LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Lefschetz operator factors through deleted Bott-Samelson subwords, with positive rescalings

Facts & Assumptions

Given: An expression $\underline{x} = s_{1} \dots s_{m}$ in simple reflections, the Bott-Samelson bimodule $B S (\underline{x})$ with its subexpression right basis ${c_{ε}}$ , top element $c_{t o p}$ , trace $Tr$ and intersection form $⟨ -, - ⟩$ of The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction, a strictly dominant $ρ$ with associated operator $L_{0} = ρ \cdot (-)$ (The reduced Soergel module and the dominant Lefschetz operator), and the auxiliary maps ${B r}_{i} (b_{1} \dots b_{m}) : = b_{1} \dots (b_{i} c_{s_{i}}) \dots b_{m}$ , $φ_{i} (b_{1} \dots b_{m}) : = b_{1} \dots μ (b_{i}) \dots b_{m}$ and $χ_{i} (b_{1} \dots \hat{b_{i}} \dots b_{m}) : = b_{1} \dots c_{s_{i}} \dots b_{m}$ , where $μ : B_{s_{i}} \to S (1)$ is the multiplication map $f \otimes g \mapsto f g$ and ${\hat{\underline{x}}}_{i}$ is the word with letter $s_{i}$ deleted. The maps ${B r}_{i}$ have internal degree two; their factorizations use degree-zero bimodule maps $φ_{i}$ and $χ_{i}$ into the displayed shifts. Thus ${B r}_{i} : B S (\underline{x}) \to B S (\underline{x})$ and $φ_{i} : B S (\underline{x}) \to B S ({\hat{\underline{x}}}_{i}) (1)$ , $χ_{i} : B S ({\hat{\underline{x}}}_{i}) \to B S (\underline{x}) (1)$ with ${B r}_{i} = χ_{i} \circ φ_{i}$ , and $Tr (χ_{i} (γ)) = {Tr}_{i} (γ)$ for all $γ \in B S ({\hat{\underline{x}}}_{i})$ , where ${Tr}_{i}$ is the trace of $B S ({\hat{\underline{x}}}_{i})$ .

[F1]

In each rank-one factor the elements $c_{i d}, c_{s}$ form a right $S$ -basis with $c_{i d} c_{i d} = c_{i d}$ , $c_{i d} c_{s} = c_{s}$ , $c_{s} c_{s} = c_{s} α_{s}$ and the left-action rule $r \cdot c_{i d} = c_{i d} \cdot s r + \partial_{s} (r) c_{s}$ , $r \cdot c_{s} = c_{s} \cdot r$ , for $r \in S$ (The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction, The Soergel bimodule $B_{i}$ of a simple reflection).

[F2]

The trace is right $S$ -linear and the intersection form is symmetric and invariant (The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction).

[F3]

Descent detection. For every $w$ and simple $s$ , $(w ρ) (α_{s}^{\lor}) > 0$ if and only if $s w > w$ (The reduced Soergel module and the dominant Lefschetz operator).

[F4]

The positive Rouquier generator complex is $F_{i} = [B_{s_{i}} \overset{}{\to} S (1)]$ with $B_{s_{i}}$ in cohomological degree $0$ ; the Rouquier complex of a word is the iterated signed tensor totalization of the $F_{i}$ (The positive and negative Rouquier generator complexes, The Rouquier complex of a braid word).

Statement

Let $\underline{x} = s_{1} \dots s_{m}$ be an expression, $B S (\underline{x})$ its real Bott-Samelson bimodule, ${B r}_{i} = χ_{i} \circ φ_{i}$ the Demazure operation at the $i$ -th letter, and $Tr, ⟨ -, - ⟩$ the trace and intersection form of The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction. Then:

(a) as endomorphisms of $B S (\underline{x})$ one has $ρ \cdot (-) = \sum_{i = 1}^{m} (s_{i - 1} \dots s_{1} ρ) (α_{s_{i}}^{\lor}) χ_{i} \circ φ_{i} + (-) \cdot x^{- 1} ρ$ (EW Lemma 3.4), where $(-) \cdot x^{- 1} ρ$ denotes right multiplication by the polynomial $x^{- 1} ρ$ ;

(b) with the rescaled forms $⟨ -, - ⟩_{i}^{'} : = (s_{i - 1} \dots s_{1} ρ) (α_{s_{i}}^{\lor}) ⟨ -, - ⟩_{i}$ on $B S ({\hat{\underline{x}}}_{i})$ and $⟨ -, - ⟩^{'}$ their direct sum, one has $⟨ b, ρ b^{'} ⟩_{\overline{B S (\underline{x})}} = ⟨ φ (b), φ (b^{'}) ⟩^{'}$ for all $b, b^{'}$ in the reduced module, where $φ = \sum_{i} φ_{i} : B S (\underline{x}) \to ⨁_{i} B S ({\hat{\underline{x}}}_{i}) (1)$ (EW Lemmas 6.14 and 6.15);

