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Coxeter, Artin, and Hecke Interfaces

1 · Prerequisites

2 · Summary

From one finite Coxeter matrix $(S, m)$ the page builds three presentation-level interfaces: the Artin monoid and group carried by the braid pairs alone, the positive section lifted from a reduced-word indexing of $W$ , and the generic Hecke algebra with its standard basis and quadratic normalizations.

The Artin monoid $A^{+}$ is the quotient of the word monoid $S^{*}$ by the smallest congruence containing every braid pair ${s t s \dots, t s t \dots}$ , and the Artin group $A$ is the quotient of the free group $F (S)$ by the normal closure of the relators $u v^{- 1}$ , so that no relation $σ_{s}^{2} = 1$ is imposed. The two words of a braid pair agree in $A$ , and minimality of the congruence descends $s \mapsto σ_{s}$ to the canonical comparison $γ : A^{+} \to A$ . The same universal property applied to $s \mapsto s$ produces the projection $π : A \to W$ with $π \circ γ = π^{+}$ , and $π$ is surjective because the images of the $σ_{s}$ generate $W$ . Quotienting $A$ by the normal closure $D$ of the squares of the generators returns the Coxeter presentation, the induced map $A / D \to W$ is an isomorphism, and $\ker π = D$ .

Reduced-word control of $W$ then produces the positive section. For each $w$ the product of the $σ_{s}$ along any reduced expression is independent of that expression by Matsumoto's theorem, giving $b_{w} \in A^{+}$ with $π^{+} (b_{w}) = w$ and $b_{1} = [ε]$ ; the map $w \mapsto b_{w}$ is injective and a set-theoretic section, and $b_{u} b_{v} = b_{u v}$ holds exactly when $ℓ (u v) = ℓ (u) + ℓ (v)$ . The additive class length $L$ on $A^{+}$ and the group degree $\deg$ on $A$ make the failure of multiplicativity explicit when $S \neq \emptyset$ : $b_{s} b_{s} = [s s] \neq [ε] = b_{s^{2}}$ for every $s \in S$ , because $L ([s s]) = 2 \neq 0 = L ([ε])$ , and the composite $γ \circ b$ is therefore not a group homomorphism either. Nothing here asserts injectivity of $γ$ or of $π^{+}$ , an Ore or Garside condition, or an embedding of $A^{+}$ into $A$ .

The Hecke side of the interface reuses the same indexing. Since the generators $T_{s}$ of the generic Hecke algebra satisfy the braid relations, the universal property of the Artin monoid gives a monoid homomorphism $Θ : A^{+} \to H$ with $Θ (σ_{s}) = T_{s}$ and $Θ (b_{w}) = T_{w}$ , so the positive section and the standard basis share one reduced-word indexing, with base-change compatibility of the specialized bases. The four normalizations $T_{s}$ , $S_{s} = v_{s} T_{s}$ , $- T_{s}$ and $- v_{s}^{- 1} T_{s}$ are interconverted by unit changes of generators, and the root-length matching clause records exactly what a supplied root system must satisfy to match the canonical form: the diagram fixes the normalized products but neither the root lengths nor the individual values of the form. For Soergel and Kazhdan–Lusztig applications the page separates a faithful canonical representation from the reflection-faithful realization required by arguments that assume reflection faithfulness; the companion page computes a degenerate rank-two instance in which the canonical representation is faithful but not reflection faithful.

For the standard type- $A_{n - 1}$ system the page records one conditional application: under AC, and only by consuming the independently published presentation-completeness theorem, the presentation-defined Artin group is identified with the geometric braid group. The companion coxeter-artin-and-hecke-interfaces-examples exercises the projection to $S_{n}$ , the positive lifts, the positive braid monoid and the quadratic Hecke conventions.

Throughout the type- $A$ discussion on this page and its companion, use the order-preserving relabelling $k \mapsto k + 1$ from the library's underlying set ${0, \dots, n - 1}$ to ${1, \dots, n}$ . Thus $(i i + 1)$ denotes the transported adjacent transposition, and one-line notation and inversion positions use the transported labels; inversion numbers are unchanged.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Definition

Let $S$ be a finite set and let $m$ be a Coxeter matrix on $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so that $m (s, s) = 1$ and $m (s, t) = m (t, s) \in {2, 3, \dots} \cup {\infty}$ for $s \neq t$ .

(1) Words. Let $S^{*}$ be the set of finite words $u = s_{1} s_{2} \dots s_{k}$ in the alphabet $S$ ( $k \geq 0$ , letters read from left to right, concatenation written by juxtaposition, empty word $ε$ ; Words in an alphabet with formal inverses, elementary cancellation, and reduced words). Concatenation makes $S^{*}$ a monoid with identity $ε$ (Semigroup and monoid).

(2) Braid pairs. Let $s \neq t$ in $S$ with $m : = m (s, t) < \infty$ . The braid words of length $m$ between $s$ and $t$ are the strictly alternating words $u = u_{1} u_{2} \dots u_{m}$ and $v = v_{1} v_{2} \dots v_{m}$ defined by $u_{i} = s$ for odd $i$ and $u_{i} = t$ for even $i$ , while $v_{i} = t$ for odd $i$ and $v_{i} = s$ for even $i$ . A braid pair is an unordered pair ${u, v}$ arising this way; $m = 2$ contributes the commuting pair ${s t, t s}$ and $m = \infty$ contributes nothing.

(3) The Artin monoid. A congruence on $S^{*}$ is an equivalence relation $\sim$ on $S^{*}$ (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ) with $u \sim v \Rightarrow x u y \sim x v y$ for all $x, y \in S^{*}$ . The total relation is a congruence and the intersection of a nonempty family of congruences is a congruence, so there is a smallest congruence $\equiv^{+}$ containing every braid pair: take the intersection of all congruences containing the braid pairs. Reflexivity, symmetry, transitivity and compatibility with concatenation hold in the intersection because they hold in each of its members. Put

$A^{+} : = A^{+} (S, m) : = S^{*} / ⁣ ⁣ \equiv^{+},$

write $[u]$ for the class of a word $u$ , and define $[u] \cdot [v] : = [u v]$ . This multiplication is well defined: if $u \equiv^{+} u^{'}$ and $v \equiv^{+} v^{'}$ , then $u v \equiv^{+} u^{'} v \equiv^{+} u^{'} v^{'}$ by compatibility and transitivity. Associativity and the identity law descend from concatenation, so $A^{+}$ is a monoid with identity $[ε]$ (Semigroup and monoid); $A^{+}$ is the Artin monoid of $(S, m)$ , and $σ_{s} : = [s]$ for $s \in S$ .

(4) The Artin group. Let $F (S)$ be the free group on $S$ (Free group on a set of generators) and, for $s \neq t$ with $m (s, t) < \infty$ , let $ρ_{s, t} : = u v^{- 1} \in F (S)$ with $u, v$ the braid words of (2). Let $N : = ⟨ ⁣ ⟨ {ρ_{s, t} : s \neq t, m (s, t) < \infty} ⟩ ⁣ ⟩_{F (S)}$ be their normal closure (The normal closure of a subset of a group, Relators and relations; finitely generated, finitely related, and finite presentations), and put

$A : = A (S, m) : = F (S) / N .$

Equivalently, $A$ is the group presented by $⟨ S ∣ u = v for every braid pair {u, v} ⟩$ in the sense of Group presentation by generators and relations; its elements are the left cosets $g N$ , and the images $σ_{s}$ of $s \in S$ generate $A$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, Group and abelian group). No relation $σ_{s}^{2} = 1$ is imposed: the element $σ_{s}^{2}$ need not be trivial in $A$ .

(5) The canonical comparison $A^{+} \to A$ . The two braid words of a braid pair have equal images in $A$ , because their difference lies in $N$ . Hence the assignment $s \mapsto σ_{s}$ is constant on braid pairs, and by minimality of $\equiv^{+}$ it induces a monoid homomorphism

$γ : A^{+} ⟶ A, γ (σ_{s}) = σ_{s},$

regarding the group $A$ as a monoid under its multiplication. It is the unique monoid homomorphism with $γ (σ_{s}) = σ_{s}$ for all $s$ . This is the construction whose universal property is proved in Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares ↗ (1); that lemma is the recorded justifier of this definition.

