Alphabeta MathCoxeter geometry and relative chambers
Connect Coxeter representations, root signs, parabolic minima, opposition and weak order with relative arrangements, chamber labels, galleries and wall crossings.
1 · Prerequisites
- Locally finite arrangements and relative face terminology
- A convex subset of ℝᵐ contains every line segment between two of its points
- Open cover, subcover, compact metric space, and compact subset of a metric space
- Linear hyperplane
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- A continuous image of a connected space is connected, and connectedness is a topological property
- Annihilators, relative hyperplanes and complexification
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Invertible linear maps, linear isomorphisms, and inverse linear maps
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Tits cone geometry with its exact interior boundary
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Definition 1.6(1), printed p. 6 (PDF p. 30)
- Connected components, quasicomponents, and totally disconnected spaces
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Sentence following Definition 1.6, printed p. 6 (PDF p. 30)
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications (author-hosted draft) · §§1.1, 1.3–1.4; Definitions 1.4–1.6 and the sentence immediately after Definition 1.6; printed pp. 3–6, especially p. 6 (PDF pp. 27–30, especially p. 30).
- Parabolic root subsystems and unique coset minima
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Definition 1.8 and Theorem 1.12, printed pp. 6–7 (PDF pp. 30–31)
- Finite parabolic longest elements and disconnected factors
- 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)
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Proposition
Finite galleries and separating hyperplanes
Lemma
Coordinate faces span their annihilator subspaces
Lemma
The relative root deletion set is the restricted-hyperplane union
Lemma
Full-dimensional J-chambers lie in the relative interior
Lemma
Deleting relative root hyperplanes removes all cone-boundary points
Lemma
The full-cone and interior arrangements have identical chambers
Theorem
Independent relative chamber labelling input
Theorem