The type-A diagrammatic Soergel category and its candidate bimodule functor

Soergel graphs. Fix $n\ge 2$ and the type-A realization of The standard type-A reflection realization and its polynomial ring, so the colors are the simple reflections $s_1,\dots ,s_{n-1}$, with $m_{st}=3$ for $\mathrm{\mid }s-t\mathrm{\mid }=1$ and $m_{st}=2$ for $\mathrm{\mid }s-t\mathrm{\mid }>1$. A Soergel graph is an isotopy class of decorated graphs embedded in the planar strip $\mathbb{R}\times [0,1]$, as in Elias–Williamson, Definition 5.1: edges are colored by simple reflections and may meet the bottom boundary $\mathbb{R}\times \{0\}$ and the top boundary $\mathbb{R}\times \{1\}$ in colored boundary points; the decorations are boxes labelled by homogeneous elements $f\in R$, and the vertices are of three types, with the degrees of Elias–Williamson, Definition 5.1:

1. univalent vertices (dots), degree $+1$;
2. trivalent vertices, all three adjoining edges of one color, degree $-1$;
3. $2m_{st}$-valent vertices, the adjoining edges alternating between two colors $s\mathrm{\ne }t$ with $m_{st}<\mathrm{\infty }$, degree $0$; in type $A$ this is a $4$-valent vertex when $\mathrm{\mid }s-t\mathrm{\mid }>1$ and a $6$-valent vertex when $\mathrm{\mid }s-t\mathrm{\mid }=1$.

The degree of a graph is the sum of the degrees of its vertices and boxes; a graph with bottom boundary $(i_1,\dots ,i_r)$ and top boundary $(j_1,\dots ,j_s)$ has boundary dots of those colors and is read from bottom to top. Composing graphs by vertical juxtaposition and tensoring them by horizontal juxtaposition is the usual pasting of boundary graphs.

The category $D$. Let $D$ (also written $D_n$) denote the $\mathbb{Q}$-linear monoidal category of Elias–Williamson, Definition 5.2, in this type-A case: objects are the words $\underset{‾}{i}=(i_1,\dots ,i_r)$ in the alphabet $\{1,\dots ,n-1\}$ — we also write $B_{\underset{‾}{i}}$ for the object, in the notation of The Bott–Samelson bimodule of a word — with monoidal structure given by concatenation and with the empty word as unit; the hom space ${\mathrm{Hom}}_D(\underset{‾}{i},\underset{‾}{j})$ is the free $\mathbb{Q}$-module on the Soergel graphs with bottom boundary $\underset{‾}{i}$ and top boundary $\underset{‾}{j}$, graded by graph degree and with degree-zero composites, modulo the homogeneous relations of the source, which in type $A$ are:

1. the polynomial relations (5.1) and (5.2) of Elias–Williamson §5.1, which slide boxes labelled by $f\in R$ across strands at the cost of the appropriate action of the Coxeter group and the Demazure operator, with the root labelling the dot on a strand of color $s$;
2. the one-color relations (5.3)–(5.5) of Elias–Williamson §5.1: the Frobenius relations among dots and trivalent vertices of a single color and the needle relation (5.5): a one-color loop attached at a trivalent vertex to a single strand is zero;
3. the two-color relations of Elias–Williamson §§5.2–5.3, in the two parities; for distant colors ($m_{st}=2$) the $4$-valent vertex is an interchange isomorphism, and for adjacent colors ($m_{st}=3$) the $6$-valent vertex satisfies the Jones–Wenzl relations, including the two ways of reading the relation that resolve the two triple products $B_i{B}_{i+1}{B}_i\cong B_{i,i+1,i}\oplus B_i$ and $B_{i+1}{B}_i{B}_{i+1}\cong B_{i,i+1,i}\oplus B_{i+1}$ of Rank-two type-A Soergel bimodule decompositions;
4. the three-color relations of Elias–Williamson §§5.4–5.5, which in type $A$ are: the $A_1\times I_2(m)$ relation (5.8) for a triple of colours consisting of two adjacent colours together with a third colour distant from both; its special case (5.9) for three pairwise distant colours; and the $A_3$ Zamolodchikov relation (5.10) on three consecutive colours. The $B_3$ and $H_3$ relations (5.11)–(5.12) of the source are not part of the type-$A$ presentation, and Elias–Williamson's Definition 5.2 lists all of (5.8), (5.9) and (5.10) before the definition is complete.