(6) Conventions and abstentions. (i) A value $m (s, t) = \infty$ imposes no braid pair and hence no relation in either $A^{+}$ or $A$ . (ii) The constructions depend only on the pair $(S, m)$ . (iii) Nothing is asserted here about injectivity of $γ$ or of $A^{+} \to W$ , about Ore localisation or a group of fractions, about torsion-freeness of $A$ , about the word problem, about $A^{+}$ embedding in $A$ , or about the $K (π, 1)$ property; in particular no multiplicativity of any lift of a reduced expression is claimed by this definition. (iv) No topological model (configuration space, hyperplane complement, Salvetti complex) is constructed or used. Part (3) establishes the congruence and quotient multiplication; Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares ↗ establishes their universal property.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares

Statement

Let $(S, m)$ be a finite Coxeter matrix, $W$ the presented Coxeter group with its universal property and length function $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $S^{*}$ , $A^{+}$ , $A$ , the classes $[u]$ , the elements $σ_{s}$ and $γ : A^{+} \to A$ be as in Artin monoid and Artin group presentations, and the canonical monoid-to-group map.

(1) Universal property of the Artin monoid. Let $M$ be a monoid (Semigroup and monoid; every group is a monoid, Group and abelian group) and let $f : S \to M$ be a map such that $f (u) = f (v)$ in $M$ for every braid pair ${u, v}$ , where $f (s_{1} \dots s_{k}) : = f (s_{1}) \dots f (s_{k})$ . Then there is a unique monoid homomorphism $\overset{ˉ}{f} : A^{+} \to M$ with $\overset{ˉ}{f} (σ_{s}) = f (s)$ for all $s \in S$ .

(2) Universal property of the Artin group. Let $G$ be a group and let $f : S \to G$ satisfy $f (u) = f (v)$ for every braid pair. Then there is a unique group homomorphism $\overset{ˉ}{f} : A \to G$ with $\overset{ˉ}{f} (σ_{s}) = f (s)$ for all $s \in S$ (Monoid homomorphism and group homomorphism). Consequently $(A, (σ_{s})_{s \in S})$ is the group presented by the Artin presentation $⟨ S ∣ u = v (braid pairs) ⟩$ in the sense of Group presentation by generators and relations and Relators and relations; finitely generated, finitely related, and finite presentations.

(3) The projection onto $W$ . The map $s \mapsto s$ from $S$ to $W$ sends every braid pair to an equality in $W$ , so (2) gives a homomorphism

$π : A ⟶ W, π (σ_{s}) = s,$

which is surjective because the images $π (σ_{s}) = s$ generate $W$ . Moreover (1) with $M = W$ gives $π^{+} : A^{+} \to W$ with $π^{+} (σ_{s}) = s$ , and $π \circ γ = π^{+}$ .

(4) Quotient by the squares. Let $D : = ⟨ ⁣ ⟨ {σ_{s}^{2} : s \in S} ⟩ ⁣ ⟩_{A}$ be the normal closure in $A$ of the squares of the generators. Then the relators $u v^{- 1}$ of $A$ are trivial in $A / D$ and $σ_{s}^{2} D = D$ for all $s$ , so the assignment $s \mapsto σ_{s} D$ induces a homomorphism $φ : W \to A / D$ with $φ (s) = σ_{s} D$ ; the induced map $\overset{ˉ}{π} : A / D \to W$ is an isomorphism with inverse $φ$ . Equivalently, imposing the relations $σ_{s}^{2} = 1$ on the Artin presentation returns the Coxeter presentation, and $\ker π = D$ .

(5) Type-A application (conditional on the independent presentation theorem). Suppose $(S, m)$ is the standard Coxeter system of type $A_{n - 1}$ for some $n \geq 2$ , with generators $s_{1}, \dots, s_{n - 1}$ , adjacent labels $3$ and all other off-diagonal labels $2$ . Under AC, the Artin group $A (S, m)$ is identified with the published geometric braid group on $n$ strands by the generator correspondence $σ_{s_{i}} \mapsto$ the standard geometric half twist. The presentation-defined group agrees with $B_{n}^{A r t i n}$ by The braid group by Artin presentation, and its isomorphism with the geometric braid group is supplied by the independently proved The Artin presentation is complete for geometric braids (whose AC hypothesis is part of this application). This is an application of that published theorem, not a topological proof here; the universal properties in (1)--(4) alone do not establish injectivity of the type-A comparison.

(6) Scope. No injectivity of $γ$ or $π^{+}$ , no torsion-freeness of $A$ , no solvability of the word problem, no Ore localisation, no monoid-to-group embedding and no general $K (π, 1)$ statement is made. The type-A identification is only the conditional application in (5).

Facts & Assumptions

Given: A finite Coxeter matrix $(S, m)$ , the presented group $W$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups with its universal property, the constructions $S^{*}, A^{+}, A, [u], σ_{s}, γ$ of Artin monoid and Artin group presentations, and the canonical monoid-to-group map, and, in (1) and (2), a monoid $M$ or a group $G$ with a map $f : S \to M$ or $f : S \to G$ whose two values on the words of every braid pair agree.

[F1]

A monoid is a set with an associative product and a two-sided identity, and a group is such a monoid in which every element is invertible. (Semigroup and monoid)

[F2]

On $S^{*}$ there is a smallest congruence $\equiv^{+}$ containing every braid pair, the classes satisfy $[u] [v] = [u v]$ , and $A^{+} = S^{*} / ⁣ \equiv^{+}$ has the elements $σ_{s} = [s]$ . (Artin monoid and Artin group presentations, and the canonical monoid-to-group map)

[F3]

In $A = F (S) / N$ the subgroup $N$ is the normal closure of the elements $ρ_{s, t} = u v^{- 1}$ for $s \neq t$ with $m (s, t) < \infty$ . (Artin monoid and Artin group presentations, and the canonical monoid-to-group map)

[F4]

Every map from a set $X$ into a group extends uniquely to a group homomorphism on the free group $F (X)$ . (Free group on a set of generators)

[F5]

The normal closure of a subset $R$ of a group is the smallest normal subgroup containing $R$ . (The normal closure of a subset of a group)

[F6]

The kernel of every group homomorphism is a normal subgroup. (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup)

[F7]

A homomorphism killing a normal subgroup factors uniquely through the quotient. (A homomorphism that kills a normal subgroup factors uniquely through the quotient group)

[F8]

The subgroup generated by a set is the smallest subgroup containing it. (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups)

[F9]

An isomorphism is a bijective group homomorphism. (Group isomorphisms, automorphisms and the set $Aut (G)$ )

[F10]

A map on the generators of a presentation that sends every relator to the identity induces a unique homomorphism on the presented group, and that homomorphism is surjective exactly when the images of the generators generate the target. (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group)

[F11]

$B_{n}^{A r t i n}$ is the group with generators $σ_{1}, \dots, σ_{n - 1}$ and the braid relations $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ and the commuting relations $σ_{i} σ_{j} = σ_{j} σ_{i}$ for $∣ i - j ∣ > 1$ . (The braid group by Artin presentation)

[F12]

Assume AC: the Artin presentation of The braid group by Artin presentation is a presentation of the geometric braid group $B_{n}^{g e o m}$ , the published surjection $φ_{n} : B_{n}^{A r t i n} \to B_{n}^{g e o m}$ being an isomorphism. (The Artin presentation is complete for geometric braids)

[F13]

AC: every family of nonempty sets has a choice function. (The Axiom of Choice)

[F14]

The group $W$ is presented by the generators $S$ and the relators $s^{2}$ ( $s \in S$ ) and $(s t)^{m (s, t)}$ ( $s \neq t$ , $m (s, t) < \infty$ ); in particular $s^{2} = 1$ and $(s t)^{m (s, t)} = 1$ in $W$ for those $s, t$ . (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups)