(c) if $\underline{x}$ is reduced then every scaling factor $γ_{i} : = (s_{i - 1} \dots s_{1} ρ) (α_{s_{i}}^{\lor})$ is strictly positive, so the rescalings change no signs in the HR statements, and the unscattered term $(-) \cdot x^{- 1} ρ$ vanishes after reduction to the Soergel module;

(d) for a reduced expression, consequently left multiplication by $ρ$ on $\overline{B S (\underline{x})}$ factors, up to positive rescalings, as the composite $B S (\underline{x}) \to ⨁_{i} B S ({\hat{\underline{x}}}_{i}) (1) \to B S (\underline{x}) (2)$ , and this composite is the first differential of the Rouquier complex $F_{s_{1}} \dots F_{s_{m}}$ followed by its dual map, up to the same positive rescalings.

Proof

technique · direct; a rank-one computation followed by iteration over the letters

1.1givenF1

The single-factor identity. Let $s$ be a simple reflection with root $α = α_{s}$ , let $B r (b) : = b c_{s}$ on $B_{s}$ , and let $σ \in S$ . For every $b \in B_{s}$ one has $σ \cdot b = b \cdot (s σ) + \partial_{s} (σ) B r (b)$ . Indeed both sides are right $S$ -linear in $b$ , so it suffices to check the two basis elements: for $b = c_{i d}$ the identity is the rule $r \cdot c_{i d} = c_{i d} \cdot s r + \partial_{s} (r) c_{s}$ of [F1] with $r = σ$ , and for $b = c_{s}$ it reads $σ \cdot c_{s} = c_{s} \cdot s σ + \partial_{s} (σ) c_{s} α_{s}$ , which is $c_{s} σ = c_{s} (s σ + α_{s} \partial_{s} (σ))$ and holds because $σ - s σ = α_{s} \partial_{s} (σ)$ and $c_{s}$ commutes with $S$ .

2.1step 1.1F1

Iteration over the letters proves (a). For $m = 0$ both outer actions are multiplication by $ρ$ on $S$ . For $m > 0$ , put $σ_{0} = ρ$ and $σ_{i} = s_{i} σ_{i - 1}$ . Apply step 1.1 in slot $i$ to the remainder $b_{1} \dots b_{i - 1} (σ_{i - 1} b_{i}) b_{i + 1} \dots b_{m}$ . This produces $\partial_{s_{i}} (σ_{i - 1}) {B r}_{i} (b)$ and a remainder with $σ_{i}$ acting on the right of slot $i$ . Balancing moves that polynomial to the left of slot $i + 1$ ; at the last slot it is the outer right action by $x^{- 1} ρ$ . Each $σ_{i - 1}$ is linear, so $\partial_{s_{i}} (σ_{i - 1}) = σ_{i - 1} (α_{s_{i}}^{\lor}) = γ_{i}$ . Also $b c_{s} = μ (b) c_{s}$ in one factor: on $c_{i d}$ and $c_{s}$ this follows from $μ (c_{i d}) = 1$ , $μ (c_{s}) = α_{s}$ and $c_{s}^{2} = c_{s} α_{s}$ . Thus ${B r}_{i} = χ_{i} φ_{i}$ with the appropriate shift of $χ_{i}$ , and the stated degree-two identity holds on every tensor monomial and hence on the whole bimodule.

3.1step 1.1step 2.1givenF1

The covariance identity. For all $b, b^{'} \in B S (\underline{x})$ and all $i$ one has $⟨ b, {B r}_{i} b^{'} ⟩ = ⟨ φ_{i} b, φ_{i} b^{'} ⟩_{i}$ . It suffices to check on tensor monomials $b = b_{1} \dots b_{m}$ , $b^{'} = b_{1}^{'} \dots b_{m}^{'}$ : the left side is $Tr ((b_{1} b_{1}^{'}) \dots (b_{i} b_{i}^{'} c_{s_{i}}) \dots (b_{m} b_{m}^{'})) = Tr ((b_{1} b_{1}^{'}) \dots μ (b_{i}) μ (b_{i}^{'}) c_{s_{i}} \dots (b_{m} b_{m}^{'}))$ , using $b_{i} c_{s_{i}} = μ (b_{i}) c_{s_{i}}$ from step 2.1, so this equals $Tr (χ_{i} ((b_{1} b_{1}^{'}) \dots μ (b_{i}) μ (b_{i}^{'}) \dots (b_{m} b_{m}^{'}))) = {Tr}_{i} ((b_{1} b_{1}^{'}) \dots μ (b_{i}) μ (b_{i}^{'}) \dots (b_{m} b_{m}^{'})) = ⟨ φ_{i} b, φ_{i} b^{'} ⟩_{i}$ , using $Tr \circ χ_{i} = {Tr}_{i}$ and the multiplicativity of the ring structure.

