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Independent finite local wall-crossing formula

Statement

Let $(W, S)$ be a finite-rank Coxeter system, with canonical real reflection representation, dual action and Tits cone $U$ , and fix $J \subseteq S$ . For $I \subseteq S$ , write $W_{I} = ⟨ I ⟩$ , put $E_{I} = {f : f (α_{i}) = 0 (i \in I)}$ , and let $C_{I}$ consist of the vectors in $E_{I}$ positive on the simple roots indexed by $S ∖ I$ . Set $E = E_{J}$ , with its Euclidean topology, $T = U \cap E$ and $T^{\circ} = {int}_{E} T$ . A $J$ -chamber label is a pair $(x, I)$ with $W_{J} x = x W_{I}$ such that $x$ has minimum Coxeter length in $x W_{I}$ ; it labels the $J$ -chamber $x C_{I}$ in the relative-interior arrangement. For a root $α$ put $H_{α} = {f : f (α) = 0}$ . The arrangement hyperplanes are the proper traces $E \cap H_{α}$ meeting $T^{\circ}$ . A wall is a nonempty intersection of a chamber's closure in $E$ with an arrangement hyperplane, of affine dimension $\dim E - 1$ .

Let $(x, I)$ be a $J$ -chamber label, let $a \in S ∖ I$ , and put $K = I \cup {a}$ . Suppose $W_{K}$ is finite. For a finite standard parabolic $W_{L}$ , write $w_{L}$ for its longest element and $ι_{L}$ for the permutation of $L$ determined by $w_{L} α_{s} = - α_{ι_{L} (s)}$ . Define $b = ι_{K} (a)$ and $(y, I^{'}) = (x w_{I} w_{K}, K ∖ {b})$ . Then $(y, I^{'})$ is a $J$ -chamber label, its chamber is adjacent to $x C_{I}$ through the common wall $\overline{x C_{K}}$ , and crossing back at $b$ returns $(x, I)$ . Here the bar denotes closure in $E$ , and $x C_{K}$ is the relative interior of that wall in its affine hull.

Call $J$ strongly Dynkin if $J$ is a proper subset of $S$ and $W_{J \cup {s}}$ is finite for every $s \in S ∖ J$ . If $J$ is strongly Dynkin, any two $J$ -chamber labels are connected by a finite sequence of the crossings just described.

Statement references

Independent relative chamber labelling input

Finite parabolic longest elements and disconnected factors

The full-cone and interior arrangements have identical chambers

Tits cone geometry with its exact interior boundary

Locally finite arrangements and relative face terminology

External input

Iyama–Wemyss, Definitions 1.8, 1.16 and 1.19 and Theorem 1.20(1)(a),(c),(d) and (2)(a), supplies the displayed crossing label, the inverse crossing, adjacency at the face $x C_{K}$ , and finite crossing connectivity under the strongly Dynkin hypothesis. The source's fixed and variable subsets are denoted $J$ and $I$ here. These assertions are imported; their source proof is not reproduced or independently reviewed. The local proof below identifies the closed wall and its relative interior under the stated portal conventions.

Facts & Assumptions

[F1]

The canonical representation, dual action, coordinate faces, minimum-length labels and their bijection with the relative-interior arrangement chambers are supplied by Independent relative chamber labelling input.

[F2]

Finite parabolics have the longest elements and opposition permutations stated above, including the empty-parabolic case, by Finite parabolic longest elements and disconnected factors.

[F3]

These chambers are the source $J$ -chambers $x C_{I} \subseteq E_{J}$ with $∣ I ∣ = ∣ J ∣$ , with the same Euclidean subspace topology. Their arrangement consists of proper root-hyperplane traces meeting $T^{\circ}$ . The full-cone and interior arrangements have identical chambers

[F4]

The Tits cone is the union of the translates of the closed fundamental chamber. The stabilizer of every $f \in C_{L}$ is $W_{L}$ , and $C_{L} \subseteq U^{\circ}$ when $W_{L}$ is finite. Tits cone geometry with its exact interior boundary

[F5]

Closures are in $E$ ; a wall is a nonempty chamber-closure intersection with an arrangement hyperplane having affine dimension $\dim E - 1$ . Distinct chambers sharing a wall are adjacent. Locally finite arrangements and relative face terminology

[F6]

The crossing-label, inverse-crossing, adjacency and strongly Dynkin connectivity assertions are the independent input recorded in External input.

Proof

Given: The system, label and subsets in the Statement, with $W_{K}$ finite for the local crossing. Goal: Establish the claimed crossing assertions, with the closed-wall convention.

1.1F1F2F3F5F6given

With the longest elements interpreted by [F2], apply [F6] to the label of [F1]. It gives the label $(y, I^{'})$ , the inverse crossing at $b$ , and distinct adjacent source chambers; it also gives the asserted finite crossing connectivity when $J$ is strongly Dynkin. By [F1] and [F3] the same labelled subsets are the relative-interior arrangement chambers. It remains to identify their common wall in the convention of [F5].