Proof

1.1F1F2

For (1), define $f (s_{1} \dots s_{k}) : = f (s_{1}) \dots f (s_{k})$ and $f (ε) : = e_{M}$ on $S^{*}$ , and let $\sim$ be the relation $u \sim v : \Leftrightarrow f (u) = f (v)$ . Then $\sim$ is an equivalence relation, and it is compatible with concatenation: if $f (u) = f (v)$ then $f (x u y) = f (x) f (u) f (y) = f (x) f (v) f (y) = f (x v y)$ for all $x, y \in S^{*}$ . By hypothesis $\sim$ contains every braid pair, so minimality of $\equiv^{+}$ gives $\equiv^{+} \subseteq \sim$ ; hence $\overset{ˉ}{f} ([u]) : = f (u)$ is well defined, and $\overset{ˉ}{f} ([u] [v]) = \overset{ˉ}{f} ([u v]) = f (u) f (v) = \overset{ˉ}{f} ([u]) \overset{ˉ}{f} ([v])$ together with $\overset{ˉ}{f} ([ε]) = e_{M}$ makes $\overset{ˉ}{f}$ a monoid homomorphism with $\overset{ˉ}{f} (σ_{s}) = f (s)$ . Conversely every element of $A^{+}$ is a class of a word, hence a product of the elements $σ_{s} = [s]$ , so a monoid homomorphism out of $A^{+}$ is determined by its values on the $σ_{s}$ and $\overset{ˉ}{f}$ is the unique homomorphism with $\overset{ˉ}{f} (σ_{s}) = f (s)$ .

1.2F14algebra

The two words of a braid pair have equal images in $W$ . Let $m = m (s, t) < \infty$ , $u = s t s \dots$ and $v = t s t \dots$ be the alternating words of length $m$ . In $W$ one has $s^{2} = t^{2} = 1$ and $(s t)^{m} = 1$ by [F14]. If $m = 2 k$ is even, then $u = (s t)^{k}$ and $v = (t s)^{k}$ ; from $(s t) (t s) = s t^{2} s = s^{2} = 1$ one gets $t s = (s t)^{- 1}$ , hence $(t s)^{k} = (s t)^{- k} = (s t)^{2 k - k} = (s t)^{k} = u$ because $(s t)^{2 k} = (s t)^{m} = 1$ . If $m = 2 k + 1$ is odd, then $u = (s t)^{k} s$ and $v = (t s)^{k} t = (s t)^{- k} t = (s t)^{m - k} t = (s t)^{k + 1} t = (s t)^{k} s = u$ , using $(s t) t = s$ and again $t s = (s t)^{- 1}$ .

1.3F3F4F5F6F7F8F10

For (2), regard $S$ as a subset of the free group $F (S)$ . By [F4] the map $f : S \to G$ extends uniquely to a homomorphism $\hat{f} : F (S) \to G$ with $\hat{f} (s) = f (s)$ ; on words this gives $\hat{f} (u) = f (u)$ for the two words of any braid pair. For every braid pair, with $ρ_{s, t} = u v^{- 1}$ as in [F3], one has $\hat{f} (ρ_{s, t}) = \hat{f} (u) \hat{f} (v)^{- 1} = f (u) f (v)^{- 1} = 1$ , so $\hat{f}$ kills the set ${ρ_{s, t}}$ ; its kernel is a normal subgroup by [F6] and hence contains the normal closure $N = ⟨ ⁣ ⟨ {ρ_{s, t}} ⟩ ⁣ ⟩_{F (S)}$ by [F5]. Thus [F7] gives a homomorphism $\overset{ˉ}{f} : A \to G$ with $\overset{ˉ}{f} (g N) = \hat{f} (g)$ , and in particular $\overset{ˉ}{f} (σ_{s}) = f (s)$ ; it is unique with this property because the elements $σ_{s}$ are the images of $S$ and generate $A$ in the sense of [F8]. Writing the relators as the equations $u = v$ exhibits $(A, (σ_{s}))$ as the presented group $⟨ S ∣ u = v (braid pairs) ⟩$ : a homomorphism out of this presentation is exactly a map $S \to G$ whose two values on each braid pair agree, and it is unique, which is the universal property just proved.

2.1F8F10step 1.1step 1.2step 1.3

For (3), step 1.2 says that the map $s \mapsto s$ sends every braid pair to an equality in $W$ , so (2) of step 1.3 yields $π : A \to W$ with $π (σ_{s}) = s$ , and (1) with $M = W$ applied to the same map yields $π^{+} : A^{+} \to W$ with $π^{+} (σ_{s}) = s$ . The homomorphism $π$ is surjective: every element of $W$ is a product of the images of elements of $S$ , and these are the images under $π$ of the generators $σ_{s}$ of $A$ , so $π$ satisfies the surjectivity criterion of [F10]; equivalently the images $π (σ_{s}) = s$ generate $W$ by [F8]. Finally $π \circ γ = π^{+}$ , because both sides are monoid homomorphisms $A^{+} \to W$ agreeing on the $σ_{s}$ , and these generate $A^{+}$ by the uniqueness clause of step 1.1.

2.2F9F10F11step 1.3

For (5), let $n \geq 2$ and let $m$ be the type- $A_{n - 1}$ matrix on $S = {s_{1}, \dots, s_{n - 1}}$ , so $m (s_{i}, s_{j}) = 3$ for $∣ i - j ∣ = 1$ and $m (s_{i}, s_{j}) = 2$ for $∣ i - j ∣ > 1$ . Its braid pairs are then exactly the pairs of alternating words of length $3$ , ${s_{i} s_{j} s_{i}, s_{j} s_{i} s_{j}}$ for $∣ i - j ∣ = 1$ , and the commuting pairs ${s_{i} s_{j}, s_{j} s_{i}}$ for $∣ i - j ∣ > 1$ ; these are precisely the two relation families of [F11], so the identity correspondence $s_{i} \leftrightarrow σ_{i}$ matches the defining relations of $A (S, m)$ with those of $B_{n}^{A r t i n}$ . By (2) of step 1.3 there is a homomorphism $A (S, m) \to B_{n}^{A r t i n}$ with $σ_{s_{i}} \mapsto σ_{i}$ , and by [F10] applied to the presentation of [F11] there is a homomorphism $B_{n}^{A r t i n} \to A (S, m)$ with $σ_{i} \mapsto σ_{s_{i}}$ ; the two are inverse on the generators, hence mutually inverse and so isomorphisms by [F9].

3.1F12F13step 2.2

Under AC, [F12] identifies $B_{n}^{A r t i n}$ with $B_{n}^{g e o m}$ by the published isomorphism $φ_{n}$ carrying $σ_{i}$ to the standard geometric half twist. Composing with the isomorphism of step 2.2 identifies $A (S, m)$ with $B_{n}^{g e o m}$ under the stated generator correspondence. This is the only place where choice is used: the applied completeness theorem is an AC-conditional statement [F13], while the constructions and arguments of (1)--(4) and step 2.2 are explicit on finite relator sets and use no choice.

