Alphabeta MathArticle items · 11 — Coxeter normaliser semidirect product
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 13–17.
1 · Prerequisites
- Root-permuting normaliser complement
- Each normaliser coset has exactly one root-permuting representative
- Every normalising element factors through the root-permuting subgroup
- The root-permuting subgroup intersects the parabolic trivially
- Normalizer quotients, kernels and conditional surjectivity
- Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent
- Real Coxeter representation and contragredient notation
- Tits cone intersections, relative roots and arrangements
- Labelling theorem for relative chambers
- Parabolic root subsystems and unique coset minima
- The root-length criterion and faithfulness of the canonical reflection representation
- Component opposition transports the retained simple roots
- Algebraic chamber-label set
- Convexity, stratification and faces of the relative cone
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- Connected component notation [I,a]
- Longest parabolic elements and opposition permutation
- Chambers, walls and adjacency
- Finite relative longest element without spherical ambient rank
- Elementary inversions detect the two initial prefixes
- Standard expressions and their lengths
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · Lemma 3.12(a), PDF pp. 42–43; §1.5, PDF p. 4
- Atomic reduced compositions and standard products use the same order
- Core and atomic double cosets
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 4 opening and Notation 4.1, Definition 4.2, PDF p. 13; Proposition 4.3, Definition 4.4, Proposition 4.5 and Lemma 4.7, p. 14; Remark 4.8 (4.8), Proposition 4.9 (4.9)–(4.13), p. 15; Definition 4.10, pp. 15–16
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 — singular-expression conventions · Section 1.2, (1.3)–(1.4), PDF p. 3; Section 1.3, p. 4; Section 2.1, paragraph on unique Bruhat extrema of finitary double cosets, p. 8; Section 2.2, Definition 2.1 and (2.4)–(2.5), p. 9; Definitions 2.2–2.3 and Proposition 2.4, p. 10; Definition 3.9, p. 13
- Coxeter-system, roots, length and weak-order conventions
- Reflecting hyperplanes, fundamental chamber and Tits cone
- Simple wall-crossing elements and maps
- Separation by a hyperplane
- Root signs, inversions and descent translation
- Finite parabolic longest elements and disconnected factors
- The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- Reduced ribbon groupoid
- Garside divisibility and lcm assertions for A+
- Reduced-ribbon lcm presentation including the empty subset
- Artin–Tits group, pure subgroup and positive lifts
- Set-sized groupoids, functors and composition conversion
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons · §2.3, printed p. 4; §4.2 opening, printed p. 20; Example 3.14, printed p. 15
- Normalisers and their quotients
- Standard parabolic subgroups and parabolic roots
- Independent spherical ribbon splitting input
- Common relative facets have one enlarged standard subset
- Adjacent relative chambers force a finite crossing component
- Componentwise crossings are inverse elements
- Parabolic coordinate subspaces and fundamental faces
- Independent relative chamber labelling input
- Connected small groupoids and vertex groups
- Transfer to the root-action Brink–Howlett groupoid
- Parallel classes of roots and finite inversion sets
- Independent arbitrary-rank factorization and simple-class inputs
- Compressed inversion count is additive for Coxeter-reduced composites
- Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws
- Interior restriction preserves chambers and adjacency
- Faithful normaliser-quotient action on the relative cone
- Stabilizer of a fundamental parabolic face
- Local finiteness of the interior relative arrangement
- Basic Tits-cone properties
- Finite galleries and separating hyperplanes
- Brink–Howlett groupoid
- Root bases and nonnegative matrices for chamber labels
- Root-sign and inversion-length facts
- Relative complexified hyperplane complement
- Complexified complements in an open convex cone
- Covering-space actions and fundamental-groupoid quotients
- The finite-negativity criterion, the reduction step, and convexity of the Tits cone
- Groupoid completion and its universal property
