Unreduced type-A Soergel bimodules and the trivial polynomial factor

Keep the ambient ring $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$, the reduced ring $R\subset R^{\mathrm{\prime }}$ generated by the differences $y_j=x_j-x_{j+1}$, the invariant rings $R^{s_i}$, the invariant coordinates $t_i=x_i+x_{i+1}$ and the reduced bimodules $B_i=R{\otimes }_{R^{s_i}}R$ of The reduced type-A polynomial ring and Soergel bimodules for the HHH construction, with the internal shift convention $(M\{r\}{)}_d=M_{d-r}$ of Associative graded algebras, bimodules, and internal shifts.

The unreduced simple-reflection bimodule. For a simple reflection $s_i$ put $R_i^{\mathrm{\prime }}:=\mathbb{Q}[x_1,\dots ,x_{i-1},x_i+x_{i+1},x_i{x}_{i+1},x_{i+2},\dots ,x_m]=(R^{\mathrm{\prime }}{)}^{s_i},$ the $s_i$-invariant subring of $R^{\mathrm{\prime }}$ (symmetric polynomials in $x_i,x_{i+1}$), and $B_i^{\mathrm{\prime }}:=R^{\mathrm{\prime }}{\otimes }_{R_i^{\mathrm{\prime }}}{R}^{\mathrm{\prime }}$ with no internal shift. Since $R^{\mathrm{\prime }}=R[t_i]$ and $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[t_i]$ by The reduced type-A polynomial ring and Soergel bimodules for the HHH construction, and $R^{\mathrm{\prime }}=R^{s_i}[t_i]\oplus y_i{R}^{s_i}[t_i]$, the bimodule $B_i^{\mathrm{\prime }}$ is free of rank two as a left and as a right $R^{\mathrm{\prime }}$-module, with left basis $1\otimes 1,1\otimes y_i$ and right basis $1\otimes 1,y_i\otimes 1$, of degrees $0,2$ on either side.

The trivial polynomial factor. Write $t=t_i$ and let $R_t\subseteq R^{\mathrm{\prime }}$ denote the polynomial ring $\mathbb{Q}[t]$ regarded as a subring of $R^{\mathrm{\prime }}$; the coordinate $t$ is $s_i$-invariant, hence central. The assignment on $R_i^{\mathrm{\prime }}$-balanced tensors $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:B_i^{\mathrm{\prime }}⟶B_i{\otimes }_{\mathbb{Q}}\mathbb{Q}[t],\mathrm{\Phi }(r{t}^a\otimes r^{\mathrm{\prime }}{t}^b)=(r\otimes r^{\mathrm{\prime }})\otimes t^{a+b}(r,r^{\mathrm{\prime }}\in R,a,b\ge 0)$ is an isomorphism of graded $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodules, where $R^{\mathrm{\prime }}=R[t]$ acts on the target by $R$ on the $B_i$-factor and by multiplication on $\mathbb{Q}[t]$. It is well defined: if $h={\sum }_a{h}_a{t}^a\in R^{s_i}[t]=R_i^{\mathrm{\prime }}$ with $h_a\in R^{s_i}$, then $\mathrm{\Phi }(h{t}^c\otimes 1)={\sum }_a(h_a\otimes 1)\otimes t^{a+c}$ and $\mathrm{\Phi }(1\otimes h{t}^c)={\sum }_a(1\otimes h_a)\otimes t^{a+c}$ agree because $h_a\otimes 1=1\otimes h_a$ in $B_i$. It is bijective because both sides are free of rank two over $R^{\mathrm{\prime }}$ on the respective generators $1\otimes 1,1\otimes y_i$ and $(1\otimes 1)\otimes 1,(1\otimes y_i)\otimes 1$, which $\mathrm{\Phi }$ matches. The factor $\mathbb{Q}[t]$ is the trivial polynomial factor: the coordinate $t$ acts by the same multiplication on both sides of the target, which is exactly the statement that it lies in the invariant ring $(R^{\mathrm{\prime }}{)}^{s_i}$ and hence balances in $B_i^{\mathrm{\prime }}$.