3.2F5F6F7constructalgebrastep 2.1

For (4), let $D_{0} : = {σ_{s}^{2} : s \in S}$ , so $D = ⟨ ⁣ ⟨ D_{0} ⟩ ⁣ ⟩_{A}$ . Since $π (σ_{s}^{2}) = π (σ_{s})^{2} = s^{2} = 1$ , each $σ_{s}^{2}$ lies in $\ker π$ , which is normal by [F6]; hence $D \subseteq \ker π$ by [F5], and [F7] gives the induced homomorphism $\overset{ˉ}{π} : A / D \to W$ with $\overset{ˉ}{π} (σ_{s} D) = s$ . For the inverse direction, put $f (s) : = σ_{s} D$ ; then $f (s)^{2} = σ_{s}^{2} D = D$ , and for $s \neq t$ with $m = m (s, t) < \infty$ the defining braid relation of $A$ and the relations $σ_{s}^{2} D = σ_{t}^{2} D = D$ give $(f (s) f (t))^{m} = D$ : writing $k : = ⌊ m / 2 ⌋$ , the braid relation reads $(σ_{s} σ_{t})^{k} = (σ_{t} σ_{s})^{k}$ when $m = 2 k$ , so that $(σ_{s} σ_{t})^{2 k} = (σ_{s} σ_{t})^{k} (σ_{s} σ_{t})^{k} = (σ_{s} σ_{t})^{k} (σ_{t} σ_{s})^{k} = (σ_{s} σ_{t})^{k - 1} σ_{s} σ_{t} σ_{t} σ_{s} (σ_{t} σ_{s})^{k - 1} = \dots = 1$ in $A / D$ by $σ_{s}^{2} = σ_{t}^{2} = 1$ , while for $m = 2 k + 1$ it reads $(σ_{s} σ_{t})^{k} σ_{s} = (σ_{t} σ_{s})^{k} σ_{t}$ , so that $(σ_{s} σ_{t})^{2 k + 1} = (σ_{s} σ_{t})^{k} σ_{s} σ_{t} (σ_{s} σ_{t})^{k} = (σ_{t} σ_{s})^{k} σ_{t}^{2} (σ_{s} σ_{t})^{k} = (σ_{t} σ_{s})^{k} (σ_{s} σ_{t})^{k} = D$ . The universal property of $W$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups) therefore gives a homomorphism $φ : W \to A / D$ with $φ (s) = σ_{s} D$ .

4.1F8F9F10step 3.2

The homomorphisms $\overset{ˉ}{π}$ and $φ$ of step 3.2 are mutually inverse: $\overset{ˉ}{π} (φ (s)) = \overset{ˉ}{π} (σ_{s} D) = s$ for all $s$ , and $φ (\overset{ˉ}{π} (σ_{s} D)) = φ (s) = σ_{s} D$ for all $s$ ; two homomorphisms out of a group generated by a set are equal when they agree on that set, while $W$ is generated by the images of $S$ and $A / D$ is generated by the $σ_{s} D$ by [F8]. Hence $\overset{ˉ}{π}$ is bijective, i.e. an isomorphism by [F9]. Since $π$ is the composite of the quotient map $A \to A / D$ with $\overset{ˉ}{π}$ and $\overset{ˉ}{π}$ is injective, an element $g \in A$ lies in $\ker π$ exactly when $g D = D$ , that is, exactly when $g \in D$ ; thus $\ker π = D$ , and imposing the relations $σ_{s}^{2} = 1$ on the Artin presentation returns the Coxeter presentation in the sense of [F7] and [F10].

5.1given∎

Scope: nothing in the proof establishes injectivity of $γ$ or of $π^{+}$ , torsion-freeness of $A$ , solvability of the word problem, an Ore localisation, an embedding of $A^{+}$ into $A$ , or a general $K (π, 1)$ statement, and the only identification with geometric braids is the conditional application of step 3.1.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The reduced positive section b_w, its length additivity, and the degree homomorphism

Statement

Let $(S, m)$ be a finite Coxeter matrix, $W$ the presented group with length function $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $S^{*}$ , $A^{+}$ , $A$ , $γ : A^{+} \to A$ , $σ_{s} = [s]$ and $π^{+} : A^{+} \to W$ be as in Artin monoid and Artin group presentations, and the canonical monoid-to-group map and Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares.

(1) The positive lift. For $w \in W$ choose a reduced expression $w = s_{1} \dots s_{k}$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups) and put

$b_{w} : = σ_{s_{1}} \dots σ_{s_{k}} \in A^{+} .$

Then $b_{w}$ depends only on $w$ , not on the chosen reduced expression; $b_{1} = 1_{A^{+}} = [ε]$ ; and $π^{+} (b_{w}) = w$ .

(2) The set-section. The map $b : W \to A^{+}$ , $w \mapsto b_{w}$ , satisfies $π^{+} \circ b = {i d}_{W}$ and $π \circ γ \circ b = {i d}_{W}$ , hence $b$ is injective and is a set-theoretic section of both $π^{+}$ and the composite $π \circ γ : A^{+} \to W$ .

(3) The positive length. There is a unique monoid homomorphism

$L : A^{+} ⟶ (N, +, 0), L (σ_{s}) = 1;$

explicitly $L ([s_{1} \dots s_{k}]) = k$ , so $L$ is well defined, $L (x y) = L (x) + L (y)$ and $L (1_{A^{+}}) = 0$ . Consequently $L (b_{w}) = ℓ (w)$ for every $w \in W$ . Moreover the same assignment defines a group homomorphism

$\deg : A ⟶ Z, \deg (σ_{s}) = 1,$

and $\deg (γ (x)) = L (x)$ for all $x \in A^{+}$ .

(4) Multiplicativity. For all $u, v \in W$ :

$b_{u} b_{v} = b_{u v} ⟺ ℓ (u v) = ℓ (u) + ℓ (v) .$

If $ℓ (u v) < ℓ (u) + ℓ (v)$ , then $b_{u} b_{v} \neq b_{u v}$ , and indeed $L (b_{u} b_{v}) = ℓ (u) + ℓ (v) > ℓ (u v) = L (b_{u v})$ .

(5) Failure of multiplicativity. If $S \neq \emptyset$ then $b$ is not a monoid homomorphism: for $s \in S$ one has $b_{s} b_{s} = [s s]$ and $b_{s^{2}} = b_{1} = [ε]$ , and these differ because $L ([s s]) = 2 \neq 0 = L ([ε])$ . Likewise $γ \circ b : W \to A$ is not a group homomorphism.

(6) Scope. No injectivity of $γ : A^{+} \to A$ or of $π^{+} : A^{+} \to W$ , no Ore or Garside condition, no embedding of $A^{+}$ into $A$ , and no topological statement is made or needed.

Facts & Assumptions

Given: A finite Coxeter matrix $(S, m)$ , the group $W$ with its length function $ℓ$ and reduced expressions, and the constructions $S^{*}$ , $A^{+}$ , $A$ , $γ$ , $σ_{s}$ and $π^{+}$ of the two items named in the statement.

[F1]

On $S^{*}$ there is a smallest congruence $\equiv^{+}$ containing every braid pair, the classes satisfy $[u] [v] = [u v]$ and $A^{+} = S^{*} / ⁣ \equiv^{+}$ has identity $[ε]$ and elements $σ_{s} = [s]$ . (Artin monoid and Artin group presentations, and the canonical monoid-to-group map)

[F2]

Every map $f : S \to G$ into a group whose values on the two words of every braid pair agree extends uniquely to a group homomorphism $\overset{ˉ}{f} : A \to G$ with $\overset{ˉ}{f} (σ_{s}) = f (s)$ . (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares)

[F3]

Every map $f : S \to M$ into a monoid whose values on the two words of every braid pair agree extends uniquely to a monoid homomorphism $\overset{ˉ}{f} : A^{+} \to M$ with $\overset{ˉ}{f} (σ_{s}) = f (s)$ . (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares)

[F4]

The homomorphism $π^{+} : A^{+} \to W$ satisfies $π^{+} (σ_{s}) = s$ and $π \circ γ = π^{+}$ . (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares)

[F5]

$N$ is a set containing $0$ on which addition is defined. (The natural numbers $N$ (von Neumann), Addition of natural numbers)

[F6]

Addition on $N$ satisfies $m + 0 = m$ and is associative. (Addition of natural numbers, Addition is associative)

[F7]

$0$ is a two-sided identity for addition on $N$ . (Left identity for addition)

[F8]

$(Z, +, \cdot, 0, 1)$ with the operations of Arithmetic on the integers is a commutative ring in which every element has an additive inverse; in particular $(Z, +, 0)$ is an abelian group, and its element $1$ is the multiplicative identity. The natural numbers embed in $Z$ by an injective map preserving addition and multiplication, and $2 \neq 0$ in $N$ (The natural numbers $N$ (von Neumann)); hence the images of the distinct naturals $2$ and $0$ differ, and since $1 + 1$ is the image of $2$ while $0$ is the image of $0$ , the element $2 : = 1 + 1$ is nonzero in $Z$ . (The integers form a commutative ring, Arithmetic on the integers, The naturals embed in the integers)

