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
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- The real Coxeter form, its radical, reflections, and form-preserving maps
- The canonical reflection homomorphism, roots, reflections, and the positive cone
- Descent of the reflection representation, unit root norms, and conjugation of reflections
- The dual action, chambers, faces, and root hyperplanes
- The dual action, the faces, and the rank-two chamber tiling
- Linear functionals and the algebraic dual V^*=L(V,F)
- Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups
- Root sign coherence and the action of simple reflections on positive roots
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- The root-length criterion and faithfulness of the canonical reflection representation
- The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Disconnected diagrams, direct products, and comparison of invariant forms
- Coxeter diagrams: edges, labels, components and finite type
- The Tits cone, its interior, and the negative-root set of a functional
- 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
- The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc
- Linear hyperplane
- Kernel and image of a linear map
- The tensor and direct-sum models of complexification are canonically complex-linearly isomorphic
- Symmetry and associativity isomorphisms for tensor products over a commutative ring
- Module homomorphisms induce tensor-product homomorphisms functorially
- Isometry, isometric embedding, and the subspace metric on a subset
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- span(S) is exactly the set of linear combinations of finite lists of elements of S, and span(∅) = {0_V}
- The quotient vector space V/W and its canonical projection
- Coset equality, well-defined quotient operations, and the canonical projection with kernel W
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Linear subspace of a vector space
- Linear map between vector spaces over the same field
- A convex subset of ℝᵐ contains every line segment between two of its points
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- 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
- Connected components, quasicomponents, and totally disconnected spaces
- Refinements, locally finite families, point-finite families, and star refinements
- Affine subspaces as translates x+U of linear subspaces
- 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
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V
- ℝⁿ is polygonally connected, connected, locally path-connected and locally connected
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Vector space over a field
- Invertible linear maps, linear isomorphisms, and inverse linear maps
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups
- The right and left weak orders, intervals, covers, and meets and joins of subsets
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
Definition
Real Coxeter representation and contragredient notation
Proposition
Parabolic root subsystems and unique coset minima
Proposition
Finite parabolic longest elements and disconnected factors
Proposition
Tits cone geometry with its exact interior boundary
Proposition
Annihilators, relative hyperplanes and complexification
Definition
Locally finite arrangements and relative face terminology
Proposition