Subsection 6.2.5. (0PGR) · ScalingStacks

6.2.5. Differential

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 ${\mathcal{S}}_{n}$ .

0P7S

Lemma 6.2.8. Let $\sigma,\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ . Given $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,I)$ with $\ell(\tau)=0$ , we have $\sigma^{\prime}<\sigma$ if and only if $\tau\circ\sigma^{\prime}<\tau\circ\sigma$ if and only if $\sigma^{\prime}\circ\tau<\sigma\circ\tau$ .

0P7T

Proof. Note that $\tau$ is an increasing bijection since $\ell(\tau)=0$ . We have $D(\tau\circ\sigma)=D(\sigma)$ and given $(i_{1},i_{2})\in D(\sigma)$ , we have $(\tau\circ\sigma)^{i_{1},i_{2}}=\tau\circ\sigma^{i_{1},i_{2}}$ . This shows the first equivalence. The second equivalence follows from the fact that $D(\sigma\circ\tau)=(\tau^{-1}\times\tau^{-1})(D(\sigma))$ and given $(i_{1},i_{2})\in D(\sigma)$ , we have $(\sigma\circ\tau)^{\tau^{-1}(i_{1}),\tau^{-1}(i_{2})}=\sigma^{i_{1},i_{2}}\circ\tau$ . ∎

0P7U

Lemma 6.2.9. Given $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ , there is a bijection

\tilde{D}(\sigma)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\},\ (i_{1},i_{2})\mapsto\sigma^{i_{1},i_{2}}.

Note that

\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\}=\{\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)\ |\ \sigma^{\prime}<\sigma,\ \deg(\sigma^{\prime})=\deg(\sigma)+1\}.

Given $(i_{1},i_{2})\in L(\sigma)$ , we have $(i_{1},i_{2})\in D(\sigma)$ if and only if $\deg_{D}(\sigma)=\deg_{D}(\sigma^{i_{1},i_{2}})-1$ for some subset (equivalently, for any subset) $D$ of $\{1,\ldots,n\}\times\{\pm 1\}$ that embeds in its projection on $\{1,\ldots,n\}$ .

0P7V

Proof. Let $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,I)$ be an increasing bijection. We have $D(\tau\circ\sigma)=D(\sigma)$ and

\{\sigma^{\prime\prime}\in\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}(I)\ |\ \sigma^{\prime\prime}<\tau\circ\sigma,\ \ell(\sigma^{\prime\prime})=\ell(\tau\circ\sigma)-1\}=\{\tau\circ\sigma^{\prime}\ |\ \sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J),\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\}

by Lemma 6.2.8. Since the first statement of the lemma holds for $\tau\circ\sigma$ by Lemma 3.2.4, it holds for $\sigma$ .

The other statements follow from Lemmas 6.2.4 and 6.2.5. ∎

0P7W

Lemma 6.2.10. Consider $\sigma^{\prime\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ and $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K)$ and let $\sigma=\sigma^{\prime}\sigma^{\prime\prime}$ . Assume $\ell(\sigma)=\ell(\sigma^{\prime})+\ell(\sigma^{\prime\prime})$ .

Let $(i_{1},i_{2})\in D(\sigma)\setminus(D(\sigma)\cap D(\sigma^{\prime\prime}))$ . Let $\alpha^{\prime\prime}=\sigma^{\prime\prime}s_{i_{1},i_{2}}$ and $\alpha^{\prime\prime}=(\sigma^{\prime})^{\sigma^{\prime\prime}(i_{1}),\sigma^{\prime\prime}(i_{2})}$ . We have $\sigma=\alpha^{\prime}\alpha^{\prime\prime}$ and $\ell(\sigma)=\ell(\alpha^{\prime})+\ell(\alpha^{\prime\prime})$ .

0P7X

Proof. Assume first $I=J=K$ . The lemma follows in that case from Lemmas 3.2.4 and 3.2.2.