[F9]

$γ : A^{+} \to A$ is the unique monoid homomorphism with $γ (σ_{s}) = σ_{s}$ for all $s \in S$ . (Artin monoid and Artin group presentations, and the canonical monoid-to-group map)

Proof

1.1F1given

The product $σ_{s_{1}} \dots σ_{s_{k}}$ is independent of the reduced expression: by Matsumoto's theorem (Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups (1)), any two reduced expressions of an element $w \in W$ are connected by finitely many replacements of an alternating subword $s t s \dots$ of length $m (s, t) < \infty$ by the other alternating subword $t s t \dots$ , inside some context $x (\cdot) y$ . Such a subword pair is exactly a braid pair of Artin monoid and Artin group presentations, and the canonical monoid-to-group map (2), so by the congruence property of $\equiv^{+}$ the two full words are equivalent and their classes in $A^{+}$ coincide; hence the product in $A^{+}$ depends only on $w$ . A reduced expression of $w$ exists because $ℓ (w)$ is a minimum over a nonempty set of word lengths (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

2.1F1F3F4F9step 1.1

The empty word is a reduced expression of $1$ because $ℓ (1) = 0$ , so $b_{1} = [ε] = 1_{A^{+}}$ by [F1]. For the projection, [F4] gives $π^{+} (σ_{s}) = s$ and $π \circ γ = π^{+}$ while [F9] makes $γ$ a monoid homomorphism, so for a reduced expression $w = s_{1} \dots s_{k}$ one has $π^{+} (b_{w}) = π^{+} (σ_{s_{1}}) \dots π^{+} (σ_{s_{k}}) = s_{1} \dots s_{k} = w$ , and composing with $π \circ γ = π^{+}$ gives $π \circ γ \circ b = π^{+} \circ b = {i d}_{W}$ .

2.2F3F5F6F7step 1.1

The length map exists and is unique: apply the monoid universal property [F3] to $M : = (N, +, 0)$ and the constant map $f (s) : = 1$ , which is legitimate because the two words of a braid pair both have length $m (s, t)$ , so their images under the word-length map agree; this gives a unique monoid homomorphism $L : A^{+} \to (N, +, 0)$ with $L (σ_{s}) = 1$ . For a word $u = s_{1} \dots s_{k}$ one has $L ([u]) = L (σ_{s_{1}} \dots σ_{s_{k}}) = 1 + \dots + 1 = k$ by additivity [F6] and $L ([ε]) = 0$ [F7], so $L$ is the word length on classes and in particular $L (b_{w}) = k = ℓ (w)$ for every $w$ .

3.1F2F3F8step 2.2

The degree map exists: apply the group universal property [F2] to $G : = (Z, +, 0)$ of [F8] and the constant map $f (s) : = 1$ , whose two values on a braid pair both equal $m (s, t)$ ; this gives a group homomorphism $\deg : A \to Z$ with $\deg (σ_{s}) = 1$ . Then $\deg \circ γ$ and $L$ are monoid homomorphisms $A^{+} \to Z$ agreeing on every generator: $\deg (γ (σ_{s})) = \deg (σ_{s}) = 1 = L (σ_{s})$ , so by the uniqueness in [F3] they agree on all of $A^{+}$ , that is, $\deg (γ (x)) = L (x)$ for every $x \in A^{+}$ .

3.2F4step 2.1

The map $b$ is injective and a section: if $b_{u} = b_{v}$ then $u = π^{+} (b_{u}) = π^{+} (b_{v}) = v$ , and $π^{+} \circ b = {i d}_{W}$ and $π \circ γ \circ b = {i d}_{W}$ were proved in step 2.1; a map with a left inverse is injective, so $b$ is a set-theoretic section of both maps.

3.3F1step 1.1step 2.2algebra

For the forward direction of (4), suppose $ℓ (u v) = ℓ (u) + ℓ (v)$ and let $u = s_{1} \dots s_{p}$ , $v = t_{1} \dots t_{q}$ be reduced expressions. The concatenated word $s_{1} \dots s_{p} t_{1} \dots t_{q}$ represents $u v$ and has length $p + q = ℓ (u) + ℓ (v) = ℓ (u v)$ , so it is a reduced expression of $u v$ ; hence by step 1.1, $b_{u v} = σ_{s_{1}} \dots σ_{s_{p}} σ_{t_{1}} \dots σ_{t_{q}} = b_{u} b_{v}$ . Conversely, if $b_{u} b_{v} = b_{u v}$ , then applying the additive map $L$ of step 2.2 gives $ℓ (u) + ℓ (v) = L (b_{u}) + L (b_{v}) = L (b_{u} b_{v}) = L (b_{u v}) = ℓ (u v)$ . In particular, if $ℓ (u v) < ℓ (u) + ℓ (v)$ then $L (b_{u} b_{v}) = ℓ (u) + ℓ (v) > ℓ (u v) = L (b_{u v})$ , so $b_{u} b_{v} \neq b_{u v}$ .

4.1F1F8step 2.1step 2.2step 3.1algebra

If $S \neq \emptyset$ , fix $s \in S$ . By part 1 of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action the sign character of $W$ satisfies $sgn (s) = - 1 \neq 1 = sgn (1)$ , so $s \neq 1$ in $W$ ; since $s$ is a word of length $1$ we have $ℓ (s) \leq 1$ , while $ℓ (s) \neq 0$ because the empty word is the only word of length $0$ and its value is $1$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so $ℓ (s) = 1$ . Hence the one-letter word $s$ is a reduced expression and $b_{s} = [s]$ by step 1.1, while $s^{2} = 1$ in $W$ with $ℓ (1) = 0$ makes the empty word a reduced expression of $s^{2}$ , so $b_{s^{2}} = b_{1} = [ε]$ by step 2.1. Hence $b_{s} b_{s} = [s] [s] = [s s]$ by [F1], whereas $L ([s s]) = 2 \neq 0 = L ([ε])$ by step 2.2, so $b_{s} b_{s} \neq b_{s^{2}}$ and $b$ is not a monoid homomorphism. For the composite, $(γ \circ b) (s)^{2} = γ ([s s]) = σ_{s}^{2}$ while $(γ \circ b) (s^{2}) = γ ([ε]) = 1_{A}$ , and these differ because $\deg (σ_{s}^{2}) = 2 \neq 0 = \deg (1_{A})$ by step 3.1, where $2 = 1 + 1 \neq 0$ in $Z$ is the nontriviality recorded in [F8], and a group homomorphism preserves the identity (Monoid homomorphism and group homomorphism); so $γ \circ b$ is not a group homomorphism.

5.1given∎

Scope and choice: injectivity of $γ$ and of $π^{+}$ , an Ore or Garside condition, an embedding of $A^{+}$ into $A$ and every topological statement are outside this result, and nothing here asserts them. No choice is used: $b_{w}$ is defined by the uniqueness proved in step 1.1, so it is a definite description rather than a selection among expressions, $\equiv^{+}$ is an intersection of a definable family of congruences, and all computations are finite.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary

Statement

Let $(S, m)$ be a finite Coxeter matrix, $W$ the presented Coxeter group with length $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $R$ , $v_{s}$ , $H$ and its generators $T_{s}$ be the generic Coxeter Hecke algebra of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy (so $R = Z [v_{1}^{\pm 1}, \dots, v_{c}^{\pm 1}]$ and $v_{s} = v_{[s]}$ ), with the standard basis elements $T_{w}$ of Reduced-word independence of T_w and the length-multiplication rules and The standard basis of the generic Hecke algebra and base change. Let $A^{+}$ , $γ$ , $σ_{s}$ be as in Artin monoid and Artin group presentations, and the canonical monoid-to-group map and $b_{w}$ as in The reduced positive section b_w, its length additivity, and the degree homomorphism. Finally let $V = R^{S}$ , $B$ and $ρ : W \to G L (V)$ be the Coxeter form and canonical reflection representation of The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, with root system $Φ$ (Descent of the reflection representation, unit root norms, and conjugation of reflections).