For type $A$ the same presentation is enumerated, with all generators, degrees and local relations, in Elias–Khovanov, Definition 3.8 and §3.4 (relations (3.1)–(3.37)); the indexing there is $S_{n+1}$ with colors $\{1,\dots ,n\}$ and polynomial ring in $n+1$ variables of degree $2$, and its internal shift is translated to the library by $M\{-1\}=M(1)$, as on this page. The Karoubi envelope $\mathrm{Kar}(D)$ is the idempotent completion of the additive graded closure of $D$, using degree-zero idempotents and morphisms there.

The candidate bimodule functor. Let $\mathcal{F}:D\to R\text{-Bim}$ be the assignment that sends a word $\underset{‾}{i}$ to the Bott–Samelson bimodule $B_{i_1}{\otimes }_R\cdots {\otimes }_R{B}_{i_r}$ of The Bott–Samelson bimodule of a word and a graph to a bimodule map by the following images on the generating vertices, all of which are the maps displayed in Elias–Williamson, p. 5, written here in the library's shifts and with ${\alpha }_s$ and ${\mathrm{\partial }}_s(f)=(f-sf)\mathrm{/}{\alpha }_s$:

1. the degree-$+1$ dot $B_s\to R$ ↦ the map $f\otimes g\mapsto fg$;
2. the degree-$+1$ dot $R\to B_s$ ↦ the map $1\mapsto \frac{1}{2}({\alpha }_s\otimes 1+1\otimes {\alpha }_s)$;
3. the degree-$-1$ trivalent vertices ↦ the maps $1\otimes g\otimes 1\mapsto {\mathrm{\partial }}_{s}g\otimes 1$ and $1\otimes 1\mapsto 1\otimes 1\otimes 1$;
4. a box labelled by homogeneous $f\in R$ ↦ multiplication by $f$ in the region of the tensor product where the box lies, that is, the $R$-bimodule structure of the corresponding Bott–Samelson bimodule;
5. a $4$-valent vertex of two distant colors ↦ the interchange isomorphism $B_i{\otimes }_R{B}_j\to B_j{\otimes }_R{B}_i$ of Distant Soergel generators commute;
6. a $6$-valent vertex of two adjacent colors ↦ the unique degree-zero bimodule map between the two alternating reduced-word products that sends their $1$-tensor to the $1$-tensor, as specified for a $2m$-valent vertex in Elias–Williamson §§5.2–5.3 (and given explicitly by Libedinsky). It factors through the common summand $B_{i,i+1,i}$ of the two triple products in Rank-two type-A Soergel bimodule decompositions; the existence of that summand alone does not specify the map's scalar.

Status. The assignment $\mathcal{F}$ is a candidate: this item defines its values on generators only, and no functoriality is asserted here. That the values respect the relations, so that $\mathcal{F}$ is a graded monoidal functor, is the content of the next item on this page; the functor is not asserted to be an equivalence before the double-leaves items.

