Subsection 4.4.3. (0PFU) · ScalingStacks

4.4.3. Relations

Let ${\mathcal{M}}$ be the strict monoidal pointed category generated by objects $a_{l}$ for $1\leq l\leq 3$ and maps $\lambda_{lm}:a_{l}a_{m}\to a_{m}a_{l}$ for $l\leq m$ with relations $\lambda_{ll}^{2}=0$ and

\lambda_{mn}l\circ m\lambda_{ln}\circ\lambda_{lm}n=n\lambda_{lm}\circ\lambda_{ln}m\circ l\lambda_{mn}\text{ for }l\leq m\leq n.

0P6B

Lemma 4.4.6. We have a pointed faithful strict monoidal functor

H:{\mathcal{M}}\to{\mathcal{U}}^{\bullet},\ a_{l}\mapsto e,\ \lambda_{lm}\mapsto\tau.

Given $l_{1},\ldots,l_{r},m_{1},\ldots,m_{r}\in\{1,2,3\}$ , the non-zero elements of $H(\operatorname{Hom}\nolimits_{\mathcal{M}}(a_{l_{1}}\cdots a_{l_{r}},a_{m_{1}}\cdots a_{m_{r}}))\subset H_{r}^{\bullet}$ are those $T_{w}$ with $w\in{\mathfrak{S}}_{r}$ such that for all $i,j\in\{1,\ldots,r\}$ with $i<j$ and $w(i)>w(j)$ , we have $l_{i}\leq l_{j}$ .

0P6C

Proof. Given the defining relations for ${\mathcal{U}}^{\bullet}$ , the construction of the lemma does define (uniquely) a monoidal functor $H$ .

Fix $l_{1},\ldots,l_{n}\in\{1,\ldots,3\}$ . Given $i\in\{1,\ldots,n-1\}$ such that $l_{i}\leq l_{i+1}$ , we put $\tilde{T}_{i}=a_{l_{1}}\cdots a_{l_{i-1}}\lambda_{l_{i},l_{i+1}}a_{l_{i+2}}\cdots a_{l_{n}}$ . Note that $\tilde{T}_{i}\tilde{T}_{i+1}\tilde{T}_{i}$ is well-defined if and only if $l_{i}\leq l_{i+1}\leq l_{i+2}$ , hence if and only if $\tilde{T}_{i+1}\tilde{T}_{i}\tilde{T}_{i+1}$ is well-defined. As a consequence, given $i_{1},\ldots,i_{r},j_{1},\ldots,j_{s}\in\{1,\ldots,n-1\}$ such that $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r}}$ and $\tilde{T}_{j_{1}}\cdots\tilde{T}_{j_{s}}$ are well-defined and $T_{i_{1}}\cdots T_{i_{r}}=T_{j_{1}}\cdots T_{j_{s}}$ , then we have $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r}}=\tilde{T}_{j_{1}}\cdots\tilde{T}_{j_{s}}$ . This shows the faithfulness of $H$ .

Consider $i_{1},\ldots,i_{r}$ such that $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r}}$ is well-defined and non-zero. Let $w=s_{i_{1}}\cdots s_{i_{r}}\in{\mathfrak{S}}_{n}$ . We show by induction on $r$ that given $(i,j)\in\tilde{L}(w)$ , we have $l_{i}\leq l_{j}$ .

Let $w^{\prime}=s_{i_{1}}\cdots s_{i_{r-1}}$ . Put $d=i_{r}$ and $w^{\prime}=ws_{d}$ . Since $T_{i_{1}}\cdots T_{i_{r}}\neq 0$ , we have $r=\ell(w)$ . We have $\tilde{L}(w)=\{(d,d+1)\}\coprod s_{d}(\tilde{L}(w^{\prime}))$ by Lemma 3.2.3. We have a well-defined map $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r-1}}$ from $a_{l_{1}}\cdots a_{l_{d-1}}a_{l_{d+1}}a_{l_{d}}a_{l_{d+2}}\cdots a_{l_{n}}$ . It follows by induction that given $(i,j)\in\tilde{L}(w^{\prime})$ , we have $l_{s_{d}(i)}\leq l_{s_{d}(j)}$ . Since $\tilde{L}(w)=\{(d,d+1)\}\coprod s_{d}(\tilde{L}(w^{\prime}))$ (Lemma 3.2.3), we deduce that $l_{i}\leq l_{j}$ for all $(i,j)\in\tilde{L}(w)$ .

