ScalingStacks

The graded \(k\)-linear category of Bott–Samelson bimodules for \(R_n\) is the graded \(k\)-linear full subcategory \(\mathrm{BSbim}^{\mathrm{gr}}_n\) of \(R_n\)-bimodules given by all graded \(R_n\)-bimodules of the form: \[ B_{\mathbf{i}}\langle j\rangle:=R\otimes_{R^{s_{i_k}}}R\otimes_{R^{s_{i_{k-1}}}}\cdots\otimes_{R^{s_{i_1}}}R\langle j-k\rangle\] for \(R=R_n\), \(j\in\mathbb{Z}\) and some \(\mathbf{i}=(i_k,i_{k-1},\ldots, i_1)\in \{1,2,\ldots n-1\}^k\) with \(k \in \mathbb{N}_0\), including the bimodules \(R\langle j\rangle\) in case \(k=0\). In the case \(k=1\) we also abbreviate: \[B_i:= R\otimes_{R^{s_{i}}}R\langle-1\rangle\]