(1) Indexing and coefficient compatibility of the Artin and Hecke interfaces. Because the generators $T_{s}$ satisfy the braid relations of $H$ , the universal property of the Artin monoid (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares (1)) gives a unique monoid homomorphism

$Θ : A^{+} ⟶ (H, \cdot, 1), Θ (σ_{s}) = T_{s} .$

It satisfies $Θ (b_{w}) = T_{w}$ for every $w \in W$ . Moreover the indexing is coefficient-compatible: $(T_{w})_{w \in W}$ is an $R$ -basis of $H$ , and for every commutative ring $R^{'}$ and ring homomorphism $R \to R^{'}$ the specialised elements $(1 \otimes T_{w})$ form an $R^{'}$ -basis of $R^{'} \otimes_{R} H$ (The standard basis of the generic Hecke algebra and base change (2),(4)). Thus the positive section $b$ and the Hecke standard basis use the same reduced-word indexing, and $Θ$ carries the monoid multiplication of $A^{+}$ to the multiplication of $H$ in that basis: $Θ (b_{u} b_{v}) = T_{u} T_{v}$ for all $u, v \in W$ .

(2) Normalization conversions. In $H$ : (i) $T_{s}^{2} = (v_{s} - v_{s}^{- 1}) T_{s} + 1$ (the normalized or $T$ -normalization); (ii) the multiplicative generators $S_{s} : = v_{s} T_{s}$ satisfy $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ , i.e. $S_{s}^{2} = (Q_{s} - 1) S_{s} + Q_{s}$ , where $Q_{s} : = v_{s}^{2} \in R$ ; for a single parameter $v_{s} = v$ and $Q = v^{2}$ this is the " $S = q T$ , $Q = q^{2}$ " convention with $q = v$ ; (iii) the opposite-sign generators $H_{s} : = - T_{s}$ satisfy $H_{s}^{2} = (v_{s}^{- 1} - v_{s}) H_{s} + 1$ ; (iv) the Soergel-calculus generators $T_{s}^{E W} : = - v_{s}^{- 1} T_{s}$ satisfy $(T_{s}^{E W} + 1) (T_{s}^{E W} - v_{s}^{- 2}) = 0$ , i.e. $(T_{s}^{E W})^{2} = (v_{s}^{- 2} - 1) T_{s}^{E W} + v_{s}^{- 2}$ .

The assignments $T_{s} \leftrightarrow S_{s} = v_{s} T_{s}$ , $T_{s} \leftrightarrow H_{s} = - T_{s}$ and $T_{s} \leftrightarrow T_{s}^{E W} = - v_{s}^{- 1} T_{s}$ , together with the identity on $T_{s}$ , extend to $R$ -algebra isomorphisms between the four corresponding presentations, with inverse displayed: $T_{s} = v_{s}^{- 1} S_{s} = - H_{s} = - v_{s} T_{s}^{E W}$ . In every case the braid relations are preserved: when $m$ is even the two alternating words contain each of $s, t$ exactly $m / 2$ times, and when $m$ is odd the generators $s, t$ are joined by an odd-labelled edge, so their parameters satisfy $v_{s} = v_{t}$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy); in both cases the product of the scaling constants attached to the letters is the same on the two sides (with a single parameter simply $v_{s}^{m}$ , $(- 1)^{m}$ or $(- v_{s}^{- 1})^{m}$ ). For the single-parameter specialization, put $H_{w} : = (- 1)^{ℓ (w)} T_{w}$ and $T_{w}^{K L} : = v^{- ℓ (w)} H_{w}$ , where each basis element is the product along a reduced expression. Thus $H_{s} = - T_{s}$ as in (iii), $T_{s}^{K L} = T_{s}^{E W}$ as in (iv), and $H_{w} = v^{ℓ (w)} T_{w}^{K L}$ . Here $T^{K L}$ denotes the original Kazhdan–Lusztig normalization with parameter $q = v^{- 2}$ ; it is distinct from the normalized $T$ -basis in (i). For any finite family of polynomials $P_{y, x} (q)$ and an element written in that normalization as

$C_{x} = v^{ℓ (x)} \sum_{y} P_{y, x} (v^{- 2}) T_{y}^{K L},$

its coefficients in $C_{x} = \sum_{y} h_{y, x} H_{y}$ are exactly $h_{y, x} = v^{ℓ (x) - ℓ (y)} P_{y, x} (v^{- 2})$ . This proves the coefficient conversion used in Remark 3.2 of the Hodge-theory source whenever such an expansion is supplied, without constructing a canonical basis or asserting that the supplied polynomials are Kazhdan–Lusztig polynomials.

(3) Root-length matching for a supplied root system. Let $β$ be a symmetric bilinear form on a real vector space $E$ and let $α_{s} \in E$ ( $s \in S$ ) be vectors with $∣ α_{s} ∣ : = β (α_{s}, α_{s})^{1 / 2} > 0$ such that

$\frac{β (α_{s}, α_{t})}{∣ α_{s} ∣ ∣ α_{t} ∣} = - c (s, t) (s \neq t),$

where $c (s, t) : = \cos (π / m (s, t))$ for finite $m (s, t)$ and $c (s, t) : = 1$ when $m (s, t) = \infty$ (the convention of the Coxeter form The real Coxeter form, its radical, reflections, and form-preserving maps, so that the right-hand side is the number $B (e_{s}, e_{t})$ ), that ${α_{s} : s \in S}$ is linearly independent, and that the assignment $v \mapsto v - \frac{2 β (v, α_{s})}{β (α_{s}, α_{s})} α_{s}$ defines a representation of $W$ on $E$ . Then the linear map

$φ : V ⟶ E, φ (e_{s}) : = \frac{α_{s}}{∣ α_{s} ∣},$

is an isomorphism of $V$ onto $span {α_{s} : s \in S}$ , it satisfies $β (φ u, φ w) = B (u, w)$ for all $u, w \in V$ , and it intertwines the canonical representation with the reflection representation generated by the $α_{s}$ : $φ (ρ (s) v) = r_{α_{s}} (φ (v))$ for all $s \in S$ , $v \in V$ . Consequently the labelled Coxeter diagram fixes $B$ and the normalized products $β (α_{s}, α_{t}) / (∣ α_{s} ∣ ∣ α_{t} ∣)$ , but neither the root lengths $∣ α_{s} ∣$ nor the individual numbers $β (α_{s}, α_{t})$ ; a supplied root-system treatment with its own root-length normalization, coroot system and lattice data matches this canonical form exactly under the displayed normalization, and such a match is a hypothesis on the supplier, not a consequence of the Coxeter matrix. No root system, coroot system, Cartan matrix or lattice is constructed or identified here.