Consider now $w\in{\mathfrak{S}}_{n}$ such that given $(i,j)\in\tilde{L}(w)$ , we have $l_{i}\leq l_{j}$ . Let $w=s_{i_{1}}\cdots s_{i_{r}}$ be a reduced decomposition of $w$ . We show by induction on $r$ that $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r}}$ is well-defined. As before, we define $d$ and $w^{\prime}$ . By induction on $r$ , the element $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r-1}}$ gives a well-defined map from $a_{l_{1}}\cdots a_{l_{d-1}}a_{l_{d+1}}a_{l_{d}}a_{l_{d+2}}\cdots a_{l_{n}}$ . Since $(d,d+1)\in\tilde{L}(w)$ , it follows that $l_{d}\leq l_{d+1}$ , hence $\tilde{T}_{d}$ is a well-defined map from $a_{l_{1}}\cdots a_{l_{n}}$ . We deduce that $\tilde{T}_{i_{1}}\cdots\tilde{T}_{i_{r}}$ . This shows that $T_{w}$ is in the image of $H$ . ∎

Given $l_{1},\ldots,l_{r},m_{1},\ldots,m_{r}\in\{1,2,3\}$ and $w\in{\mathfrak{S}}_{r}$ satisfying the assumptions of Lemma 4.4.6, we put $\lambda_{w}=H^{-1}(T_{w})$ .

We denote by ${\mathcal{M}}^{\prime}$ the strict monoidal $k$ -linear category obtained from $k[{\mathcal{M}}]$ by adding maps $\varepsilon:a_{1}a_{3}\to 1$ and $\eta:1\to a_{3}a_{1}$ and relations

a_{3}\varepsilon\circ\eta a_{3}=\operatorname{id}\nolimits,\ \varepsilon a_{1}\circ a_{1}\eta=\operatorname{id}\nolimits

\lambda_{23}=a_{3}a_{2}\varepsilon\circ a_{3}\lambda_{12}a_{3}\circ\eta a_{2}a_{3},\ \lambda_{13}=\varepsilon a_{3}a_{1}\circ a_{1}\lambda_{33}a_{1}\circ a_{1}a_{3}\eta

\lambda_{11}=\varepsilon a_{1}^{2}\circ a_{1}\varepsilon a_{3}a_{1}^{2}\circ a_{1}^{2}\lambda_{33}a_{1}^{2}\circ a_{1}^{2}a_{3}\eta a_{1}\circ a_{1}^{2}\eta.

There is a monoidal duality, i.e. a monoidal equivalence ${\mathcal{M}}^{\prime{\operatorname{opp}\nolimits}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{M}}^{\prime}$ given by

a_{1}\mapsto a_{3},\ a_{2}\mapsto a_{2},\ a_{3}\mapsto a_{1},\ \lambda_{12}\mapsto\lambda_{23},\ \lambda_{23}\mapsto\lambda_{12},\ \lambda_{13}\mapsto\lambda_{13}

\lambda_{11}\mapsto\lambda_{33},\ \lambda_{22}\mapsto\lambda_{22},\ \lambda_{33}\mapsto\lambda_{11},\ \varepsilon\mapsto\eta,\ \eta\mapsto\varepsilon.

0P6D

Lemma 4.4.7. Let $G_{1},\ldots,G_{n}\in\{a_{1},a_{2},a_{3}\}$ . We have

\lambda_{(1\cdots n+1)}\circ G_{1}\cdots G_{n}\eta=\lambda_{(n+2\cdots 2)}\circ\eta G_{1}\cdots G_{n}:G_{1}\cdots G_{n}\to a_{3}G_{1}\cdots G_{n}a_{1}

and

\varepsilon G_{1}\cdots G_{n}\circ\lambda_{(2\cdots n+2)}=G_{1}\cdots G_{n}\varepsilon\circ\lambda_{(n+1\cdots 1)}:a_{1}G_{1}\cdots G_{n}a_{3}\to G_{1}\cdots G_{n}.

0P6E

Proof. We have

	$\displaystyle a_{3}\lambda_{13}\circ\eta a_{3}$	$\displaystyle=a_{3}\varepsilon a_{3}a_{1}\circ a_{3}a_{1}\lambda_{33}a_{1}\circ a_{3}a_{1}a_{3}\eta\circ\eta a_{3}$
		$\displaystyle=a_{3}\varepsilon a_{3}a_{1}\circ\eta a_{3}^{2}a_{1}\circ\lambda_{33}a_{1}\circ a_{3}\eta$
		$\displaystyle=\lambda_{33}a_{1}\circ a_{3}\eta$

