The standard type-A reflection realization and its polynomial ring

The ring and its grading. Fix $k=\mathbb{Q}$ and $n\ge 2$, and let $R=\mathbb{Q}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ commuting indeterminates (Polynomial rings in finitely many commuting indeterminates by iteration). Grade it by $\mathrm{deg}{x}_i=2$: $R_d$ is the $\mathbb{Q}$-span of the monomials of total degree $d\mathrm{/}2$ when $d$ is even, and $R_d=0$ when $d$ is odd. Thus $R$ is a nonnegatively graded commutative $\mathbb{Q}$-algebra with $R_0=\mathbb{Q}$, and every element of $R$ is a finite sum of homogeneous elements.

The reflection representation. Put $\mathfrak{h}:=R_2={⨁}_{i=1}^n\mathbb{Q}{x}_i$, so ${\dim }_{\mathbb{Q}}\mathfrak{h}=n$, and let $S_n$ act on $\mathfrak{h}$ by place permutations, $w\cdot x_i:=x_{w(i)},$ extended to a $\mathbb{Q}$-algebra automorphism of $R$; on polynomial functions this is the substitution $(w\cdot f)(x_1,\dots ,x_n)=f(x_{w(1)},\dots ,x_{w(n)})\mathrm{.}$ This is a left action: $v\cdot (w\cdot x_i)=v\cdot x_{w(i)}=x_{v(w(i))}=x_{(vw)(i)}$, and the substitution formula is the same statement read on functions, since $x_i\mapsto x_{w(i)}$ for all $i$ is the substitution by the tuple $(x_{w(1)},\dots ,x_{w(n)})$; the linear extension of $x_i\mapsto x_{w(i)}$ to $R$ is an automorphism because it merely permutes indeterminates. By Reduced adjacent-transposition words have well-defined positive lifts, $S_n$ is generated by the adjacent transpositions $s_i=(ii+1)$.

Two normalizations of the simple roots. Equip $\mathfrak{h}$ with the symmetric bilinear form $(x_a,x_b)={\delta }_{ab}$ and use it to identify ${\mathfrak{h}}^{\ast }$ with $\mathfrak{h}$.

(a) Coordinate roots. For $1\le i\le n-1$ put ${\beta }_i:=x_i-x_{i+1}\in {\mathfrak{h}}^{\ast }$ and ${\beta }_i^{\vee }:={\beta }_i\in \mathfrak{h}$, with $⟨{\beta }_i^{\vee },{\beta }_j⟩:=({\beta }_i,{\beta }_j)=2{\delta }_{ij}-{\delta }_{\mathrm{\mid }i-j\mathrm{\mid },1}\mathrm{.}$ Then $s_i(v):=v-⟨v,{\beta }_i⟩{\beta }_i^{\vee }$ is the place permutation of $x_i$ and $x_{i+1}$ on the basis: $s_i(x_a)=x_a-({\delta }_{ai}-{\delta }_{a,i+1})(x_i-x_{i+1})$ exchanges $x_i$ and $x_{i+1}$ and fixes the remaining $x_a$. We write $\mathrm{\Phi }$ for the root system $\{\pm (x_a-x_b):a\mathrm{\ne }b\}$, ${\mathrm{\Phi }}^{+}$ for the roots ${\beta }_{ab}:=x_a-x_b$ with $a<b$, and ${\mathrm{\Phi }}_{st}^{+}:={\mathrm{\Phi }}^{+}\cap (\mathbb{Q}{\beta }_s+\mathbb{Q}{\beta }_t)$ for the positive roots of the rank-two parabolic subsystem generated by $s,t$. With this choice the published length criterion of Finite Weyl strong exchange and deletion applies verbatim: $\mathrm{\ell }(x{s}_i)=\mathrm{\ell }(x)-1$ if and only if $x{\beta }_i<0$ in the positive system ${\mathrm{\Phi }}^{+}$. We call $(\mathfrak{h},\{{\beta }_i^{\vee }\},\{{\beta }_i\})$ the length normalization of the realization.