(a) Degree bookkeeping. The degrees above are the degrees of the Elias–Williamson graphs, in which a polynomial box labelled by $f$ has degree $\mathrm{deg}f$ with $\mathrm{deg}{x}_i=2$, a dot has degree $+1$ and a trivalent vertex has degree $-1$. Because the library's bimodules carry the shift convention $M(1)=M\{-1\}$ of The Soergel bimodule $B_i$ of a simple reflection, in which $1\otimes 1\in B_s$ has degree $-1$, a graph of degree $d$ is sent to a bimodule map of degree $d$: for example the degree-$+1$ dot $B_s\to R$ sends the degree-$-1$ element $1\otimes 1$ to the degree-$0$ element $1\in R$; the degree-$-1$ splitting vertex $1\otimes 1\mapsto 1\otimes 1\otimes 1$ sends the degree-$-1$ element $1\otimes 1\in B_s$ to the degree-$-2$ element $1\otimes 1\otimes 1\in B_s{B}_s$; and the degree-$-1$ merging vertex $1\otimes g\otimes 1\mapsto {\mathrm{\partial }}_{s}g\otimes 1$ sends the degree-$0$ element $1\otimes {\alpha }_s\otimes 1$ to $2(1\otimes 1)$ of degree $-1$. The element $1\otimes 1\otimes 1$ is not a witness for the merging vertex: it has degree $-2$ and ${\mathrm{\partial }}_s(1)=0$, so this instance of the merge is the zero map. Scalars: the sources fix the normalisation $\delta ={\alpha }_s\mathrm{/}2$ and send the second dot to $1\mapsto \frac{1}{2}({\alpha }_s\otimes 1+1\otimes {\alpha }_s)=\delta \otimes 1+1\otimes \delta$, which is Elias–Williamson's image $1\mapsto {\mathrm{\Delta }}_s$ of Definition 5.12 in the balanced type-$A$ case, where ${\mathrm{\Delta }}_s=\frac{1}{2}({\alpha }_s\otimes 1+1\otimes {\alpha }_s)$ because $\frac{1}{2}$ exists in $k=\mathbb{Q}$. The unscaled element $Z_s={\alpha }_s\otimes 1+1\otimes {\alpha }_s=2{\mathrm{\Delta }}_s$ used in Rank-two type-A Soergel bimodule decompositions and in the rank-two example is not the image of this dot: it differs from it by the invertible scalar $2$, and the rank-two splitting maps built from $Z_s$ differ from the fixed diagrammatic generators by that factor, so the relation checks and the six-valent generator are not invariant under this rescaling without adjustment. When the balanced realization of The standard type-A reflection realization and its polynomial ring is used, the translation is by the alternating signs ${\epsilon }_s,{\epsilon }_t$ with ${\epsilon }_s{\epsilon }_t=-1$: the insertion maps rescale as ${\mu }_t^{a,\alpha }={\epsilon }_t{\mu }_t^{a,\beta }$ and the contractions as ${\kappa }_s^{\alpha }={\epsilon }_s{\kappa }_s^{\beta }$, the zig-zag composite changes sign, and the idempotent and the summand it defines do not. Throughout this item the normalisation used is the displayed one, $1\mapsto \frac{1}{2}({\alpha }_s\otimes 1+1\otimes {\alpha }_s)$.

(b) Small $n$. For $n=2$ there is one color, no pair of distinct colors, and so no $4$-valent or $6$-valent vertices; $D_2$ is generated by the one-color graphs, and the rank-one square relation of The rank-one Soergel bimodule square splits is the corresponding idempotent decomposition. For $n\le 1$ there are no colors at all. The word category $D$ has only the empty word, with $\mathrm{End}(\mathrm{\varnothing })=R$; its additive graded and Karoubi closures also contain sums, shifts and their summands. No vertex, and in particular no $2m$-valent vertex, is used.

(c) What is not claimed. Nothing here asserts that the relations of the sources list every relation valid in $\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}$, nor that $\mathcal{F}$ descends to $D$, nor that $D$ has finite-dimensional hom spaces. Those are separate items of this page, proved from the sources quoted there.

(d) Imported statements of the sources, recorded for the later items. The later items of this page cite the following results of Elias–Williamson and Elias–Khovanov through this definition; we record their content here, in the notation fixed above, so that every later citation has a textual anchor in this item. Write ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},\underset{‾}{y}}$ for a set consisting of one fixed choice of double leaf ${\overline{\mathrm{L}\mathrm{L}}}_{\underset{‾}{y},f}\circ {\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$, the light leaf from $\underset{‾}{x}$ followed by the flipped light leaf towards $\underset{‾}{y}$, so that the composite is a morphism $\underset{‾}{x}\to \underset{‾}{y}$ for each pair of subexpressions $e,f$ of $\underset{‾}{x},\underset{‾}{y}$ expressing a common element $w$. Fix one reduced word $\underset{‾}{w}$ for each such $w$, shared by both light leaves in each composite; their common intermediate object is that word, not the permutation itself. Let $I_w$ for the two-sided ideal spanned by double leaves factoring through elements $y\ge ̸w$ in Bruhat order (Elias–Williamson, §6.4). Thus $D_{\ge w}=D\mathrm{/}{I}_w$ retains the cells indexed by $y\ge w$; for maps into a reduced-word object $B_{\underset{‾}{w}}$, this amounts to quotienting by terms indexed by $y<w$.

