For \(n \geq 2\) and \(R=R_n\) as above and \(w\in S_n\) we let \(R_{\circlearrowleft w}\) denote the graded \(R\)-bimodule which is isomorphic to \(R\) as left \(R\)-module and with right-action twisted by \(w\): i.e. \(r\in R\) acts on \(R_w\) from the right as multiplication by \(w(r)\). We emphasize that for non-trivial \(w\), this \(R_n\)-bimodule \(R_{\circlearrowleft w}\) is not an object of \(\mathrm{Sbim}_n\).
However, for \(1\leq i\leq n-1\), the bimodule morphism \(R_{\circlearrowleft s_i}\langle 1\rangle \rightarrow B_i\) determined by \(1\mapsto x_i\otimes 1 - 1\otimes x_i\) induces a quasi-isomorphism \(R_{\circlearrowleft s_i}\langle 1\rangle \rightarrow F(\sigma_i)\). Likewise, the multiplication map \(B_i \rightarrow R_{\circlearrowleft s_i}\langle -1 \rangle\) determined by \(1\otimes 1 \mapsto 1\) induces a quasi-isomorphism \(F(\sigma^{-1})\rightarrow R_{\circlearrowleft s_i}\langle -1\rangle\).
Up to a grading shift, the generating Rouquier complexes, and more generally, the Rouquier complexes of positive resp. negative permutation braids, can hence be identified with permutation bimodules upon proceeding to the derived category \(\mathrm{D}^b({}_R\mathrm{grbmod}_{R})\) of graded \(R\)-\(R\)-bimodules. To obtain an interesting (non-symmetric) braiding, it is thus essential to work up-to-chain-homotopy, rather than up-to-quasi-isomorphism. Nevertheless, the comparison with permutation bimodules is important in this paper and the grading shifts in the following definition are motivated by it.