\{i_{1}<\cdots<i_{r}\}$ a non-empty subset of $\{1,\ldots,n\}$ , we put

c_{I}=(cs_{n-1}\cdots s_{i_{1}+r-1})(cs_{n-1}\cdots s_{i_{2}+r-2})\cdots(cs_{n-1}\cdots s_{i_{r}})\in\hat{{\mathfrak{S}}}_{n}.

We have $c_{I}(i_{l})=n+l$ for $1\leq l\leq r$ and $c_{I}(j)=j+r-k$ if $i_{k}<j<i_{k+1}$ (where we put $i_{0}=0$ and $i_{r+1}=n+1$ ).

Let $E$ be the set of families $(I_{1},\ldots,I_{m})$ where $m\geq 0$ , $I_{1}\subset\{1,\ldots,n\}$ and $I_{r}\subset\{1,\ldots,|I_{r-1}|\}$ for $1<r\leq m$ .

Given $w\in{\mathfrak{S}}_{n}$ and $(I_{1},\ldots,I_{m})\in E$ , we have $\rho(t_{w}\gamma_{I_{1}}\cdots\gamma_{I_{m}})=T_{w}T_{c_{I_{1}}}\cdots T_{c_{I_{m}}}$ and that element is either $T_{wc_{I_{1}}\cdots c_{I_{m}}}$ or $0$ .

We define a map $\phi:({\mathbf{Z}}_{\geq 0})^{n}\to E$ . Let $a\in({\mathbf{Z}}_{\geq 0})^{n}$ . Let $m=\max\{a(i)\}_{1\leq i\leq n}$ . We put $I_{1}=a^{-1}({\mathbf{Z}}_{\geq 1})$ and we define inductively $I_{r}$ for $2\leq r\leq m$ by $I_{r}=c_{I_{r-1}}\cdots c_{I_{1}}(a^{-1}({\mathbf{Z}}_{\geq r}))$ . We put $\phi(a)=(I_{1},\ldots,I_{m})$ . We have

c_{I_{m}}\cdots c_{I_{1}}(i)=na(i)+|a^{-1}({\mathbf{Z}}_{>a(i)})|+(\text{position of }i\text{ in }a^{-1}(a(i))).

We define a map $\psi:E\to({\mathbf{Z}}_{\geq 0})^{n}$ . Let $(I_{1},\ldots,I_{m})\in E$ . We define $a\in({\mathbf{Z}}_{\geq 0})^{n}$ by $a(i)=\lfloor\frac{c_{I_{m}}\cdots c_{I_{1}}(i)-1}{n}\rfloor$ and we put $\psi(I_{1},\ldots,I_{m})=a$ . The maps $\psi$ and $\phi$ are inverse bijections. We deduce that the map $E\to({\mathfrak{S}}_{n}\setminus\hat{{\mathfrak{S}}}_{n}^{+})$ sending $(I_{1},\ldots,I_{m})$ to the class of $c_{I_{m}}\cdots c_{I_{1}}$ is bijective. It follows that the map ${\mathfrak{S}}_{n}\times E\to\hat{{\mathfrak{S}}}_{n}^{+},\ (w,(I_{1},\ldots,I_{m}))\mapsto wc_{I_{m}}\cdots c_{I_{1}}$ is bijective.

If $\rho(t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}})=T_{wc_{I_{m}}\cdots c_{I_{1}}}=0$ for some $w\in{\mathfrak{S}}_{n}$ and $(I_{1},\ldots,I_{m})\in E$ , then the bijectivty of the map above shows that the image of $\rho$ is the span of a proper subset of a basis of $\hat{H}_{n}^{+}$ , contradicting the surjectivity of $\rho$ .

This shows that the elements $\rho(t_{w}\gamma_{I_{m}}\cdots\gamma_{I_{1}})$ are distinct basis elements of $\hat{H}_{n}^{+}$ , hence $\rho$ is an isomorphism. ∎

0P5E

Remark 3.2.13. The same method as the one used in the proof of Proposition 3.2.9 shows that $\hat{{\mathfrak{S}}}_{n}^{+}$ is the free $({\mathfrak{S}}_{n},{\mathfrak{S}}_{n})$ -monoid on a generator $c$ with relations $c\cdot s_{r}=s_{r+1}\cdot c$ for $r\in\{1,\ldots,n-1\}$ and $c^{2}\cdot s_{n-1}=s_{1}\cdot c^{2}$ .

3.2.7. Pointed versions

Given $n\geq 0$ , we put $H_{n}^{\bullet}=({\mathfrak{S}}_{n})^{\mathrm{nil}}$ . This is the quotient of the free pointed monoid generated by $T_{1},\ldots,T_{n-1}$ by the relations (3.2.1). The differential is given by $d(T_{i})=1$ . Note that $k[H_{n}^{\bullet}]=H_{n}$ and $H_{n}^{\bullet}=\{0\}\cup\{T_{w}\}_{w\in{\mathfrak{S}}_{n}}$ .

We define $\hat{{\mathfrak{S}}}_{n}^{\operatorname{nil}\nolimits}$ to be the differential graded pointed monoid with underlying differential pointed set $\{T_{\sigma}\}_{\sigma\in\hat{{\mathfrak{S}}}_{n}}\coprod\{0\}$ and multiplication, grading and differential that of $\hat{H}_{n}$ .

We define $\hat{{\mathfrak{S}}}_{n}^{+,\operatorname{nil}\nolimits}$ to be its differential graded pointed submonoid with non-zero elements those that stabilize ${\mathbf{Z}}_{>0}$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2