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Wall-crossing notation and Brink–Howlett convention

PDF p. 11: Remarks 3.18–3.19

Statement

Use the simple wall-crossing convention $v_{a, I} = w_{D ∖ {a}} w_{D}$ , where $D = [I, a]$ is the connected component containing $a$ in the diagram on $I \cup {a}$ , $a \notin I$ , and $W_{D}$ is finite.

The article's Remarks 3.18–3.19 say that $ω_{a, I}$ may be abbreviated to $ω_{a}$ when the source is clear, following its reference [24, Definition 1.28], and report that $v_{a, I}$ is the inverse of the element denoted $v [a, I]$ by Brink–Howlett, whose Section 2 also treats $v [I, J]$ for certain disjoint subsets. This sentence records the original authors' attribution; the 1999 original has not been read here.

Independently, the general Coxeter-system convention of Dyer–Gimenez gives $ν (a, I) = w_{D} w_{D ∖ {a}} = v_{a, I}^{- 1} .$ Thus the inverse product comparison holds whenever the connected support $D$ is spherical, without assuming that $W$ , $W_{I}$ , or $W_{I \cup {a}}$ is finite. This verified comparison uses Dyer–Gimenez's symbol $ν (a, I)$ ; it does not claim direct verification of the historical symbol $v [a, I]$ .

Statement references

Simple wall-crossing elements and maps

External input

Dyer–Gimenez, Lemma 1.25, uses subsets of the simple-root set $Π$ , identified with vertices by $i \mapsto α_{i}$ . Given $J \subseteq K \subseteq Π$ , suppose that finitely many components of $K$ meet $K ∖ J$ and each such component is spherical. If $L$ is their union, the stated external formula is $ν (K ∖ J, J) = w_{L} w_{L \cap J}$ . Section 1.27 abbreviates a singleton first argument, and §1.4 permits infinite rank. Only this product formula and singleton notation are imported; their proof is not reconstructed here.

Verification

Given: A Coxeter system, a vertex subset $I$ , and $a \notin I$ with finite $W_{D}$ , where $D = [I, a]$ .

Goal: Verify the independent inverse product comparison, keeping the article's historical attribution separate.

[F1] Simple wall-crossing elements and maps defines the connected support $D$ , the longest elements of its finite standard parabolics, and $v_{a, I} = w_{D ∖ {a}} w_{D}$ .

[F2] The external formula above, from Dyer–Gimenez Lemma 1.25 and §1.27, defines $ν$ by the displayed product when the components meeting the first argument are spherical. This is an independent external input; the original article is not its proof source.

[F3] Longest parabolic elements and opposition permutation supplies the unique maximum-length element of every finite standard parabolic, with length measured in the ambient Coxeter system. Its Justification obtains uniqueness from the finite-opposition interface after identifying the finite parabolic Coxeter system and its length, so no finite-rank assumption on the ambient system is needed.

1.1F1F2given

In [F2] take $J = I$ and $K = I \cup {a}$ , identifying vertices and simple roots. Exactly one connected component of $K$ meets $K ∖ J = {a}$ : it is $D$ . By the given finiteness of $W_{D}$ , this component is spherical, so [F2] applies with $L = D$ and $L \cap J = D ∖ {a}$ . Hence $ν (a, I) = w_{D} w_{D ∖ {a}}$ . No condition on the other components of $I$ enters this application.

2.1F1F3step 1.1algebra∎

A longest element $w_{L}$ of a finite standard parabolic is an involution: reversing a shortest word preserves its length, so $ℓ (w_{L}^{- 1}) = ℓ (w_{L})$ , and uniqueness of the maximum-length element supplied by [F3] gives $w_{L}^{- 1} = w_{L}$ . Apply this to $L = D$ and $L = D ∖ {a}$ ; both parabolics are finite by [F1]. Taking the inverse of the product in [F1] now gives $v_{a, I}^{- 1} = w_{D} w_{D ∖ {a}} = ν (a, I)$ , as asserted.

Source notes

The article's $v [a, I]$ and $v [I, J]$ comparison is retained solely as its explicit historical attribution. The independently read Dyer–Gimenez source supports the product convention in general Coxeter type and even infinite rank, but writes $ν$ rather than the Latin $v$ of that attribution. The 1999 Brink–Howlett full text was inaccessible and was not read. The imported lemma's proof was not read either; the Verification checks the applicability of its stated formula and the inverse calculation only. No conclusion about an error in the original article follows from this reading limit.

Depends on

Dependency tree · 2 levels

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External dependencies

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.4, PDF p.4; §1.24–§1.27, PDF pp.9–10, especially Lemma 1.25 and Remark 1.26

For a Coxeter system of possibly infinite rank, and simple-root subsets J subset K, the reflections in W_K minus W_J are finite exactly when finitely many connected components of K meet K minus J and all these components are spherical. If L is their union, nu(K minus J,J)=w_L w_(L intersect J). The singleton case K minus J={alpha} is written nu(alpha,J) and gives the Brink–Howlett generators.

Hypotheses: Any Coxeter system with a based root system; J subset K subset Pi; finitely many connected components of K meet K minus J and each has finite Coxeter group. Applied only with J=I, K=I union {a}, and the one component D=[I,a] spherical. Neither W nor W_I is assumed finite.

Reading limits: Read extracted text of §1.4 and §§1.24–1.27 and visually inspected PDF p.10. The proof of Lemma 1.25 and the remainder of the paper were not read. Its stated product formula is an external input, not a locally reconstructed theorem. This source uses Greek nu with parentheses, and does not certify the literal notation in the unread 1999 original. The rank-two paragraph of §1.27 is not imported.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 11, Remarks 3.18–3.19
Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.4, PDF p.4; §1.24–§1.27, PDF pp.9–10, especially Lemma 1.25 and Remark 1.26