- Path presentations, congruences and formal inverses
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Tits cone geometry with its exact interior boundary
- Deleting relative root hyperplanes removes all cone-boundary points
- D4 diagram and generator conventions
- Seven relative D4 hyperplanes and thirty-two chambers
- Standard expressions are minimum generator words
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.27, PDF pp. 10–11, rank-two relations paragraph
- Normaliser quotient exact sequence
- Eilenberg--Mac Lane space
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- The Salvetti order complex and its finite CW case
- Independent Salvetti homotopy-model input
- Universal covering spaces
- Nullhomotopic maps and contractible spaces
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications · §§1.1–1.3, printed pp.3–5 / PDF pp.27–29
- Luis Paris, K(pi,1) conjecture for Artin groups · §1, printed pp.363–366 and 368; §2.2, Theorem 2.5, p.379; §3.1, construction pp.385–386 and Theorem 3.1, p.386; §3.2, Theorem 3.3 and Corollary 3.4, pp.393–394; §3.3, cellular quotient construction p.398; §4.1, Proposition 4.2, p.401
- OpenAI, Harmonic heights and the Artin K(pi,1) conjecture · September 23, 2026; Theorem 1.1, PDF p.2; §§2.1–2.4, PDF pp.6–9; §9.4, PDF p.58
- MathVet, openai-math challenge status and fidelity table · HarmonicArtin row; corresponding family 254 row in fidelity-table.csv at the same commit
- Carlos Toledo, cert-machine openai-math kernel audit · HarmonicArtin entry, run38019899278 at 2026-10-10T14:00:12Z; top-level what and decider
- Complete-local smooth threefold flops and their slice data
- Generic slicing identifies flop stability walls with restricted ADE walls
- Chambers label smooth minimal models and their NCCR categories
- Natural isomorphism
- Based loops and the fundamental group
- Hirano–Wemyss, Faithful actions from hyperplane arrangements · Definition 2.7 and Theorem 2.8, printed p.5; Corollary 2.12 and its proof, p.6; Theorem 1.4(1), pp.2--3, and Theorem 6.9(1), pp.18--19; Definition 6.4, Theorem 6.5 and Corollary 6.6, pp.17--18; Setup 4.1 and Remark 4.4, pp.8--9
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
CorollaryDependency level 13
Coxeter normaliser semidirect product
ExampleDependency level 13
A3 relative arrangement and its six labelled chambers
LemmaDependency level 13
A finite connected component gives the opposite relative chamber
LemmaDependency level 13
Simultaneous elementary prefixes force the relative longest prefix
LemmaDependency level 13
Rotation parameters and the two switchback completions
LemmaDependency level 13
Length additivity and the base-chamber half-space
PropositionDependency level 13
Two ribbon-lcm expressions and presentation
RemarkDependency level 13
Minimal coset representatives remove chamber-label ambiguity
TheoremDependency level 13
Spherical Artin normaliser splitting
PropositionDependency level 14
Adjacent chambers and inverse wall-crossing elements
DefinitionDependency level 15
Chamber paths, concatenation and wall words
DefinitionDependency level 15
Universal wall groupoid on relative chambers
LemmaDependency level 15
Every standard expression counts the same parallel classes
LemmaDependency level 15
Finite wall-crossing connectivity
LemmaDependency level 15
Normaliser action on labels, orbits and adjacency
CorollaryDependency level 16
Brink–Howlett generators and invariance of standard length
DefinitionDependency level 16
Geodesic chamber paths
PropositionDependency level 16
Correspondence between chamber labels and Brink–Howlett morphisms
PropositionDependency level 16
Relative free action, normal cover and exact sequence
PropositionDependency level 16
The universal wall groupoid is generated by adjacent crossings
RemarkDependency level 16
The relative normaliser action need not be chamber-transitive
DefinitionDependency level 17
Positive arrangement category and its groupoid completion
ExampleDependency level 17
D4 chamber labels and complete adjacency graph
LemmaDependency level 17
Exactly two standard words in the finite corank-two residue
LemmaDependency level 17
Translation between relative arrangements in the geodesic argument
Open problemDependency level 17
K(pi,1) conjecture for arbitrary normaliser quotients
PropositionDependency level 17