(4) The reflection-faithfulness boundary for Soergel applications. (a) For a field $k$ , a realization of $(W, S)$ in the sense of Elias–Williamson is a free finite-rank $k$ -module $h$ with subsets ${α_{s}^{\lor}} \subset h$ , ${α_{s}} \subset h^{*}$ satisfying $⟨ α_{s}^{\lor}, α_{s} ⟩ = 2$ , the reflection assignment $v \mapsto v - ⟨ v, α_{s} ⟩ α_{s}^{\lor}$ defining a representation of $W$ , and the following condition on each distinct pair with finite $m = m (s, t)$ . Put $x : = ⟨ α_{s}^{\lor}, α_{t} ⟩$ and $y : = ⟨ α_{t}^{\lor}, α_{s} ⟩$ , and define $p_{0} = q_{0} = 0$ , $p_{1} = q_{1} = 1$ , $p_{j + 1} = x q_{j} - p_{j - 1}$ and $q_{j + 1} = y p_{j} - q_{j - 1}$ for $j \geq 1$ . Require $p_{m} = q_{m} = 0$ (the two-colored quantum-number condition (3.3), with the recursion of Definition 3.5 of the source). An infinite label imposes no such condition. Such a realization is faithful when the action of $W$ on $h$ is faithful, and reflection faithful when it is faithful and the assignment sending each reflection of $W$ (each conjugate of a simple reflection) to its full fixed space $Fix (w) : = {v : w (v) = v}$ is a bijection onto ${Fix (w) : w \in W, codim Fix (w) = 1}$ (the source's Definition 3.8). In the canonical case $k = R$ , $h = ⨁_{s} R α_{s}^{\lor}$ and $⟨ α_{t}^{\lor}, α_{s} ⟩ = - 2 \cos (π / m (s, t))$ , with the source's convention $π / \infty : = 0$ under which the value for $m (s, t) = \infty$ is $- 2$ . (b) Every element of $W$ fixes the radical $rad (B) = {v \in V : B (v, e_{s}) = 0 \forall s \in S}$ pointwise: $ρ$ preserves $B$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (2)) and $r_{a} (v) = v$ whenever $B (v, a) = 0$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)), so each generator $r_{e_{s}}$ and hence $ρ (W)$ fixes $rad (B)$ pointwise. (c) Suppose $\dim rad (B) = \dim V - 1 \geq 1$ . Then $rad (B)$ is a hyperplane, and for every root $α \in Φ$ the reflection $r_{α}$ has fixed hyperplane $\ker B (-, α)$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2), Descent of the reflection representation, unit root norms, and conjugation of reflections (4)), which contains $rad (B)$ and therefore equals it; moreover the simple reflections $ρ (s) \neq ρ (t)$ ( $s \neq t$ ) are distinct reflections with the same fixed hyperplane. Hence the correspondence "reflection $\mapsto$ its fixed hyperplane" fails to be injective, and the canonical realization is not reflection faithful in the sense of Elias–Williamson even though $ρ$ is faithful (The root-length criterion and faithfulness of the canonical reflection representation (3)). (d) Therefore a faithful canonical representation may not be substituted for a reflection-faithful realization in an argument that assumes reflection faithfulness. The source records Soergel's construction of a reflection-faithful representation for every Coxeter group over $R$ (Example 3.2(2) and the discussion after Definition 3.8). Reflection faithfulness is a sufficient hypothesis for the classical Soergel theory over an infinite field of characteristic different from $2$ , but is not necessary for every Soergel application: the same discussion records Libedinsky's extension to the geometric realization and defines a Soergel realization as a faithful realization to which Soergel's techniques apply. That property is not asserted to be equivalent to reflection faithfulness. A concrete rank-two instance is computed on the companion page. (e) No Kazhdan–Lusztig canonical basis, bar-invariant basis, cell theory, Soergel bimodule or positivity statement is constructed here; those use the standard basis and bar involution of The standard basis of the generic Hecke algebra and base change and The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization together with the conversions of (2), and remain in their designated proof homes.

Facts & Assumptions

Given: A finite Coxeter matrix $(S, m)$ , the group $W$ with length $ℓ$ , the generic Hecke algebra $H$ with its generators $T_{s}$ , basis $(T_{w})$ and parameters $v_{s}$ , and the canonical data $V = R^{S}$ , $B$ , $ρ$ , $Φ$ of the items named in the statement; in (3), a symmetric bilinear form $β$ on a real vector space $E$ with vectors $α_{s}$ satisfying the stated hypotheses.

[F1]

A bilinear form on $V$ over $F$ is a function $B : V \times V \to F$ that is linear in each variable separately. (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms)

[F2]

The left and right radicals of a bilinear form $B$ are ${rad}_{L} (B) = {u : B (u, v) = 0 for every v}$ and ${rad}_{R} (B) = {v : B (u, v) = 0 for every u}$ , and in the symmetric case they coincide. (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space)

[F3]

A function $T : V \to W$ between vector spaces over one field is linear when $T (a u + b v) = a T (u) + b T (v)$ for all $a, b \in F$ , $u, v \in V$ ; a linear map vanishing on a spanning set vanishes identically, and a linear map sending a basis to linearly independent vectors is injective. (Linear map between vector spaces over the same field, Linear subspace of a vector space, Kernel and image of a linear map)

[F4]

A linear map has trivial kernel exactly when it is injective. (Kernel and image of a linear map)

[F5]

An $R$ -algebra homomorphism $f : A \to B$ is a unital ring homomorphism satisfying $f \circ η_{A} = η_{B}$ , and such a map is automatically $R$ -linear; two such homomorphisms agreeing on a set of generators agree everywhere. (Algebras over a commutative ring, central structure maps, and algebra homomorphisms)

Proof

1.1givenconstructalgebra

(1) The Hecke generators $T_{s}$ satisfy the braid relations $T_{s} T_{t} T_{s} \dots = T_{t} T_{s} T_{t} \dots$ of the presentation of $H$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy), and these are exactly the braid-pair identities of the Artin monoid, so the monoid universal property (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares (1)) applied to $f (s) : = T_{s}$ , with target the multiplicative monoid of $H$ , gives a unique monoid homomorphism $Θ : A^{+} \to H$ with $Θ (σ_{s}) = T_{s}$ . For a reduced expression $w = s_{1} \dots s_{k}$ one has $Θ (b_{w}) = Θ (σ_{s_{1}} \dots σ_{s_{k}}) = Θ (σ_{s_{1}}) \dots Θ (σ_{s_{k}}) = T_{s_{1}} \dots T_{s_{k}} = T_{w}$ , the last equality and the independence of $T_{w}$ of the reduced expression being the defining properties of the standard basis (Reduced-word independence of T_w and the length-multiplication rules (1)-(2), whose proof consumes Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups). The $R$ -basis property of $(T_{w})$ and the base-change compatibility of the specialised elements are clauses (2) and (4) of The standard basis of the generic Hecke algebra and base change, quoted here as the interface they describe.

1.2algebra

(2) All four quadratic relations are substitutions in the normalized relation (i), which is part of the presentation of $H$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy). (ii) With $S_{s} = v_{s} T_{s}$ and $Q_{s} = v_{s}^{2}$ : $S_{s}^{2} = v_{s}^{2} T_{s}^{2} = v_{s}^{2} ((v_{s} - v_{s}^{- 1}) T_{s} + 1) = (v_{s}^{2} - 1) v_{s} T_{s} + v_{s}^{2} = (Q_{s} - 1) S_{s} + Q_{s}$ , which is equivalent to $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ . (iii) With $H_{s} = - T_{s}$ : $H_{s}^{2} = T_{s}^{2} = (v_{s} - v_{s}^{- 1}) T_{s} + 1 = - (v_{s} - v_{s}^{- 1}) H_{s} + 1 = (v_{s}^{- 1} - v_{s}) H_{s} + 1$ . (iv) With $U = T_{s}^{E W} = - v_{s}^{- 1} T_{s}$ : $U^{2} = v_{s}^{- 2} T_{s}^{2} = v_{s}^{- 2} ((v_{s} - v_{s}^{- 1}) T_{s} + 1) = v_{s}^{- 2} (- (v_{s} - v_{s}^{- 1}) v_{s} U + 1) = (v_{s}^{- 2} - 1) U + v_{s}^{- 2}$ , which is $(U + 1) (U - v_{s}^{- 2}) = 0$ . The relation (ii) is also recorded as part 4 of The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization.

