Subsection 6.2.2. (0PGN) · ScalingStacks

6.2.2. Length

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

L(\sigma)=\{(i,i^{\prime})\in\tilde{I}^{2}\ |\ i<i^{\prime},\ \sigma(i)>\sigma(i^{\prime})\}

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 $\ell(\sigma)=|\tilde{L}(\sigma)|$ .

0P7F

Lemma 6.2.1. We have $\ell(\sigma^{\prime}\circ\sigma)\leq\ell(\sigma^{\prime})+\ell(\sigma)$ for all $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ and $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K)$ .

0P7G

Proof. We have

L(\sigma^{\prime}\circ\sigma)=\{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})>\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\sqcup\\ \{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})<\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\\ =\{(i_{1},i_{2})\in L(\sigma)\ |\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\sqcup(\sigma^{-1}\times\sigma^{-1})\bigl(\{(j_{1},j_{2})\in L(\sigma^{\prime})\ |\ \sigma^{-1}(j_{1})<\sigma^{-1}(j_{2})\}\bigr).

It follows that

\ell(\sigma^{\prime})+\ell(\sigma)-\ell(\sigma^{\prime}\circ\sigma)=2|\{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})>\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})<\sigma^{\prime}\circ\sigma(i_{2})\}/n{\mathbf{Z}}|\geq 0.

∎

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.

0P7H

Lemma 6.2.2. Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ . We have

\ell(\sigma)=\sum_{\begin{subarray}{c}0\leq i_{1}<i_{2}<n\\ i_{1},i_{2}\in\tilde{I}\end{subarray}}\bigl|{\lfloor\frac{\sigma(i_{2})-\sigma(i_{1})}{n}\rfloor}\bigr|.

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

0P7I

Lemma 6.2.3. Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ where $I=\{i_{1}+n{\mathbf{Z}},i_{2}+n{\mathbf{Z}}\}$ and $J=\{j_{1}+n{\mathbf{Z}},j_{2}+n{\mathbf{Z}}\}$ with $1\leq i_{1}\neq i_{2}\leq n$ , $1\leq j_{1}\neq j_{2}\leq n$ and $\sigma(i_{r})=j_{r}\pmod{n}$ for $r\in\{1,2\}$ . Fix $\beta:\{i_{1},i_{2},j_{1},j_{2}\}\to{\mathbf{R}}$ increasing with $|\beta(u)-\beta(v)|<1$ for all $u,v$ .

Consider $\gamma_{r}:[0,1]\to{\mathbf{R}}$ continuous with $\gamma_{r}(0)=\beta(i_{r})$ and $\gamma_{r}(1)=\beta(j_{r})+\frac{\sigma(i_{r})-j_{r}}{n}$ for $r\in\{1,2\}$ . We have

\ell(\sigma)\leq|\{t\in[0,1]\ |\ e^{2i\pi\gamma_{1}(t)}=e^{2i\pi\gamma_{2}(t)}\}|

with equality if, for all $r\in\{1,2\}$ , the map $\gamma_{r}$ is affine.

0P7J

Proof. Without loss of generality, we can assume $i_{1}<i_{2}$ . The lemma follows by applying the intermediate value theorem to $\gamma_{2}(t)-\gamma_{1}(t)$ and using Lemma 6.2.2, considering four cases according to the signs of $j_{2}-j_{1}$ and $\sigma(i_{2})-\sigma(i_{1})$ . ∎

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2