positive integers).

Soergel bimodules {#s3}
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Soergel bimodules for finite Coxeter groups {#s3.1}
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Let $(W,S)$ be a finite irreducible Coxeter system and $\varphi:W\to\mathrm{GL}(V)$ be the geometric representation of $W$ as in [@Hu Section 5.3]. Here $V$ is a real vector space. We denote by $\leq$ the Bruhat order and by $\mathbf{l}:W\to\mathbb{Z}_{\geq 0}$ the length function. For $w\in W$, we denote by $\underline{w}$ the corresponding element in the Kazhdan-Lusztig basis of $\mathbb{Z}[W]$, see [@KaLu; @So2; @EW]. Each $\underline{w}$ is a linear combination of elements in $W$ with non-negative integer coefficients.

Let $\mathtt{C}$ be the (complexified) coinvariant algebra associated to $(W,S,V)$. For $s\in S$, we denote by $\mathtt{C}^s$ the subalgebra of $s$-invariants in $\mathtt{C}$. A [*Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodule*]{} is a $\mathtt{C}\text{-}\mathtt{C}$–bimodule isomorphic to a bimodule from the additive closure of the monoidal category of $\mathtt{C}\text{-}\mathtt{C}$–bimodules generated by $\mathtt{C}\otimes_{\mathtt{C}^s}\mathtt{C}$, where $s$ runs through $S$, see [@So; @So2; @Li]. We also note that by the [*additive closure*]{} of some $X$ we mean the the full subcategory whose objects are isomorphic to finite direct sums of direct summands of $X$. Isomorphism classes of indecomposable Soergel bimodules are naturally indexed by $w\in W$, we denote by $B_w$ a fixed representative from such a class. There is 