4.1step 2.1step 3.1F1algebra

The rescaled form identity (b). With $φ = \sum_{i} φ_{i}$ and $⟨ -, - ⟩^{'} = ⨁_{i} ⟨ -, - ⟩_{i}^{'}$ , the direct sum decomposition is orthogonal, so for all $b, b^{'}$ one has $⟨ φ (b), φ (b^{'}) ⟩^{'} = \sum_{i} γ_{i} ⟨ b, {B r}_{i} b^{'} ⟩$ ; substituting step 3.1 gives $\sum_{i} γ_{i} ⟨ b, {B r}_{i} b^{'} ⟩ = ⟨ b, \sum_{i} γ_{i} {B r}_{i} b^{'} ⟩$ , and substituting the factorization of step 2.1 gives $⟨ b, ρ b^{'} ⟩ - ⟨ b, b^{'} \cdot x^{- 1} ρ ⟩$ . The second summand is $- ⟨ b, b^{'} ⟩ x^{- 1} ρ$ by right $S$ -linearity of the form, and it vanishes after reduction because $x^{- 1} ρ$ has positive degree; hence $⟨ b, ρ b^{'} ⟩_{\overline{B S (\underline{x})}} = ⟨ φ (b), φ (b^{'}) ⟩^{'}$ for $b, b^{'}$ in the reduced module, as claimed in (b).

5.1step 4.1F1F2F3

Positivity for reduced words and the vanished remainder (c). Let $\underline{x}$ be reduced. For each $i$ , the prefix $w_{i} : = s_{i - 1} \dots s_{1}$ has length $i - 1$ and $s_{i} w_{i}$ has length $i$ ; the descent criterion [F3] therefore gives $γ_{i} = (w_{i} ρ) (α_{s_{i}}^{\lor}) > 0$ . All rescalings are by strictly positive real numbers, so no sign in a definite form changes; and the polynomial $x^{- 1} ρ$ lies in $S_{2} \subset S_{+}$ whose right multiplication lands in $B S (\underline{x}) S_{+}$ , which becomes zero after reduction, so the remainder vanishes on the reduced module. This proves (c).

6.1step 2.1step 4.1step 5.1F2F4∎

Identification with the Rouquier differential (d). By [F4] the first differential of the Rouquier complex is the sum, over the letters, of the maps $φ_{i}$ from the cohomological degree $0$ term $B S (\underline{x})$ to the degree $1$ term $⨁_{i} B S ({\hat{\underline{x}}}_{i}) (1)$ , and its dual maps decompose the generator $c_{s_{i}}$ back, i.e. are the maps $χ_{i}$ with the displayed internal shifts. Composing the differential with the rescaled dual maps $γ_{i} χ_{i}$ and summing over $i$ gives $\sum_{i} γ_{i} {B r}_{i}$ , which by step 2.1 equals left multiplication by $ρ$ modulo the right-multiplication remainder that vanishes on the reduced module by step 5.1. Hence left multiplication by $ρ$ on the reduced Bott-Samelson module factors, up to the positive rescalings $γ_{i}$ , as the composite $B S (\underline{x}) \overset{φ}{\to} ⨁_{i} B S ({\hat{\underline{x}}}_{i}) (1) \overset{γ χ}{\to} B S (\underline{x}) (2)$ built from the first Rouquier differential and its dual; all first-differential Koszul signs are $+ 1$ , because every preceding tensor factor is in cohomological degree zero, so this is exactly $ρ \cdot (-)$ after right reduction. This proves (d) and completes the lemma; no positivity of characters and no character conjecture is used.

Remarks

The remainder term is genuinely present before reduction: right multiplication by $x^{- 1} ρ$ does not vanish on the raw Bott-Samelson bimodule, and the formula (2) is stated on the reduced module, where it does.
The rescalings $γ_{i}$ are positive precisely for reduced $\underline{x}$ ; for nonreduced words the factorization (1) still holds as an identity of endomorphisms, but no positivity claim about the scalings is made.

Depends on

Used by

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017) (standard reference, not scraped)
Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525) (standard reference, not scraped)