Alphabeta MathArticle items · 06 — Tits cone geometry with its exact interior boundary
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 0–2.
1 · Prerequisites
- The Tits cone, its interior, and the negative-root set of a functional
- The dual action, chambers, faces, and root hyperplanes
- The finite-negativity criterion, the reduction step, and convexity of the Tits cone
- The interior of the Tits cone, finite parabolic stabilizers, and local finiteness
- Chamber collisions, point stabilizers, and the intersection rule
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- The right and left weak orders, intervals, covers, and meets and joins of subsets
- Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics
- Locally finite arrangements and relative face terminology
- The tensor and direct-sum models of complexification are canonically complex-linearly isomorphic
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Covering-space actions by disjoint translates of neighbourhoods
- Complexification as ℂ⊗_ℝV with its canonical real-linear embedding
- The direct-sum model V⊕ iV of a complexification
- Left group actions, transitive actions, and faithful actions
- Paths, path-connected spaces and path components
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- Real Coxeter representation and contragredient notation
- Divisibility and atomicity in a category
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Natural transformation and its components
- Lattices, distributive lattices, and order ideals
- Semigroup and monoid
- Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why CAT is not formed
- Covariant functor, identity functor, composite functor, and contravariant functor
- Garside categories, periodic loops and cyclic sets · §2, basic triple and Definitions 2.2–2.5, footnote 3; printed pp. 9–10
- Set-sized groupoids, functors and composition conversion
- Linear hyperplane
- Linear subspace of a vector space
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Vector space over a field
- The quotient vector space V/W and its canonical projection
- Finite directed paths form the free category on a quiver
- Equivalence relation, equivalence class, and the quotient set A/∼
- 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
- Real and complex inner product spaces, with the inner product linear in the first argument
- Orthonormal families, complete orthonormal systems and Hilbert bases
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Positive degree and atoms of the presented Artin monoid
- Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- Reduced-word lifts and the prefix-order convention
- Complete-local smooth threefold flops and their slice data
- Finite Weyl root system, lattice and chamber conventions
- Wemyss, Flops and clusters in the homological minimal model programme · §1.6, printed p.9; Setup 2.9, p.12; Theorem 5.24(1)--(3), pp.39--40; Corollary 5.25(1)--(3), p.41
- Wemyss, A lockdown survey on cDV singularities · §3.1, printed pp.13--14; §3.3, printed p.15
- Hirano–Wemyss, Faithful actions from hyperplane arrangements · §1.3, printed p.3; Setup 4.1, pp.8--9; §4.1, p.9, before Example 4.2
- Greuel–Pham, Finite determinacy of matrices and ideals · Definition 4.1, printed p.11; Theorem 4.6(2)(i),(iii), printed p.12 (PDF pages 11--12)
- 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
- 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
- Root sign coherence and the action of simple reflections on positive roots
- The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange
- The root-length criterion and faithfulness of the canonical reflection representation
- Descent of the reflection representation, unit root norms, and conjugation of reflections
- The dual action, the faces, and the rank-two chamber tiling
- The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1}
- Linear functionals and the algebraic dual V^*=L(V,F)
- 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
- The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected
- Fundamental groupoid of a space
- Existence and uniqueness of path lifts through a covering map
- Existence and uniqueness of homotopy lifts through a covering map
- A covering map induces an injective homomorphism on fundamental groups
- Semilocally simply connected spaces with explicit basepoint convention
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Deck transformations and the deck-transformation group of a covering
- Based loops and the fundamental group
- Universal covering spaces
- The homomorphism on fundamental groups induced by a pointed continuous map
- The kernel and image of a group homomorphism
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications · §1.1, printed p. 3 / PDF p. 27; §1.2 and §1.3, printed p. 4 / PDF p. 28; §1.3, printed p. 5 / PDF p. 29
- The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X
- The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}
- The unordered pair {x,y} and the singleton {x} = {x,x}
- Subset x ⊆ y, proper subset x ⊊ y, and the separation notation { z ∈ x : φ(z) }
- ⋃ ∅ = ∅, ⋃ {a} = a, ⋃ {a,b} = a ∪ b, ⋂ {a} = a, and ⋂ {a,b} = a ∩ b
- Roots of the infinite dihedral geometric representation
- Refinements, locally finite families, point-finite families, and star refinements
- Coset equality, well-defined quotient operations, and the canonical projection with kernel W
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Orthogonality and the orthogonal complement
- Internal direct sum V = ⨁_i<n Uᵢ: the sum is everything and each summand meets the sum of the others only in 0_V
- For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥
- The fixed-point sets Xᵍ and X^G of a group action
- The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- The Coxeter elements of the classical types Aₙ, Bₙ, Dₙ and I₂(m): characteristic polynomials, orders and spectral exponents from their reflection models
- The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices
- The Coxeter plane, ordered-root enumeration, and invertibility of the Coxeter action minus the identity
- Classification of finite Coxeter systems, including the H and dihedral families
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
PropositionDependency level 0
Tits cone geometry with its exact interior boundary
PropositionDependency level 0
Bounded weak-order joins and finite parabolic criteria
DefinitionDependency level 1
Complexified complements in an open convex cone
DefinitionDependency level 1
Coxeter-system, roots, length and weak-order conventions
DefinitionDependency level 1
Categorical Garside structures and simple arrows
DefinitionDependency level 1
Connected small groupoids and vertex groups
DefinitionDependency level 1
Linear hyperplane arrangements and local finiteness
DefinitionDependency level 1
Path presentations, congruences and formal inverses
LemmaDependency level 1
Coordinate faces span their annihilator subspaces
LemmaDependency level 1
Coordinate vertices inside a relative open ball
LemmaDependency level 1
Reduced Coxeter lifts in the presented positive monoid
PropositionDependency level 1
Generic slicing identifies flop stability walls with restricted ADE walls
PropositionDependency level 1
Finite galleries and separating hyperplanes
PropositionDependency level 1
Root signs, inversions and descent translation
PropositionDependency level 1
Sign cells and local finite-arrangement reductions
PropositionDependency level 1
Covering-space actions and fundamental-groupoid quotients
DefinitionDependency level 2
Chambers, walls and adjacency
DefinitionDependency level 2
Reflecting hyperplanes, fundamental chamber and Tits cone
DefinitionDependency level 2
D4 diagram and generator conventions
DefinitionDependency level 2
Groupoid completion and its universal property
DefinitionDependency level 2
Parabolic coordinate subspaces and fundamental faces
DefinitionDependency level 2
Subset and vertex-complement notation
LemmaDependency level 2
The infinite dihedral Tits cone is an open half-plane with its vertex
LemmaDependency level 2
A hyperplane near the centre separates opposite coordinate vertices
LemmaDependency level 2
Identifying the presented monoid with its positive image
LemmaDependency level 2
Invariant metric, product arrangements and essentiality
LemmaDependency level 2