The real type-A Soergel category by scalar extension

Scalar extension of the published realization. The published type-A page works over $k=\mathbb{Q}$ with the polynomial ring $R=\mathbb{Q}[x_1,\dots ,x_n]$ and the two normalizations of the simple roots of The standard type-A reflection realization and its polynomial ring: the coordinate roots ${\beta }_i=x_i-x_{i+1}$ and the alternating balanced roots ${\alpha }_i^{\mathrm{b}\mathrm{a}\mathrm{l}}={\epsilon }_i{\beta }_i$, ${\epsilon }_i=(-1{)}^{i-1}$. For the Hodge-theoretic statements of this page we extend scalars along $\mathbb{Q}\hookrightarrow \mathbb{R}$ and set $S:=\mathbb{R}[x_1,\dots ,x_n],\mathrm{deg}{x}_i=2,$ with $S_0=\mathbb{R}$ and finite-dimensional homogeneous pieces $S_d$; the symmetric group $S_n$ acts by the place permutations $w\cdot x_i:=x_{w(i)}$ (The standard type-A reflection realization and its polynomial ring). On this page ${\alpha }_i$ always means the coordinate root ${\alpha }_i:={\beta }_i=x_i-x_{i+1}$, and ${\alpha }_i^{\vee }:={\beta }_i^{\vee }$ is the coordinate coroot, so that under the standard form the Cartan entries are $⟨{\alpha }_i^{\vee },{\alpha }_j⟩=2{\delta }_{ij}-{\delta }_{\mathrm{\mid }i-j\mathrm{\mid },1}$ and adjacent entries are $-1$; the reflection is $s_i(v)=v-⟨v,{\alpha }_i⟩{\alpha }_i^{\vee }$. The alternating balanced roots ${\alpha }_i^{\mathrm{b}\mathrm{a}\mathrm{l}}$ are retained by the published page for its diagrammatic presentation and are not used for positivity, Demazure traces or descent signs on this page; both choices define the same reflections, the same reflecting hyperplanes, the same invariant rings $S^{s_i}$ and the same underlying bimodules, and they differ only by the unit $\pm 1$ in front of the root, which changes no tensor bimodule but does change the normalization of ${\mathrm{\partial }}_i$ and of the trace.

The real generators and the real category. For $1\le i\le n-1$ set $B_i^R:=S{\otimes }_{S^{s_i}}S(1),$ the balanced tensor product of the $(S,S^{s_i})$-bimodule $S$ with the $(S^{s_i},S)$-bimodule $S$, shifted in the Elias-Williamson convention $M(1{)}_d=M_{d+1}$ used on this page (The Soergel bimodule $B_i$ of a simple reflection, Associative graded algebras, bimodules, and internal shifts). Its left basis $\{1\otimes 1,1\otimes {\alpha }_i\}$ and right basis $\{1\otimes 1,{\alpha }_i\otimes 1\}$ have degrees $-1$ and $+1$, and $B_i^R$ is finite free of rank two on each side (Soergel generators and Bott–Samelson products are finite free on both sides). The real type-A Soergel category ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n^R$ is the idempotent completion (The type-A Soergel category ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$) of the full subcategory of graded $(S,S)$-bimodules whose objects are the finite direct sums of shifts $B_{\underset{‾}{i}}^R(d)=B_{i_1}^R{\otimes }_S\cdots {\otimes }_S{B}_{i_r}^R(d)$ of real Bott-Samelson products (The Bott–Samelson bimodule of a word), with the same internal shifts as the rational category, i.e. $M(d{)}_e=M_{d+e}$ in the Elias-Williamson convention. Its unit is the graded ring $S$ itself, regarded as an object in cohomological degree zero; it is not a polynomial ring concentrated in internal degree zero. Categorically, ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n^R=\mathrm{K}\mathrm{a}\mathrm{r}({\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n^R)$ has the same formal properties as the rational Soergel category: it is graded, additive, idempotent-complete and monoidal with unit $S$, and every object is a finite direct sum of graded shifts of indecomposables.

Scalar extension of objects. For a graded $(R,R)$-bimodule $M$ over the rational polynomial ring put $M^R:=M{\otimes }_{\mathbb{Q}}\mathbb{R}$, with the $S$-actions induced from the $R$-actions and the tensor grading. Since $\mathbb{Q}\to \mathbb{R}$ is flat, scalar extension is exact on graded pieces; it carries the rational generator $B_i=R{\otimes }_{R^{s_i}}R(1)$ to $B_i^R$, the rational Bott-Samelson construction to the real one, and the rational idempotent completion to the real one. Standard graph supports, the $\mathrm{\Delta }$- and $\mathrm{\nabla }$-flags with their characters, and all formulas of the rational page are read over $\mathbb{R}$ after extension. The Hecke algebra used for characters is the integral normalized algebra $H_v(n)$ over $\mathbb{Z}[v^{\pm 1}]$ with basis $H_x$ of Split Grothendieck rings of the type-A Soergel categories; it is the same algebra before and after the scalar extension, since it is defined over $\mathbb{Z}$.

The dominant chamber. The dominant chamber on this page is the set of strictly dominant degree-two elements $\mathrm{\Theta }:=\{\rho ={\sum }_{j=1}^n{c}_j{x}_j:c_j\in \mathbb{R},c_j>c_{j+1}\text{ for }1\le j\le n-1\}\subset S_2\mathrm{.}$ It is nonempty, for example for $c_j=(n+1)\mathrm{/}2-j$, which gives $c_j-c_{j+1}=1$ for every $j$. For $\rho ={\sum }_j{c}_j{x}_j$ the coordinate evaluation is $\rho ({\alpha }_i^{\vee })=\rho (x_i-x_{i+1})=c_i-c_{i+1}$, so $\rho \in \mathrm{\Theta }$ is exactly a degree-two element with $\rho ({\alpha }_i^{\vee })>0$ for every simple coroot; no alternating-sign chamber and no replacement of $\rho$ is used anywhere on this page.

Conventions. All tensor products are over $S$ unless indicated, all shifts are the internal shifts just fixed, "form" on this page always means a graded symmetric form that is $S$-bilinear for the right action and invariant under the left action in the sense fixed by the intersection-form definition below, and dimensions of reduced modules are always real dimensions. The rational comparison statements needed for labels, characters and forms are collected in the comparison lemma later on this page.