1. Double leaves. (Elias–Williamson, Theorem 6.11 with Proposition 6.12 and Corollary 6.13.) The set ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},\underset{‾}{y}}$ forms a free $R$-basis for $\mathrm{Hom}(B_{\underset{‾}{x}},B_{\underset{‾}{y}})$ in $D$; the light leaves ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$ with $e$ expressing the identity form a free $R$-basis for $\mathrm{Hom}(B_{\underset{‾}{x}},\mathbf{1})$ in $D$; and hom spaces in $D$ are free graded $R$-modules. This is proved from two inputs: by Elias–Williamson, Proposition 6.9, after localization the space of maps is a direct sum of terms indexed by pairs of subsequences carrying a partial order with respect to which the double-leaf maps are upper triangular with an invertible diagonal, so the double leaves are linearly independent over the fraction field; and by the result of item 3 below they span.
2. Construction of light leaves. (Elias–Williamson, Construction 6.1 with Figure 2.) For every pair $(\underset{‾}{x},e)$ expressing $w$ and every $k\le \mathrm{\ell }(\underset{‾}{x})$, the truncated light leaf on the first $k$ letters is obtained from the truncated light leaf on the first $k-1$ letters by a rex move $\beta$ when the $k$-th letter is a down step, followed by exactly one of four local moves: the dot $U_0$ of degree $+1$, the identity $U_1$ of degree $0$, the merging trivalent vertex $D_0$ of degree $-1$, and the cap $D_1$ of degree $0$. The degree of the morphism ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$ is $+1$ for each $U_0$ and $-1$ for each $D_0$, and hence agrees with the defect $d(e)=\mathrm{\#}{U}_0-\mathrm{\#}{D}_0$ of the subexpression.
3. Basis modulo lower terms. (Elias–Williamson, §7, Proposition 7.6, with the maxwidth induction of §7: for arbitrary $\underset{‾}{x}$ and a reduced word $\underset{‾}{w}$ for $w$, the maps ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$, one for each subexpression $e$ of $\underset{‾}{x}$ expressing $w$, form a basis for $\mathrm{Hom}(B_{\underset{‾}{x}},B_{\underset{‾}{w}})\mathrm{/}{I}_w$ under the action of $R$ on the left.) Linear independence is already known by localization, so it is enough to show that these maps span; the proof represents the identity of a word as a negative-positive decomposition, factors an arbitrary morphism through a reduced word, and pushes the lower terms through by induction on the maximum width of a graph, using only the relations of $D$. Consequently every morphism between Bott–Samelson objects lies in the span of the double leaves modulo the ideal of strictly lower terms.
4. Krull–Schmidt. (Elias–Williamson, Lemma 6.24.) If $k$ is a complete local ring then the category $\mathrm{Kar}(D)$ is Krull–Schmidt.
5. Classification of indecomposables. (Elias–Williamson, Theorem 6.25.) Assume that $k$ is a complete local ring; then for all $w\in W$ there exists a unique summand $B_w$ of $B_{\underset{‾}{w}}$ which is not isomorphic to the shift of a summand of $B_{\underset{‾}{v}}$ for any reduced expression $\underset{‾}{v}$ for $v<w$; the object $B_w$ does not depend on the reduced expression $\underset{‾}{w}$ up to isomorphism; every indecomposable object of $\mathrm{Kar}(D)$ is isomorphic to a shift of $B_w$ for some $w\in W$; hence the passage to isomorphism classes up to shift is a bijection from $W$ onto the indecomposable objects of $\mathrm{Kar}(D)$ up to shifts and isomorphism.
6. Relation checks for the bimodule functor. (Elias–Khovanov, §5.1 with Definition 3.8 and Claim 5.1.) Each of the finitely many relations (3.1)–(3.37) of the type-A presentation holds for the images of the generating morphisms in Soergel bimodules, and the verification is finite; by Claim 5.1 a Bott–Samelson bimodule $\underset{‾}{i}$ of length $d$ with $m$ distinct colours is generated as an $R$-bimodule by any set of $2^{d-m}$ linearly independent tensors that is in bijection with the power set of the set $X$ of repeated-colour pairs and realises each pair by a linear factor inside that pair, so the two-color and three-color relations may be checked on $2^{d-m}$ generators.
