The Soergel bimodule $B_i$ of a simple reflection

The bimodule. Keep the ring $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$, the simple reflections $s_i$ and the invariants $R^{s_i}$ of The standard type-A reflection realization and its polynomial ring, and recall that $R$ is a graded $(R,R)$-bimodule and that shifts are defined as in Associative graded algebras, bimodules, and internal shifts. For $1\le i\le n-1$ put $B_i:=R{\otimes }_{R^{s_i}}R(1),$ the balanced tensor product (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums) of the $(R,R^{s_i})$-bimodule $R$ with the $(R^{s_i},R)$-bimodule $R$, equipped with the total internal grading of the tensor product and then shifted so that $(B_i{)}_d=(R{\otimes }_{R^{s_i}}R{)}_{d+1}(d\in \mathbb{Z})\mathrm{.}$ In the notation of Elias–Williamson the shift $(r)$ has $M(r{)}_d=M_{d+r}$, so $M(1{)}_d=M_{d+1}$; in the library convention $M\{r{\}}_d=M_{d-r}$ this is $M(1)=M\{-1\}$, i.e. $M(r)=M\{-r\}$ for every integer $r$. Both conventions appear on this page, and every shifted formula is read through this dictionary; when no ambiguity is possible we write $M(r)$.

The two outer actions. The left $R$-action is $r^{\mathrm{\prime }}(r\otimes r^{\mathrm{\prime }\mathrm{\prime }})=r^{\mathrm{\prime }}r\otimes r^{\mathrm{\prime }\mathrm{\prime }}$ and the right $R$-action is $(r\otimes r^{\mathrm{\prime }\mathrm{\prime }})r^{\mathrm{\prime }}=r\otimes r^{\mathrm{\prime }\mathrm{\prime }}{r}^{\mathrm{\prime }}$; both descend to the balanced tensor product, they commute, they are homogeneous of degree $0$, and they satisfy $R$-linearity on the relevant side, so $B_i$ is a graded $(R,R)$-bimodule over $R^{s_i}$-balanced tensors. Explicitly, the $(R,R)$-bimodule $B_i$ is generated by the two elements $1\otimes 1$ and $1\otimes {\alpha }_i$, which are homogeneous of degrees $-1$ and $+1$ respectively: ${\alpha }_i$ is a degree-$2$ element of $R$ and the external shift moves degrees by $-1$.

Elementary structure. Since $2$ is invertible in $\mathbb{Q}$, we have the decomposition $R=R^{s_i}\oplus {\alpha }_i{R}^{s_i}$ (the averaging idempotent $\frac{1}{2}(1+s_i)$ and its complement), so as a graded left $R$-module and again as a graded right $R$-module $B_i\cong R(-1)\oplus R(1),$ and $B_i$ is finite free of rank two on each side. The Demazure operator ${\mathrm{\partial }}_i$ of The standard type-A reflection realization and its polynomial ring is $R^{s_i}$-linear and surjective onto $R^{s_i}$, and ${\mathrm{\partial }}_i({\alpha }_{i}h)=2h$ for $h\in R^{s_i}$, a fact used when the exact sequences of $B_i$ are displayed later on this page.

Small $n$. For $n=0$ or $n=1$ there are no simple reflections and no $B_i$. The only word is the empty word, with bimodule $R$; the additive graded Bott–Samelson category contains all finite direct sums of shifts of $R$ and all degree-zero bimodule maps between them.