(b) Balanced roots. Put ${\epsilon }_i:=(-1{)}^{i-1}$ for $1\le i\le n-1$ and ${\alpha }_i:={\epsilon }_i{\beta }_i\in {\mathfrak{h}}^{\ast },{\alpha }_i^{\vee }:={\epsilon }_i{\beta }_i^{\vee }\in \mathfrak{h},$ so that the Cartan matrix of the realization is $⟨{\alpha }_i^{\vee },{\alpha }_j⟩={\epsilon }_i{\epsilon }_j({\beta }_i,{\beta }_j)=2{\delta }_{ij}+{\delta }_{\mathrm{\mid }i-j\mathrm{\mid },1}:$ adjacent pairings are $+1$ and distant pairings are $0$. Because ${\epsilon }_i^2=1$, the reflection $s_i(v)=v-⟨v,{\alpha }_i⟩{\alpha }_i^{\vee }=v-⟨v,{\beta }_i⟩{\beta }_i^{\vee }$ coincides with the transposition computed in (a); hence the two choices of simple roots define the same reflection representation, the same reflecting hyperplanes, the same invariant rings $R^{s_i}$ and the same root system $\mathrm{\Phi }$ as a set of elements of ${\mathfrak{h}}^{\ast }$. We call $(\mathfrak{h},\{{\alpha }_i^{\vee }\},\{{\alpha }_i\})$ the balanced normalization; it is the one attached to the diagrammatic calculus of The type-A diagrammatic Soergel category and its candidate bimodule functor. The two normalizations are related by the dictionary ${\alpha }_i={\epsilon }_i{\beta }_i,{\alpha }_i^{\vee }={\epsilon }_i{\beta }_i^{\vee },{\mathrm{\partial }}_i={\epsilon }_i{\mathrm{\partial }}_i^{\beta },{\mathrm{\partial }}_i^{\beta }:=\frac{f-s_i(f)}{{\beta }_i},{\mathrm{\partial }}_i:=\frac{f-s_i(f)}{{\alpha }_i},$ and every identity below is read in the normalization displayed in it; in particular the descent criterion in the balanced labeling reads $\mathrm{\ell }(x{s}_i)=\mathrm{\ell }(x)-1$ if and only if ${\epsilon }_{i}x{\alpha }_i<0$.

Invariants and the Demazure operator. Let $R^{s_i}$ denote the subring of $s_i$-invariants and, in the balanced normalization, define ${\mathrm{\partial }}_i(f):=\frac{f-s_i(f)}{{\alpha }_i}(f\in R),$ so that ${\mathrm{\partial }}_i={\epsilon }_i{\mathrm{\partial }}_i^{\beta }$ with ${\mathrm{\partial }}_i^{\beta }(f)=(f-s_i(f))\mathrm{/}{\beta }_i$; the kernel and the surjectivity below are the same for both operators, and the algebraic items of this page may use either. This is well defined and lands in $R^{s_i}$: $R$ is free over $R^{s_i}$ with basis $\{1,{\alpha }_i\}$, because $2$ is invertible in $\mathbb{Q}$ and every polynomial can be written uniquely as $f=g+{\alpha }_{i}h$ with $g=\frac{1}{2}(f+s_i(f))$ and $h=\frac{1}{2}{\mathrm{\partial }}_i(f)$; the two summands are independent over $R^{s_i}$ since ${\alpha }_i$ is not a zero divisor. Hence $s_i({\alpha }_{i}h)=-{\alpha }_i{s}_i(h)$ forces the ${\alpha }_i$-part of $f-s_i(f)$ to be $2{\alpha }_{i}h$, so ${\mathrm{\partial }}_i(g)=0$ for $g\in R^{s_i}$ and ${\mathrm{\partial }}_i({\alpha }_{i}h)=2h(h\in R^{s_i})\mathrm{.}$ The factor $2$ is the one displayed: applying ${\mathrm{\partial }}_i$ to ${\delta }_{i}h$ with ${\delta }_i:={\alpha }_i\mathrm{/}2$ returns $h$, so ${\mathrm{\partial }}_i$ is $R^{s_i}$-linear and surjective onto $R^{s_i}$ (and the same operator annihilates $R^{s_i}$, so its kernel is exactly $R^{s_i}$).

The stated properties of the realization. For this data:

1. it is faithful: if $w\in S_n$ fixes every element of $\mathfrak{h}$ then $x_{w(i)}=w\cdot x_i=x_i$ for all $i$, so $w(i)=i$ for all $i$ and $w=e$;
2. it is reflection faithful: every reflection of $S_n$ is a transposition $(ab)$, and its fixed space on $\mathfrak{h}$ is the hyperplane $x_a-x_b=0$; distinct transpositions have distinct fixed hyperplanes, since the hyperplane determines the unordered pair $\{a,b\}$;
3. it satisfies Demazure surjectivity (Elias–Williamson, Assumption 3.7): ${\alpha }_s\in {\mathfrak{h}}^{\ast }$ is a nonzero functional on $\mathfrak{h}$, hence surjective onto $\mathbb{Q}$, evaluation at ${\alpha }_s^{\vee }\mathrm{\ne }0$ is surjective on ${\mathfrak{h}}^{\ast }$, and ${\mathrm{\partial }}_s$ is surjective onto $R^{s_s}$ because ${\mathrm{\partial }}_s({\delta }_{s}h)=h$ for $h\in R^{s_s}$ with ${\delta }_s={\alpha }_s\mathrm{/}2$;
4. it is balanced in the sense of Elias–Williamson, Definition 3.6. Recall from Definition 3.5 that the two-colored quantum numbers are determined by $[0{]}_x=[0{]}_y=0$, $[1{]}_x=[1{]}_y=1$, $[2{]}_x=x$, $[2{]}_y=y$ together with the recursions $[2{]}_x[k{]}_y=[k+1{]}_x+[k-1{]}_x$ and $[2{]}_y[k{]}_x=[k+1{]}_y+[k-1{]}_y$. Fix $s\mathrm{\ne }t$ and put $x:=⟨{\alpha }_s^{\vee },{\alpha }_t⟩$, $y:=⟨{\alpha }_t^{\vee },{\alpha }_s⟩$. If $\mathrm{\mid }s-t\mathrm{\mid }>1$ then $m_{st}=2$ and $x=y=0$, so the technical condition (3.3) reads $[m_{st}{]}_x=[m_{st}{]}_y=[2{]}_0=0$ and $[m_{st}-1{]}_x=[m_{st}-1{]}_y=[1]=1$. If $\mathrm{\mid }s-t\mathrm{\mid }=1$ then $m_{st}=3$ and $x=y=1$; the recursion at $k=2$ gives $[3{]}_x=x[2{]}_y-[1{]}_x=1\cdot 1-1=0$ and likewise $[3{]}_y=0$, so (3.3) holds, while $[m_{st}-1{]}_x=[2{]}_1=1=[m_{st}-1{]}_y$. Thus $[m_{st}-1{]}_x=[m_{st}-1{]}_y=1$ for every pair $s,t$, which is balancedness, and the technical condition (3.3) also holds throughout. This is the hypothesis under which Elias–Williamson §§5.1–5.3 present the diagrammatic category with the simple relation families used on this page, and it is why the balanced normalization (b), rather than the coordinate roots of (a), carries the diagrammatic data. Every finite dihedral integer $2m_{st}$ is $4$ or $6$ here, hence invertible in $\mathbb{Q}$, and so is $2$.

Remark on the length normalization. In the coordinate-root normalization (a) the same realization has adjacent Cartan entries $⟨{\beta }_s^{\vee },{\beta }_t⟩=-1$, so $[m_{st}-1{]}_x=[2{]}_{-1}=-1\mathrm{\ne }1$ for adjacent $s,t$: that normalization is odd-unbalanced, and Elias–Williamson's balanced presentation does not apply to it as written. Since the reflections, the invariant rings, the root hyperplanes and the bimodules $B_i$ of this page depend only on the reflection representation and not on which linear forms represent the simple roots, the page uses the coordinate roots (a) for the length-theoretic and rank-two algebraic statements, where the published length criterion is stated in that notation, and the balanced roots (b) for the diagrammatic statements of Elias–Williamson §§5–7.

Small $n$. For $n=0$ or $n=1$, take $R=\mathbb{Q}[x_1,\dots ,x_n]$ (with $R=\mathbb{Q}$ when $n=0$). The group $S_n$ is trivial and there are no simple reflections. The empty-word bimodule is $R$; the additive graded Bott–Samelson category consists of finite direct sums of shifts of $R$, with degree-zero bimodule maps. Its Soergel category is its idempotent completion. In particular it contains the zero object, all shifts and all scalar maps; ${\mathrm{End}}^0(R)=\mathbb{Q}$. Statements involving a simple reflection have no instances in these cases.

Shift dictionary. Following Elias–Williamson we write $M(1)$ for the graded shift with $M(1{)}_d=M_{d+1}$. In the library convention for graded modules and their internal shifts, in which the shift $\{r\}$ has $(M\{r\}{)}_d=M_{d-r}$, this is the internal shift $M\{-1\}$, and $M(r{)}_d=M_{d+r}=M\{-r{\}}_d$ for all $r\in \mathbb{Z}$. This dictionary is used without further comment in every shifted formula on this page.