Consider now the general case. There are increasing bijections $\tau:J\to I$ and $\tau^{\prime}:K\to J$ . We have $D(\sigma)=\tau^{-1}(D(\tau^{\prime}\sigma\tau))$ and $D(\sigma^{\prime\prime})=\tau^{-1}(D(\sigma^{\prime\prime}\tau))$ (proof of Lemma 5.4.7). The lemma follows now from the previous case applied to the decomposition $\tau^{\prime}\sigma\tau=(\tau^{\prime}\sigma^{\prime})(\sigma^{\prime\prime}\tau)$ . ∎

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

d(\sigma)=\sum_{(i_{1},i_{2})\in\tilde{D}(\sigma)}\sigma^{i_{1},i_{2}}\in\operatorname{Hom}\nolimits_{{\mathbf{F}}_{2}[{\mathcal{H}}_{n}]}(I,J).

0P7Y

Proposition 6.2.11. The maps $d$ equip the ${\mathbf{F}}_{2}$ -linear $\Gamma_{n}$ -graded category ${\mathbf{F}}_{2}[{\mathcal{H}}_{n}]$ with a differential $\Gamma_{n}$ -graded structure, hence equip ${\mathcal{H}}_{n}$ with a differential $\Gamma_{n}$ -graded pointed structure.

Given $I\subset{\mathbf{Z}}/n$ , the morphism $F_{I}$ induces an isomorphism of differential ${\mathbf{Z}}$ -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).

0P7Z

Proof. Note that Lemma 6.2.9 shows that $d$ is homogeneous of degree $1$ . The compatibility of $d$ with $F_{I}$ follows from Lemma 3.2.4.

Consider now $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}}(I,J)$ non-zero. There exists $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}}(J,I)$ with $\ell(\tau)=0$ . We have $d(\tau\circ\sigma)=\tau\circ d(\sigma)$ , hence $d^{2}(\tau\circ\sigma)=\tau\circ d^{2}(\sigma)$ . The compatibility of $F_{I}$ with $d$ shows that $d^{2}(\tau\circ\sigma)=0$ . Since $\tau$ is invertible, we deduce that $d^{2}(\sigma)=0$ .

Consider finally $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}}(J,K)$ and fix $\tau^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}}(K,J)$ with $\ell(\tau^{\prime})=0$ . We have $d(\tau^{\prime}\circ\sigma^{\prime}\circ\sigma\circ\tau)=\tau^{\prime}\circ d(\sigma^{\prime}\circ\sigma)\circ\tau$ and it follows from the compatibility of $F_{J}$ with $d$ that

	$\displaystyle d(\tau^{\prime}\circ\sigma^{\prime}\circ\sigma\circ\tau)$	$\displaystyle=F_{J}\bigl(d(F_{J}^{-1}(\tau^{\prime}\circ\sigma^{\prime}\circ\sigma\circ\tau))\bigr)=F_{J}\bigl(d(F_{J}^{-1}(\tau^{\prime}\circ\sigma^{\prime})\circ F_{J}^{-1}(\sigma\circ\tau))\bigr)$
		$\displaystyle=F_{J}\bigl(d(F_{J}^{-1}(\tau^{\prime}\circ\sigma^{\prime}))\circ F_{J}^{-1}(\sigma\circ\tau)\bigr)+F_{J}\bigl(F_{J}^{-1}(\tau^{\prime}\circ\sigma^{\prime}))\circ d(F_{J}^{-1}(\sigma\circ\tau)\bigr)$
		$\displaystyle=d(\tau^{\prime}\circ\sigma^{\prime})\circ\sigma\circ\tau+\tau^{\prime}\circ\sigma^{\prime}\circ d(\sigma\circ\tau).$

∎

0P80

Example 6.2.12. Elements of $\tilde{L}(\sigma)$ correspond to intersections in a representing diagram. Given $(i_{1},i_{2})\in\tilde{L}(\sigma)$ , the element $\sigma^{i_{1},i_{2}}$ correspond to the diagram obtained by smoothing the intersection point corresponding to $(i_{1},i_{2})$ . If $(i_{1},i_{2}){\not\in}\tilde{D}(\sigma)$ , the element associated to the diagram will vanish in ${\mathcal{H}}_{n}$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2