7. The diagrammatic character. (Elias–Williamson, §6.5: Definition 6.23 with equations (6.3) and (6.4), and Corollaries 6.26 and 6.27.) For $w\in S_n$ let $D_{\ge w}:=D\mathrm{/}{I}_w$ be the quotient retaining the Bruhat cells $y\ge w$ as just defined. For a reduced word $\underset{‾}{w}$ for $w$ the images of $B_{\underset{‾}{w}}$ in $D_{\ge w}$ are canonically isomorphic and ${\mathrm{End}}_{D_{\ge w}}(B_{\underset{‾}{w}})=R$; by imported result 1 above the module ${\mathrm{Hom}}_{D_{\ge w}}(B_{\underset{‾}{x}},B_{\underset{‾}{w}})$ is free with basis the light leaves ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$, $e$ a subexpression of $\underset{‾}{x}$ expressing $w$, so that in the Hecke algebra of The type-A Hecke algebra in Soergel normalization $H_{\underset{‾}{x}}={\sum }_{w\in S_n}\mathrm{grk}{\mathrm{Hom}}_{D_{\ge w}}(B_{\underset{‾}{x}},B_{\underset{‾}{w}}){\tilde{T}}_w,$ where $H_{\underset{‾}{x}}$ denotes the product of the $H_{x_a}$ along $\underset{‾}{x}$, the right-hand coefficients are expanded in the standard normalized basis $\{{\tilde{T}}_x=v^{\mathrm{\ell }(x)}{T}_x\}$ of The standard basis of the type-A Hecke algebra and its multiplication rule (Elias–Williamson's basis element $H_w$), and $\mathrm{grk}$ the graded rank. Since $k=\mathbb{Q}$ is a local ring, direct summands of free graded $R$-modules are graded free by Nakayama's lemma, so the diagrammatic character $\mathrm{c}\mathrm{h}:[\mathrm{Kar}(D)]⟶H_n,B\mapsto {\sum }_{w\in S_n}\mathrm{grk}{\mathrm{Hom}}_{\mathrm{Kar}(D_{\ge w})}(B,B_w){\tilde{T}}_w,$ is a well-defined homomorphism of $\mathbb{Z}[v^{\pm 1}]$-modules with $\mathrm{c}\mathrm{h}(v[B])=\mathrm{c}\mathrm{h}(B(1))=v\mathrm{c}\mathrm{h}(B)$; it satisfies $\mathrm{c}\mathrm{h}(B_{\underset{‾}{x}}{B}_{\underset{‾}{y}})=\mathrm{c}\mathrm{h}(B_{\underset{‾}{x}})\mathrm{c}\mathrm{h}(B_{\underset{‾}{y}})$ for words $\underset{‾}{x},\underset{‾}{y}$ and $\mathrm{c}\mathrm{h}(B_s)=H_s$ for every simple reflection $s$, hence is multiplicative on the classes of Bott–Samelson objects. Moreover $\mathrm{c}\mathrm{h}$ is an isomorphism of $\mathbb{Z}[v^{\pm 1}]$-algebras: the classes $[B_{\underset{‾}{x}}]$ span the split Grothendieck group, the classes $[B_w]$ of the distinguished indecomposable summands of reduced-word objects are a basis, and $\mathrm{c}\mathrm{h}(B_w)={\tilde{T}}_w+{\sum }_{y<w}{g}_{y,w}{\tilde{T}}_y$ with $g_{w,w}=1$ and the sum over the strictly lower elements $y<w$ of the Bruhat order, so triangularity with unit diagonal in the standard basis $\{{\tilde{T}}_y\}$ makes $\mathrm{c}\mathrm{h}$ a bijection. A reduced-word object itself may have lower summands, as $B_{sts}\cong B_{w_0}\oplus B_s$ in rank two. Finally the assignment $H_s\mapsto [B_s]$ defines a homomorphism $H_n\to [\mathrm{Kar}(D)]$.
8. The light-leaf basis of the hom space to the unit. (Libedinsky, Sur la catégorie des bimodules de Soergel, arXiv:0707.3603, §§4.4–4.6 and Théorème 5.1, with its proof in §5.) For a reflection-faithful representation over a field of characteristic different from two, put ${\theta }_s=R{\otimes }_{R^s}R$, so the library's $B_s={\theta }_s(1)$. From $R=R^s\oplus x_s{R}^s$, §4.4 defines $R^s$-linear coefficient operators $P_s,I_s,I_s^{\mathrm{\prime }}$ on $R$ (these are not $R$-bimodule endomorphisms) and the bimodule maps $m_s,i_s^0,i_s^1$ on the unshifted tensor products. Sections 4.5–4.6 fix braid paths and recursively construct leaves by the four up/down rules. Théorème 5.1 states that the leaves $A_r^{\mathrm{\prime }}$ ending at $R$ form an $R$-basis of ${\mathrm{Hom}}_{R\text{-}R}({\theta }_{s_1}\cdots {\theta }_{s_r},R)$. Its proof evaluates these leaves on the normal tensor basis, obtains a triangular matrix with diagonal entries one, and compares finite-dimensional graded pieces using Corollary 4.2 and Lemma 5.6. In ordinary map-degree notation a leaf using $h$ contractions has degree $-2h$ on the unshifted source and degree $r-2h$ on $B_{s_1}\cdots B_{s_r}=({\theta }_{s_1}\cdots {\theta }_{s_r})(r)$. Thus the unshifted Hecke count is ${\sum }_{a\in A_r^{\mathrm{\prime }}}{q}^{-\mathrm{deg}(a)\mathrm{/}2}=p_r^1$, where ${\prod }_j(1+T_{s_j})={\sum }_w{p}_r^w{T}_w$ and $q=v^{-2}$. The minus sign is necessary with ordinary map degrees: for the word $ss$ the degrees are $0,-2$ and $p_2^1=1+q$. The printed positive sign in Definition 5.4 and the degree-increase wording in Lemma 5.6 are inconsistent with the explicit contraction of degree $-2$ and the computation on p. 19; they are not adopted here. This source uses its own braid-map normalization; the evaluated Elias–Williamson leaves are handled separately by result 11.
