Subsection 6.1.1. (0PGH) · ScalingStacks

6.1.1. Bimodules

Fix $r,n\geq 0$ . We define some bimodules $L^{\pm}(r,n)$ and $R^{\pm}(r,n)$ with underlying differential graded module $H_{r+n}$ , following §3.1.3 and Proposition 3.1.6.

We endow $L^{+}(r,n)$ (resp. $L^{-}(r,n)$ ) with a structure of differential graded $(H_{r}\otimes H_{n},H_{r+n})$ -bimodule where

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$H_{r+n}$ acts by right multiplication
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$h\in H_{r}$ acts by left multiplication by $h$ (resp. by $f_{n}(h)$ )
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$h\in H_{n}$ acts by left multiplication by $f_{r}\circ\iota_{n}(h)$ (resp. by $h$ ).

We endow $R^{+}(r,n)$ (resp. $R^{-}(r,n)$ ) with a structure of differential graded $(H_{r+n},H_{r}\otimes H_{n})$ -bimodule where

•

$H_{r+n}$ acts by left multiplication
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$h\in H_{r}$ acts by right multiplication by $h$ (resp. by $f_{n}(h)$ )
•

$h\in H_{n}$ acts by right multiplication by $f_{r}\circ\iota_{n}(h)$ (resp. by $h$ ).

0P77

Example 6.1.1. Elements of $L^{\pm}(r,n)$ and $R^{\pm}(r,n)$ can be represented by good strand diagrams in a rectangle, as in the examples below.

The actions are obtained by concatenation of diagrams (note that a diagram that is not good represents $0$ ), as in the example below, where we first apply the reflection of the rectangle swapping the top and the bottom, then rotate $90$ degrees anticlockwise the diagram of $h^{\prime}$ :

These bimodules coincide with (the nil version of) the bimodules introduced in §3.1.3, after restricting the action of $H_{r}\otimes H_{n}$ to $H_{r}$ :

L^{\pm}(r,n)=L^{\pm}(I,S)\text{ and }R^{\pm}(r,n)=L^{\pm}(S,I)\text{ where }S=\{s_{1},\ldots,s_{r+n-1}\}\text{ and }I=\{s_{1},\ldots,s_{r-1}\}.

Given $m\geq 0$ , we denote by $w_{m}\in{\mathfrak{S}}_{m}$ the longest element, i.e., $w_{m}(i)=m-i+1$ . We have two morphisms of differential graded ${\mathbf{F}}_{2}$ -modules (cf Proposition 3.1.6)

t_{r+n,r}^{\pm}=t_{S,I}^{\pm}:H_{r+n}\to H_{r}\langle\frac{1}{2}n(2r+n-1)\rangle

given by

t_{r+n,r}^{+}(T_{w})=\begin{cases}T_{w_{r}w_{r+n}w}&\text{ if }w\in w_{r+n}{\mathfrak{S}}_{r}\\ 0&\text{ otherwise}\end{cases}\text{ and }t_{r+n,r}^{-}(T_{w})=\begin{cases}T_{ww_{r+n}w_{r}}&\text{ if }w\in{\mathfrak{S}}_{r}w_{r+n}\\ 0&\text{ otherwise}\end{cases}

0P78

Example 6.1.2. Let us describe some examples of $t_{7,4}^{\pm}(T_{w})$ :

It is immediate that there is an isomorphism of differential graded $(H_{r+n},H_{r}\otimes H_{n})$ -modules

\operatorname{Hom}\nolimits_{H_{r+n}^{\operatorname{opp}\nolimits}}(L^{\pm}(r,n),H_{r+n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R^{\pm}(r,n),f\mapsto f(1)

and it follows from Proposition 3.1.6 that there is an isomorphism of differential graded $(H_{r}\otimes H_{n},H_{r+n})$ -modules

L^{\mp}(r,n)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{H_{r}^{\operatorname{opp}\nolimits}}(R^{\pm}(r,n),H_{r})\langle\frac{1}{2}n(2r+n-1)\rangle,\ h\mapsto(h^{\prime}\mapsto t_{n+r,r}^{\pm}(hh^{\prime})).

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2