The reduced maps and their trivial extensions. Define degree-zero maps of graded $(R,R)$-bimodules $b{r}_i\text{\hspace{0.17em}⁣}:B_i\to R,b{r}_i(a\otimes b)=ab,$ $r{b}_i\text{\hspace{0.17em}⁣}:R\{2\}\to B_i,r{b}_i(1)=y_i\otimes 1+1\otimes y_i\mathrm{.}$ Both are well defined: $b{r}_i$ kills the balancing relation $h\otimes 1-1\otimes h$ for $h\in R^{s_i}$, and $r{b}_i$ is left $R$-linear by construction while the identity $r(y_i\otimes 1+1\otimes y_i)=(y_i\otimes 1+1\otimes y_i)r$ holds for every invariant generator $z_j$ because it balances, and for $r=y_i$ because $y_i^2\in R^{s_i}$. These elements generate $R$, as proved in The reduced type-A polynomial ring and Soergel bimodules for the HHH construction. Both maps have degree zero in the internal grading, since $\mathrm{deg}{y}_i=2$ and $\mathrm{deg}1\in R\{2\}=2$. Likewise $b{r}_i^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:B_i^{\mathrm{\prime }}\to R^{\mathrm{\prime }},b{r}_i^{\mathrm{\prime }}(a\otimes b)=ab,$ $r{b}_i^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:R^{\mathrm{\prime }}\{2\}\to B_i^{\mathrm{\prime }},r{b}_i^{\mathrm{\prime }}(1)=(x_i-x_{i+1})\otimes 1+1\otimes (x_i-x_{i+1}),$ are well-defined degree-zero $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodule maps by the same argument with $R^{\mathrm{\prime }}$ in place of $R$ and $(R^{\mathrm{\prime }}{)}^{s_i}$ in place of $R^{s_i}$. Under $\mathrm{\Phi }$ the elements $1\otimes 1\mapsto (1\otimes 1)\otimes 1$ and $y_i\otimes 1+1\otimes y_i\mapsto r{b}_i(1)\otimes 1$ correspond, so $b{r}_i^{\mathrm{\prime }}$ and $r{b}_i^{\mathrm{\prime }}$ restrict to the trivial-factor extensions of $b{r}_i$ and $r{b}_i$.

Products over the layers. Let $D$ be a marked MOY resolution with $r$ wide edges, at positions $s_1,\dots ,s_r$ in the layer order, and put $B^{\mathrm{\prime }}(D):={⨂}_{j=1}^r{B}_{s_j}^{\mathrm{\prime }},B(D):={⨂}_{j=1}^r{B}_{s_j},$ the balanced tensor products over $R^{\mathrm{\prime }}$ respectively over $R$ in the layer order. Choose the common invariant coordinate $t=(x_1+\cdots +x_m)\mathrm{/}m$; then $R^{\mathrm{\prime }}=R[t]$ and $t$ is fixed by every simple reflection. Applying the preceding single-factor construction with this $t$ gives an isomorphism of graded $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodules $B^{\mathrm{\prime }}(D)⟶B(D){\otimes }_{\mathbb{Q}}\mathbb{Q}[t]\mathrm{.}$ Indeed, each factor is $B_{s_j}\otimes \mathbb{Q}[t]$, and balancing over $R[t]$ combines the polynomial factors by multiplication. The inverse sends $(b_1{\otimes }_R\cdots {\otimes }_R{b}_r)\otimes t^a$ to the tensor with $t^a$ in its first factor. For $r=0$ this is the ring identity $R^{\mathrm{\prime }}=R[t]$. The two maps preserve the left and right actions and grading. An arbitrary coordinate such as $x_1$ need not act equally on both sides; its left and right actions are recovered from $x_1=t+(x_1-t)$, with $x_1-t\in R$.

Normalization dictionary with the published bimodule. The published The Soergel bimodule $B_i$ of a simple reflection has $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}(1)=B_i^{\mathrm{\prime }}\{-1\}$ because the Elias-Williamson shift $(1)$ is the internal shift $\{-1\}$ in the library convention, that is $M(r)=M\{-r\}$; equivalently $B_i^{\mathrm{\prime }}=B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}(-1)=B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}\{1\}\mathrm{.}$ The dictionary $M(r)=M\{-r\}$ is used throughout this page, and the generator $1\otimes 1$ of $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}$ has degree $-1$ while the generator $1\otimes 1$ of $B_i^{\mathrm{\prime }}$ has degree $0$.

Source convention. Khovanov writes the polynomial factor as $\mathbb{Q}[x_1]$. The ring splitting $R^{\mathrm{\prime }}=R[x_1]$ is valid, but a literal trivial factor for the bimodule actions uses a common invariant coordinate $t$ as above. The change of coordinates $x_1=t+(x_1-t)$ translates the source's ring notation into this bimodule description.

No choice principle is used in this definition.