9. Adjunction and double leaves. (Elias–Williamson, Remark 6.10 and §6.7, especially Remark 6.29.) Vertical flipping and rotating a light leaf give different constructions; no entrywise identification of a vertically flipped double leaf with a bent unit-target light leaf is asserted. Cups and caps make each generating strand self-biadjoint by planar isotopy. Evaluation sends these cups and caps to the degree-zero Frobenius coevaluation and evaluation of the bimodule $B_s$. Consequently bending boundary strands commutes with evaluation. An isomorphism on Hom spaces to the unit therefore implies an isomorphism on Hom spaces between words. The image of a diagrammatic double-leaf basis is then a basis under this isomorphism, independently of the basis obtained by bending light leaves.
10. The defect expansion of a product of generators. (Elias–Williamson, §2.4, Lemma 2.10 with Corollary 2.11, in the normalization $H_s=v{T}_s+v$, ${\tilde{T}}_w=v^{\mathrm{\ell }(w)}{T}_w$ of The type-A Hecke algebra in Soergel normalization.) A subexpression of a word $\underset{‾}{x}=(x_1,\dots ,x_m)$ is a $01$-sequence $e$, with Bruhat stroll $x_0=e$, $x_k=x_{k-1}s$ if the letter is kept and $x_k=x_{k-1}$ otherwise; each index carries a token $U_0$, $U_1$, $D_0$ or $D_1$ according to whether the letter is kept and whether the stroll moved up or down, and the defect is $d(e)=\mathrm{\#}{U}_0-\mathrm{\#}{D}_0$. Then $H_{x_1}\cdots H_{x_m}={\sum }_e{v}^{d(e)}{\tilde{T}}_{w_e},$ the sum over all subexpressions $e$ of $\underset{‾}{x}$, where $w_e$ is the element expressed by $e$. Equivalently, for a Bott–Samelson bimodule $B_{\underset{‾}{x}}$ the $\mathrm{\Delta }$-multiplicity of the standard bimodule ${\mathrm{\Delta }}_w(d)$ is the number of subexpressions of $\underset{‾}{x}$ expressing $w$ with defect $d$, $(B_{\underset{‾}{x}}:{\mathrm{\Delta }}_w(d))=\mathrm{\#}\{e:w_e=w,d(e)=d\}$.
11. Localised independence of the evaluated light leaves. (Elias–Williamson, §6.7, Remark 6.29, read together with the localisation argument of the proof of Corollary 6.8 of the same section.) Fix an expression $\underset{‾}{x}$ and a reduced word $\underset{‾}{w}$ for an element $w\in S_n$, and let ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},\underset{‾}{w}}$ consist of one light leaf ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}:B_{\underset{‾}{x}}\to B_{\underset{‾}{w}}$ for each subexpression $e$ of $\underset{‾}{x}$ expressing $w$, chosen once and for all, so that each $\mathcal{F}({\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e})$ is a composition of the images of dots, trivalent vertices and $2m_{st}$-valent vertices. Then the images $\mathcal{F}({\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e})$ are linearly independent over $R$ as elements of ${\mathrm{Hom}}_{\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}(B_{\underset{‾}{x}},B_{\underset{‾}{w}})$: the same localisation argument as in the proof of Corollary 6.8, carried out in the localised category of Soergel bimodules, exhibits the images of the light leaves as upper triangular with invertible diagonal, and localisation is injective on hom spaces. Comparing the degrees of the light leaves (the defects of result 2 above) with the graded dimension of $\mathrm{Hom}(B_{\underset{‾}{x}},R)$ then shows that the images of the light leaves ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$ with $e$ expressing the identity span $\mathrm{Hom}(B_{\underset{‾}{x}},R)$ as a graded $R$-module.