2.1F1F3step 1.1algebra

For any $L \subseteq S$ , let $D_{L} = {f \in E_{L} : f (α_{s}) \geq 0 (s \notin L)}$ . In dual coordinates, $C_{L}$ spans $E_{L}$ : the vector with coordinates $1$ off $L$ and $0$ on $L$ , and its sums with each dual basis vector off $L$ , lie in $C_{L}$ , and their differences give that basis. The inequalities defining $D_{L}$ are closed; adding $ε$ to each coordinate off $L$ shows every point of $D_{L}$ is a limit of points of $C_{L}$ . Thus $D_{L}$ is the closure of $C_{L}$ in the full dual space, and these assertions include $L = S$ . By [F3], $x C_{I} \subseteq E$ and $∣ I ∣ = ∣ J ∣$ , so spanning and equality of dimensions give $x E_{I} = E$ ; likewise $y E_{I^{'}} = E$ . Finite-dimensional linear automorphisms and their inverses are continuous, so closures commute with $x$ and $y$ . Consequently $\overline{x C_{I}} = x D_{I}$ and $\overline{y C_{I^{'}}} = y D_{I^{'}}$ . Put $H = x E_{K}$ . Since $K = I \cup {a}$ , the dual action gives $H = E \cap H_{x α_{a}}$ , a codimension-one subspace of $E$ .

3.1F4step 2.1algebra

Put $q = w_{I} w_{K} \in W_{K}$ , so $y = x q$ . By [F4], $W_{K}$ fixes every point of $C_{K}$ ; the spanning argument in step 2.1 shows that it fixes $E_{K}$ pointwise. Hence $y E_{K} = x E_{K} = H$ and $y D_{K} = x D_{K}$ . Since $K = I^{'} \cup {b}$ as well as $K = I \cup {a}$ , the coordinate descriptions give $x D_{I} \cap H = x D_{K} = y D_{K} = y D_{I^{'}} \cap H$ . Thus both chamber closures meet $H$ in $F = x D_{K} = \overline{x C_{K}}$ . The affine hull of $D_{K}$ is $E_{K}$ , because it contains $0$ and spans $E_{K}$ . In its free coordinates the relative interior of the nonnegative orthant $D_{K}$ is exactly the strictly positive orthant $C_{K}$ : positive coordinates persist in a sufficiently small ball, whereas a zero free coordinate can be perturbed negatively in every ball. If there are no free coordinates, both sets are ${0}$ in the zero space. Transporting by $x$ gives $aff (F) = H$ , $\dim F = \dim E - 1$ , and relative interior $x C_{K}$ .

4.1F1F3F4F5step 1.1step 2.1step 3.1∎

By [F4] and finiteness of $W_{K}$ , $C_{K} \subseteq U^{\circ}$ . Also $x U = U$ , since left multiplication by $x$ permutes the translates defining $U$ . As $x$ is a linear homeomorphism, $x C_{K} \subseteq U^{\circ}$ . Step 3.1 places this nonempty set in $E \cap H$ , so it lies in $T^{\circ}$ and $H$ meets $T^{\circ}$ . Since $x α_{a}$ is a root and $H$ is proper in $E$ , [F3] makes $H$ an arrangement hyperplane. Step 3.1 and [F5] now identify $F$ as a common wall of the two distinct chambers. Its relative interior and all remaining assertions were established in steps 3.1 and 1.1.

Source notes

The crossing and connectivity assertions are imported from the independent Iyama–Wemyss source. The coordinate argument is locally authored and adapts the wall terminology; it makes no claim to reconstruct that source theorem. The local hypothesis remains finiteness of the whole parabolic $W_{K}$ ; no componentwise weakening or weaker connectivity hypothesis is asserted.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Tits Cone Intersections and Applications · Definitions 1.8, 1.16 and 1.19 and Theorem 1.20(1)(a),(c),(d) and (2)(a), printed pp. 6, 8–9 (PDF pp. 30, 32–33)

For a chamber label (x,I) and a outside I with W_K finite, K=I union {a}, the simple crossing is (x w_I w_K, K minus {iota_K(a)}). This is a chamber label, crossing at iota_K(a) returns (x,I), and the chambers are adjacent at the face x C_K. If the fixed subset J is strongly Dynkin, any two labels are connected by finitely many simple crossings.

Hypotheses: A Coxeter graph and its canonical representation; J is the fixed subset and I is the variable subset in the chamber label. The local crossing requires W_{I union {a}} finite. Strongly Dynkin means J is proper and W_{J union {s}} is finite for every s outside J.

Reading limits: Checked the web-extracted text of Definitions 1.8, 1.16 and 1.19 and Theorem 1.20(1)(a),(c),(d),(2)(a) in this pass. PDF screenshots were unavailable; no visual reading or review of the source proof is claimed. The fixed and variable subsets are renamed J and I. The local proof below establishes the closed-wall and relative-interior identification under the portal conventions; it does not reconstruct the imported theorem.

Sources

Tits Cone Intersections and Applications · Definitions 1.8, 1.16 and 1.19 and Theorem 1.20(1)(a),(c),(d) and (2)(a), printed pp. 6, 8–9 (PDF pp. 30, 32–33)