ScalingStacks

Fix a positive integer $n^{\prime}\leq n$ and an increasing injection $\alpha:\{1,\ldots,n^{\prime}\}\hookrightarrow\{1,\ldots,n\}$. We extend $\alpha$ to an increasing injection ${\mathbf{Z}}\to{\mathbf{Z}}$ by $\alpha(r+dn^{\prime})=\alpha(r)+dn$ for $r\in\{1,\ldots,n^{\prime}\}$ and $d\in{\mathbf{Z}}$.

Consider $\alpha:\{1,\ldots,n^{\prime}\}\hookrightarrow\{1,\ldots,n\}$ an increasing injection as in §6.2.1. We define two injective morphisms of groups

for $1\leq i\leq n^{\prime}$. We have commutative diagrams

As a consequence, we have two injective morphisms of groups

the last of which induces an injective morphism of groups $\Gamma_{D}\to\Gamma_{(\alpha\times\operatorname{id}\nolimits)(D)}$, for $D$ a subset of $\{1,\ldots,n^{\prime}\}\times\{\pm 1\}$ that embeds in its projection on $\{1,\ldots,n^{\prime}\}$.

We define now a fully faithful functor $F=F_{\alpha}:{\mathcal{S}}_{n^{\prime}}\to{\mathcal{S}}_{n}$. Given $I$ a subset of ${\mathbf{Z}}/n^{\prime}$, we define $F(I)$ to be the image of $\alpha(\tilde{I}\cap[1,n^{\prime}])$ in ${\mathbf{Z}}/n$. Given $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n^{\prime}}}(I,J)$, we put $F(\sigma)=\alpha\circ\sigma\circ\alpha^{-1}$.

Note that the isomorphism of groups $\hat{{\mathfrak{S}}}_{n^{\prime}}=\operatorname{End}\nolimits_{{\mathcal{S}}_{n^{\prime}}}({\mathbf{Z}}/n^{\prime})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}(F({\mathbf{Z}}/n^{\prime}))$ induced by $F$ coincides with $F_{F({\mathbf{Z}}/n^{\prime})}$ defined in §6.2.1.

As a consequence, $F_{\alpha}$ induces a fully faithful graded functor ${\mathcal{H}}_{n^{\prime}}\to{\mathcal{H}}_{n}$.