Alphabeta MathArticle items · 09 — Simple wall-crossing elements and maps
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 9–10.
1 · Prerequisites
- Connected component notation [I,a]
- Longest parabolic elements and opposition permutation
- Algebraic chamber-label set
- Root signs, inversions and descent translation
- D4 diagram and generator conventions
- The twelve D4 positive roots
- Tits cone intersections, relative roots and arrangements
- Annihilators, relative hyperplanes and complexification
- Real Coxeter representation and contragredient notation
- Reflecting hyperplanes, fundamental chamber and Tits cone
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Connected components, quasicomponents, and totally disconnected spaces
- Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
- Linear hyperplane arrangements and local finiteness
- Tits cone geometry with its exact interior boundary
- Only finitely many hyperplanes separate two Tits-cone points
- Coordinate vertices inside a relative open ball
- A hyperplane near the centre separates opposite coordinate vertices
- Coxeter-system, roots, length and weak-order conventions
- Standard parabolic subgroups and parabolic roots
- Parabolic root subsystems and unique coset minima
- Nonnegative inverse matrices preserve extremal basis rays
- Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
- Normalisers and their quotients
- Artin–Tits group, pure subgroup and positive lifts
- Normalizer quotients, kernels and conditional surjectivity
- Reduced-word lifts and the prefix-order convention
- Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups
- Descent reduction, minimum-length elements, and the additive factorization in a double coset
- The parabolic intersection W_I∩ dW_Jd⁻¹ for d∈^IW^J
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Positive Artin and parabolic monoids
- Independent spherical Artin Garside input
- Positive degree and atoms of the presented Artin monoid
- Reduced Coxeter lifts in the presented positive monoid
- Identifying the presented monoid with its positive image
- Order embedding and order isomorphism
- Free actions and quotient groupoids
- Groupoid coverings, universal and normal covers
- Free object actions give orbit coverings
- Parabolic coordinate subspaces and fundamental faces
- Chambers, walls and adjacency
- Subset and vertex-complement notation
- Independent relative chamber labelling input
- The full-cone and interior arrangements have identical chambers
- Chamber collisions, point stabilizers, and the intersection rule
- Covering-space actions and fundamental-groupoid quotients
- Covering-space actions by disjoint translates of neighbourhoods
- Fundamental groupoid of a space
- Semilocally simply connected spaces with explicit basepoint convention
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Connected small groupoids and vertex groups
- Independent groupoid covering and universal-cover input
- Vertex-group exact sequence of a normal groupoid cover
- Set-sized groupoids, functors and composition conversion
- Braid congruence preserves word length and support
- Rigidity of two-letter words below braid length
- Exactly two words for a two-generator braid element
- The two-generator braid element is the lcm
- The empty-object ribbon component and its braid relations
- Opposition as conjugation by a longest Artin lift
- Elementary ribbons have explicitly determined endpoints
- The reduced ribbon subgroupoid uses exactly the exterior atoms
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Example 3.14 (printed p. 15); §4.2 opening notation and Theorem 4.14 relation (3) (printed pp. 20–22); Corollary 4.18 (printed pp. 23–24); Proposition 4.19 (printed p. 24)
- Irreducible rank-three reflection-arrangement counts
- Relative complexified hyperplane complement
- Locally finite arrangements and relative face terminology
- Sign cells and local finite-arrangement reductions
- Finite galleries and separating hyperplanes
- Brink–Howlett groupoid
- The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- The root-length criterion and faithfulness of the canonical reflection representation
- Associates of a parabolic subset
- Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order
- Stereographic projection and its inverse are explicit in coordinates
- Generic slicing identifies flop stability walls with restricted ADE walls
- Basic Tits-cone properties
- Cartan matrix of a based root system
- Dynkin diagram with edge multiplicity and arrow convention
- Finite Weyl positive roots and simple reflections
- The Axiom of Choice
- Outside parabolic roots form an invariant set
- Connected propagation spans the component
- The symmetric group Sym(X): the bijections of a set X under composition
- Core and atomic double cosets
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- CW complex with closure finiteness and weak topology
- Finite parabolic longest elements and disconnected factors
- Stabilizer of a fundamental parabolic face
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
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
DefinitionDependency level 9
Simple wall-crossing elements and maps
ExampleDependency level 9
Seven relative D4 hyperplanes and thirty-two chambers
ExampleDependency level 9
Failure of local finiteness on the full cone
LemmaDependency level 9
Finite pair separation yields a finite relative ball
LemmaDependency level 9
Root bases and nonnegative matrices for chamber labels
LemmaDependency level 9
Normaliser quotient exact sequence
PropositionDependency level 9
Divisibility of positive lifts and weak order
PropositionDependency level 9
Quotient groupoids give normal covers
PropositionDependency level 9
Convexity, stratification and faces of the relative cone
PropositionDependency level 9
Fundamental groupoids of free topological quotients
RemarkDependency level 9
Distinct roots can define the same relative hyperplane
RemarkDependency level 9
Existence of universal connected groupoid covers
TheoremDependency level 9
Reduced-ribbon lcm presentation including the empty subset
CounterexampleDependency level 10
A relative arrangement need not be a reflection arrangement
DefinitionDependency level 10
Chamber open sets used for the topological equivalence
DefinitionDependency level 10
Root-permuting normaliser complement
DefinitionDependency level 10
Ribbon groupoid in spherical type
ExampleDependency level 10
Spherical and stereographic pictures of the D4 arrangement
LemmaDependency level 10
Common relative facets have one enlarged standard subset
LemmaDependency level 10
The flop restriction complement is the finite ADE relative complement
LemmaDependency level 10
Finite outside roots give a finite complementary parabolic
LemmaDependency level 10
Shrink the finite relative ball to a central arrangement
LemmaDependency level 10
Minimal representatives identify core cosets with root-transport arrows
LemmaDependency level 10
The seven D4 planes give a finite spherical cellulation
LemmaDependency level 10
Simplified wall crossing for a finite parabolic
LemmaDependency level 10
Faithful normaliser-quotient action on the relative cone
LemmaDependency level 10