The reduced type-A polynomial ring and Soergel bimodules for the HHH construction

Fix $m\ge 1$ and let $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$ be the polynomial ring of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution in $m$ commuting indeterminates, with the place-permutation action of the symmetric group $S_m$, the grading $\mathrm{deg}{x}_i=2$, the adjacent transpositions $s_i=(ii+1)$ and the invariant rings $(R^{\mathrm{\prime }}{)}^{s_i}$ of The standard type-A reflection realization and its polynomial ring. Write $x_i$ for the $i$-th coordinate function and let $S_m$ act by $w\cdot x_i=x_{w(i)}$.

The reduced ring. Put $y_i:=x_i-x_{i+1}$ for $1\le i\le m-1$ and $R:=\mathbb{Q}[y_1,\dots ,y_{m-1}]=\mathbb{Q}[x_1-x_2,x_2-x_3,\dots ,x_{m-1}-x_m]\subset R^{\mathrm{\prime }},$ the reduced polynomial ring of the HHH construction, with the restricted $S_m$-action and the induced grading $\mathrm{deg}{y}_i=2$. On the displayed generators the restricted action is $s_i(y_i)=-y_i,s_i(y_{i+1})=y_i+y_{i+1},s_i(y_{i-1})=y_{i-1}+y_i,s_i(y_j)=y_j(\mathrm{\mid }i-j\mathrm{\mid }\ge 2),$ and it extends uniquely to a $\mathbb{Q}$-algebra automorphism of $R$, because the displayed polynomials lie in $R$. Write $R^{s_i}\subset R$ for the invariant subring of the involution $s_i$; put $z_j=y_j+\frac{1}{2}{y}_i$ when $\mathrm{\mid }j-i\mathrm{\mid }=1$ and $z_j=y_j$ when $\mathrm{\mid }j-i\mathrm{\mid }\ge 2$. Then $s_i$ fixes each $z_j$ and sends $y_i$ to $-y_i$; the linear change of variables is invertible, and explicitly $R^{s_i}=\mathbb{Q}[z_j(j\mathrm{\ne }i),y_i^2]$.

The invariant coordinate. For $1\le i\le m-1$ put $t_i:=x_i+x_{i+1}\in R^{\mathrm{\prime }}$. Then $s_i$ fixes $t_i$, and the substitution $x_i=\frac{1}{2}(t_i+y_i),x_{i+1}=\frac{1}{2}(t_i-y_i),x_{i+2}=x_{i+1}-y_{i+1},x_{i-1}=x_i+y_{i-1},\dots$ expresses every coordinate as a polynomial in $t_i$ and the $y_j$ with coefficients in $\mathbb{Q}$ ($2$ is invertible). The substitution $\phi \text{\hspace{0.17em}⁣}:\mathbb{Q}[y_1,\dots ,y_{m-1},T]\to R^{\mathrm{\prime }}$ with $y_j\mapsto y_j$ and $T\mapsto t_i$ is the linear change of variables $(x_1,\dots ,x_m)\leftrightarrow (y_1,\dots ,y_{m-1},t_i)$, which is invertible over $\mathbb{Q}$ by the preceding display, hence an isomorphism of graded rings onto $R^{\mathrm{\prime }}$; consequently $R^{\mathrm{\prime }}=R[t_i]=R{\otimes }_{\mathbb{Q}}\mathbb{Q}[t_i]$ as graded rings. This polynomial extension is a free graded $R$-module on the infinite basis $1,t_i,t_i^2,\dots$. The extension of the invariant subring instead has rank two: $R^{\mathrm{\prime }}$ is free over $(R^{\mathrm{\prime }}{)}^{s_i}$ on $1,y_i$ (or on $1,x_i$), since $R=R^{s_i}\oplus y_i{R}^{s_i}$. Since $s_i$ fixes $t_i$ and acts on the coefficients through $R$, $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[t_i]=R^{s_i}{\otimes }_{\mathbb{Q}}\mathbb{Q}[t_i]\mathrm{.}$ Concretely $(R^{\mathrm{\prime }}{)}^{s_i}=\mathbb{Q}[x_1,\dots ,x_{i-1},t_i,x_i{x}_{i+1},x_{i+2},\dots ,x_m]$ and $x_i{x}_{i+1}=\frac{1}{4}(t_i^2-y_i^2)$; both displays generate the same subring because $y_i^2=t_i^2-4x_i{x}_{i+1}$.

The reduced simple-reflection bimodule. Since $2$ is invertible in $\mathbb{Q}$, the averaging idempotent $\frac{1}{2}(1+s_i)$ splits $R=R^{s_i}\oplus y_i{R}^{s_i}$ as graded $R^{s_i}$-modules: for $f\in R$ the element $\frac{1}{2}(f-s_i(f))$ equals $y_{i}g$ with $g=\frac{1}{2}(f-s_i(f))\mathrm{/}{y}_i\in R^{s_i}$. Hence the balanced tensor product of the graded $(R,R^{s_i})$-bimodule $R$ with the graded $(R^{s_i},R)$-bimodule $R$ $B_i:=R{\otimes }_{R^{s_i}}R$ is free of rank two as a left $R$-module and as a right $R$-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; we use the library's internal shift convention $(M\{r\}{)}_d=M_{d-r}$ for graded modules, in which $B_i$ carries no internal shift.

Comparison with the published type-A bimodule. The published type-A Soergel bimodule of a simple reflection works with the ambient ring $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$ in place of the reduced ring and with the shifted bimodule $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}:=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}(1)=B_i^{\mathrm{\prime }}\{-1\},$ where $B_i^{\mathrm{\prime }}:=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}$ is the corresponding unshifted balanced tensor; the dictionary is extended on the next item of this page. In particular the element $1\otimes 1$ of the published bimodule has degree $-1$, while the element $1\otimes 1$ of $B_i$ has degree $0$.

Source convention. Khovanov writes $R^{\mathrm{\prime }}=R{\otimes }_{\mathbb{Q}}\mathbb{Q}[x_j]$ for the coordinate $x_j$ and chooses $j=1$. Read with $x_1$ this is exact as a statement about graded rings, but the identification of invariant subrings $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[x_1]$ is a literal equality of subrings of $R^{\mathrm{\prime }}$ only when $s_i$ fixes $x_1$, that is for $i\ge 2$; for $i=1$ the $s_1$-invariant coordinate is $t_1=x_1+x_2$, and all statements of this page are therefore stated with the invariant coordinate $t_i$, the two forms being related by the substitution above. This is a convention correction, not a change of the source's construction.

Small cases. For $m=1$ there are no differences and $R=\mathbb{Q}$; there are no simple reflections, and the empty-word bimodule is $R$ itself. For $m=2$ one has $R=\mathbb{Q}[y_1]$, $R^{s_1}=\mathbb{Q}[y_1^2]$ and $B_1=\mathbb{Q}[y_1]{\otimes }_{\mathbb{Q}[y_1^2]}\mathbb{Q}[y_1]$.

No choice principle is used in this definition.