	$\displaystyle\lambda_{13}a_{1}\circ a_{1}\eta$	$\displaystyle=\varepsilon a_{3}a_{1}^{2}\circ a_{1}\lambda_{33}a_{1}^{2}\circ a_{1}a_{3}\eta a_{1}\circ a_{1}\eta$
		$\displaystyle=\varepsilon a_{3}a_{1}^{2}\circ a_{1}a_{3}^{2}\lambda_{11}\circ a_{1}a_{3}\eta a_{1}\circ a_{1}\eta$
		$\displaystyle=a_{3}\lambda_{11}\circ\varepsilon a_{3}a_{1}^{2}\circ a_{1}a_{3}\eta a_{1}\circ a_{1}\eta$
		$\displaystyle=a_{3}\lambda_{11}\circ\eta a_{1}\circ\varepsilon a_{1}\circ a_{1}\eta$
		$\displaystyle=a_{3}\lambda_{11}\circ\eta a_{1}$

	$\displaystyle\lambda_{23}a_{1}\circ a_{2}\eta$	$\displaystyle=a_{3}a_{2}\varepsilon a_{1}\circ a_{3}\lambda_{12}a_{3}a_{1}\circ\eta a_{2}a_{3}a_{1}\circ a_{2}\eta$
		$\displaystyle=a_{3}a_{2}\varepsilon a_{1}\circ a_{3}a_{2}a_{1}\eta\circ a_{3}\lambda_{12}\circ\eta a_{2}$
		$\displaystyle=a_{3}\lambda_{12}\circ\eta a_{2}$

It follows that the first statement of the lemma holds when $n=1$ . Consider now $n\geq 2$ . We prove the first statement of the lemma by induction on $n$ . We have

	$\displaystyle\lambda_{(n+2\cdots 2)}\circ\eta G_{1}\cdots G_{n}$	$\displaystyle=\lambda_{(n+2\cdots 3)}\circ(\lambda_{(23)}\circ\eta G_{1})G_{2}\cdots G_{n}$
		$\displaystyle=\lambda_{(n+2\cdots 3)}\circ(\lambda_{(12)}\circ G_{1}\eta)G_{2}\cdots G_{n}$
		$\displaystyle=\lambda_{(12)}\circ G_{1}(\lambda_{(n+1\cdots 2)}\circ\eta G_{2}\cdots G_{n})$
		$\displaystyle=\lambda_{(12)}\circ G_{1}(\lambda_{(1\cdots n)}\circ G_{2}\cdots G_{n}\eta)$
		$\displaystyle=\lambda_{(1\cdots n+1)}\circ G_{1}\cdots G_{n}\eta$

The second statement of the lemma follows by applying the duality of ${\mathcal{M}}^{\prime}$ . ∎

Lemmas 4.4.5 and 4.4.6 show that there is a $k$ -linear monoidal functor $R:{\mathcal{M}}^{\prime}\to{\mathcal{W}}$

a_{1}\mapsto F_{1},\ a_{2}\mapsto E_{2},\ a_{3}\mapsto E_{1},\ \lambda_{12}\mapsto\lambda,\lambda_{23}\mapsto\sigma,\ \lambda_{13}\mapsto\rho,\ \lambda_{11}\mapsto\tau_{1},\ \lambda_{22}\mapsto\tau_{2},\ \lambda_{33}\mapsto\tau_{1}

\eta\mapsto\eta_{1},\ \varepsilon\mapsto\varepsilon_{1}.

Given $l_{1},\ldots,l_{r},m_{1},\ldots,m_{r}\in\{1,2,3\}$ and $w\in{\mathfrak{S}}_{r}$ satisfying the assumptions of Lemma 4.4.6, we still denote by $\lambda_{w}$ the element $R(\lambda_{w})$ .

Lemma 4.4.7 has the following consequence.

0P6F

Lemma 4.4.8. Let $G_{1},\ldots,G_{n}\in\{E_{1},E_{2},F_{1}\}$ . We have

\lambda_{(1\cdots n+1)}\circ G_{1}\cdots G_{n}\eta_{1}=\lambda_{(n+2\cdots 2)}\circ\eta_{1}G_{1}\cdots G_{n}:G_{1}\cdots G_{n}\to E_{1}G_{1}\cdots G_{n}F_{1}

and

\varepsilon_{1}G_{1}\cdots G_{n}\circ\lambda_{(2\cdots n+2)}=G_{1}\cdots G_{n}\varepsilon_{1}\circ\lambda_{(n+1\cdots 1)}:F_{1}G_{1}\cdots G_{n}E_{1}\to G_{1}\cdots G_{n}.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2