Indecomposable type-A diagrammatic Soergel objects are indexed by permutations and shifts

The coefficient ring is complete local: by [F1] the zero ideal of $\mathbb{Q}$ is its unique maximal ideal and $\mathbb{Q}$ is isomorphic to its own $\mathfrak{m}$-adic completion, so the hypothesis of the imported theorem [F4] is satisfied.

The realization hypotheses and the categorical setting: by [F2] the standard type-A realization is a Soergel realization in the sense of the sources, so the diagrammatic results [F3] and [F4] apply to $D$ over $k=\mathbb{Q}$; the Karoubi envelope $\mathrm{Kar}(D)$ is by construction a $k$-linear idempotent complete additive category.

Imported classification: by [F4] each $w\in S_n$ has the summand ${\underset{‾}{B}}_w$ of the Bott–Samelson object of any reduced word, characterised by excluding the shifts of summands of strictly lower elements, and every indecomposable of $\mathrm{Kar}(D)$ is a shift of one of these; the subexpression indexing is that of the double leaves of [F3], and the elements $w$ with their Bruhat order are those of [F5].

Finiteness input for the imported lemma: by [F3] the hom space of any two Bott–Samelson objects is a free graded $R$-module on finitely many homogeneous generators. Each fixed-degree piece of $R=\mathbb{Q}[x_1,\dots ,x_n]$ is finite-dimensional over $k=\mathbb{Q}$, because it has only finitely many monomials of that degree; hence the degree-zero part of a finite direct sum of shifts of $R$ is finite-dimensional over $k$. Every object of $\mathrm{Kar}(D)$ is a summand of a finite direct sum of shifts of Bott–Samelson objects, so its degree-zero endomorphism space is a direct summand of such a finite-dimensional space. Together with the completeness of $k$ from step 1.1 and the $k$-linearity and idempotent completeness of step 1.2, this verifies the hypothesis set of the imported Krull–Schmidt lemma [F4].

Existence of $D_w$: for $w\in S_n$ and a reduced word $\underset{‾}{w}$ for $w$, the uniqueness and exclusion clauses of the imported classification statement [F4] produce the summand ${\underset{‾}{B}}_w$ of $B_{\underset{‾}{w}}$ that is not a shift of a summand of $B_{\underset{‾}{v}}$ for $v<w$; by the subexpression language of [F5] this $w$ is a well-defined element of $S_n$ and the comparison of elements in Bruhat order is meaningful. We rename ${\underset{‾}{B}}_w$ as $D_w$ in the library's notation.

Krull–Schmidt: the imported Krull–Schmidt statement of [F4] applies because of steps 1.1–1.3; hence every object of $\mathrm{Kar}(D)$ is a finite direct sum of indecomposables with local endomorphism rings, uniquely up to isomorphism and order.

Reduced-word independence: by the independence clause of the imported classification statement [F4] the summand ${\underset{‾}{B}}_w$ of $B_{\underset{‾}{w}}$ does not depend on the reduced word $\underset{‾}{w}$ for $w$ up to isomorphism, so the object $D_w$ of step 2.2 is well defined; its uniqueness clause likewise rules out a second, non-isomorphic summand of $B_{\underset{‾}{w}}$ with the same exclusion property.

Classification: let $B$ be an indecomposable object of $\mathrm{Kar}(D)$; by [F4] it is isomorphic to a shift of some ${\underset{‾}{B}}_w$, i.e. to $D_w(d)$ for some $d\in \mathbb{Z}$; conversely every $D_w(d)$ is indecomposable because $D_w$ has a local endomorphism ring by step 3.1 and shifts are equivalences, hence preserve indecomposability. The pair $(w,d)$ is unique: injectivity of $w\mapsto D_w$ up to isomorphism and shift is part of the bijection statement of [F4]. Once $w$ is fixed, let $X=B_{\underset{‾}{w}}$ and realize $D_w=(X,e_w)$ for a nonzero degree-zero idempotent $e_w$. By [F3], ${\mathrm{End}}_D^{\bullet }(X)$ is a finite direct sum of shifts of the nonnegatively graded ring $R$, hence has a lower degree bound. The graded corner $e_w{\mathrm{End}}_D^{\bullet }(X)e_w={\mathrm{End}}^{\bullet }(D_w)$ has the same lower bound. If a degree-zero isomorphism $D_w\cong D_w(d)$ existed for $d\mathrm{\ne }0$, it and its inverse would give mutually inverse homogeneous elements of this corner of degrees $d$ and $-d$. Powers of the element of negative degree would be nonzero in arbitrarily negative degrees, a contradiction. Thus no nonzero shift fixes $D_w$, proving uniqueness of $d$ entirely inside the diagrammatic category.

Conclusion: $\mathrm{Kar}(D)$ is Krull–Schmidt, the objects $D_w$ for $w\in S_n$ are the indecomposables up to shift, and the assignment $w\mapsto D_w$ is the stated bijection; the case $n\le 1$ has a single indecomposable up to shift, the unit object (the empty Bott–Samelson object), and the statements hold trivially there. ∎