1.3F1F3algebra

(3) The map $φ$ is linear by construction on the basis $(e_{s})$ of $V$ (The real Coxeter form, its radical, reflections, and form-preserving maps), and it is injective with image $span {α_{s}}$ because the vectors $α_{s}$ are linearly independent by hypothesis [F3]. Fix $s$ and $t \neq s$ , put $c : = c (s, t)$ (so $c = \cos (π / m (s, t))$ for finite $m (s, t)$ and $c = 1$ when $m (s, t) = \infty$ ), and recall $B (e_{s}, e_{t}) = - c$ and $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ (The real Coxeter form, its radical, reflections, and form-preserving maps (2),(3)); then the canonical representation satisfies $ρ (s) e_{t} = r_{e_{s}} (e_{t}) = e_{t} - 2 B (e_{t}, e_{s}) e_{s} = e_{t} + 2 c e_{s}$ for finite and infinite $m (s, t)$ alike, so $φ (ρ (s) e_{t}) = \frac{α_{t}}{∣ α_{t} ∣} + 2 c \frac{α_{s}}{∣ α_{s} ∣}$ , while the reflection $r_{α_{s}}$ of the hypothesis gives $r_{α_{s}} (φ (e_{t})) = \frac{α_{t}}{∣ α_{t} ∣} - \frac{2 β (α_{t}, α_{s})}{∣ α_{t} ∣ β (α_{s}, α_{s})} α_{s} = \frac{α_{t}}{∣ α_{t} ∣} + 2 c \frac{α_{s}}{∣ α_{s} ∣}$ by the normalized-product hypothesis; for $t = s$ both sides equal $- \frac{α_{s}}{∣ α_{s} ∣}$ because $ρ (s) e_{s} = - e_{s}$ and $r_{α_{s}} (α_{s}) = - α_{s}$ . Since the $e_{s}$ form a basis and both sides are linear, the intertwining identity $φ (ρ (s) v) = r_{α_{s}} (φ (v))$ holds for all $v \in V$ [F3]; the hypothesis that the assignment defines a representation of $W$ is what makes the reflection representation $s \mapsto r_{α_{s}}$ well defined on $W$ . Finally $β (φ e_{s}, φ e_{t}) = \frac{β (α_{s}, α_{t})}{∣ α_{s} ∣ ∣ α_{t} ∣} = B (e_{s}, e_{t})$ for all $s, t$ — the case $s \neq t$ is the hypothesis and the case $s = t$ gives $1 = B (e_{s}, e_{s})$ — so by bilinearity $β (φ u, φ w) = B (u, w)$ for all $u, w \in V$ [F1].

1.4F2given

(4)(b) Let $v \in rad (B)$ , so $B (v, e_{s}) = 0$ for every $s \in S$ . The canonical generators are the reflections $ρ (s) = r_{e_{s}}$ , and $r_{a} (u) = u$ whenever $B (u, a) = 0$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)); hence $ρ (s) v = v$ for every generator $s$ , so every element of $W$ , being a composite of generators and their inverses, fixes $v$ ; thus $rad (B)$ is fixed pointwise by $ρ (W)$ .

1.5givenalgebra

The canonical data form a realization as defined in (4)(a): take $h = V$ , $α_{s}^{\lor} = e_{s}$ and $α_{s} = 2 B (-, e_{s})$ . The pairing on the diagonal is $2$ , and the reflection formula is $ρ (s) v = v - α_{s} (v) e_{s}$ , so it defines the supplied representation of $W$ . For a finite label $m = m (s, t)$ put $θ = π / m$ ; then $x = y = - 2 \cos θ$ . The recursions defining $p_{j}, q_{j}$ give $p_{j} = q_{j} = (- 1)^{j - 1} \sin (j θ) / \sin θ$ for $j \geq 1$ : the cases $j = 1, 2$ and the induction follow from $2 \cos θ \sin (j θ) = \sin ((j + 1) θ) + \sin ((j - 1) θ)$ (The addition formulas for sine and cosine). The denominator is nonzero because $0 < θ \leq π / 2$ (The zero sets of sine and cosine and the least positive common period 2 pi), and $\sin (m θ) = \sin π = 0$ (Quarter-turn values and shifts by pi/2 and pi), proving $p_{m} = q_{m} = 0$ . Infinite labels require no check.

2.1F5step 1.2

(2) The four presentations. For each of the three displayed substitutions let $H^{∙}$ denote the $R$ -algebra presented by generators $X_{s}$ subject to the braid relations and the $∙$ -quadratic relation of step 1.2. The assignment $X_{s} \mapsto$ the corresponding element of $H$ respects the $∙$ -relation by step 1.2 and the braid relations: for even $m$ the two alternating words contain each of the two letters exactly $m / 2$ times, and for odd $m$ the two letters are joined by an odd-labelled edge, so $v_{s} = v_{t}$ by the parameter convention of the Hecke presentation; hence the products of the scaling constants $v_{s_{i}}$ , $- 1$ or $v_{s_{i}}^{- 1}$ attached to their letters coincide, and the assignment extends to an $R$ -algebra homomorphism $H^{∙} \to H$ by the presentation's universal property and [F5]. The inverse assignment $T_{s} \mapsto$ the corresponding element of $H^{∙}$ is well defined by the same computation with the roles reversed, since the displayed inverse formulas have the same shape; for instance $T_{s} = v_{s}^{- 1} S_{s}$ satisfies $(v_{s}^{- 1} S_{s})^{2} = v_{s}^{- 2} ((Q_{s} - 1) S_{s} + Q_{s}) = (v_{s} - v_{s}^{- 1}) (v_{s}^{- 1} S_{s}) + 1$ , which is the $T$ -relation. The two assignments are inverse on generators, so they are mutually inverse $R$ -algebra isomorphisms by the uniqueness clause of [F5].

2.2F2F3F4step 1.4

(4)(c) Assume $\dim rad (B) = \dim V - 1 \geq 1$ . Then $rad (B)$ is a hyperplane. For a root $α \in Φ$ the reflection $r_{α}$ is defined with $B (α, α) = 1$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (3)) and its fixed space is the kernel of the nonzero linear functional $B (-, α)$ , a hyperplane (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)) [F3]; as $rad (B)$ is fixed pointwise by step 1.4 and contained in that kernel, the fixed hyperplane of $r_{α}$ is exactly $rad (B)$ . If $s \neq t$ then $ρ (s) \neq ρ (t)$ : otherwise $B (v, e_{s}) e_{s} = B (v, e_{t}) e_{t}$ for all $v \in V$ , and evaluating at $v = e_{s}$ gives $e_{s} = B (e_{s}, e_{t}) e_{t}$ , contradicting the linear independence of $e_{s}, e_{t}$ . So two distinct reflections share the same fixed hyperplane and the correspondence "reflection $\mapsto$ its fixed hyperplane" is not injective, although $ρ$ is faithful with trivial kernel (The root-length criterion and faithfulness of the canonical reflection representation (3)) [F4].

2.3step 1.1step 1.2algebra

For the single-parameter coefficient conversion, a reduced word $w = s_{1} \dots s_{k}$ gives $H_{w} = H_{s_{1}} \dots H_{s_{k}} = (- 1)^{k} T_{w}$ and $T_{w}^{K L} = T_{s_{1}}^{K L} \dots T_{s_{k}}^{K L} = v^{- k} H_{w}$ by the substitutions in step 1.2; their independence of the reduced word follows from that of $T_{w}$ , since $k = ℓ (w)$ . Their scaling factors are units, so both families are bases by the standard-basis property in step 1.1. Substituting $T_{y}^{K L} = v^{- ℓ (y)} H_{y}$ in the given finite expansion gives $C_{x} = \sum_{y} v^{ℓ (x) - ℓ (y)} P_{y, x} (v^{- 2}) H_{y}$ ; uniqueness of coefficients in the $H$ -basis proves the stated formula. The original-normalization parameter is $q = v^{- 2}$ because the generator $T_{s}^{K L}$ satisfies $(T_{s}^{K L} + 1) (T_{s}^{K L} - q) = 0$ by step 1.2. This is a change-of-basis computation for any supplied finite expansion, not an existence proof for the canonical basis or its polynomials.

3.1given∎

Scope and choice: clauses (4)(d) and (4)(e) record the interface and its boundary — the fact that a reflection-faithful realization is additional data not supplied by the Coxeter matrix, and the abstention from every Kazhdan–Lusztig, cell-theoretic, Soergel-bimodule and positivity construction, which belong to their designated proof homes. No such object is constructed here, and no choice is used: all four normalizations are explicit substitutions with constants in $R$ or $R$ , and the radical computation uses only the displayed defining formulas.

5 · Examples, counterexamples and false statements

None yet.

Coxeter, Artin, and Hecke Interfaces

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Definition

Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares

Statement

Facts & Assumptions

Proof

The reduced positive section b_w, its length additivity, and the degree homomorphism

Statement

Facts & Assumptions

Proof

Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources