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 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
- Invertible matrices and the general linear group GLₙ(F)
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- The change-of-basis matrix P_C leftarrowB=[id_V]_B^C
- The reals form a totally ordered field
- 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 normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup
- C_G(x) and N_G(H) are subgroups of G
- The quotient group G/N and coset product (gN)(hN)=ghN
- First isomorphism theorem for groups: G/ker f congimf
- Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent
- Monoid homomorphism and group homomorphism
- A group homomorphism automatically satisfies f(e) = e' and f(g⁻¹) = f(g)⁻¹, and f(gⁿ) = f(g)ⁿ for every n ∈ ℤ; for monoid homomorphisms preservation of the identity must be assumed
- The kernel and image of a group homomorphism
- Normal subgroup: invariance under conjugation
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
- Group and abelian group
- Subgroup
- Group isomorphisms, automorphisms and the set Aut(G)
- The external semidirect product N⋊_α H
- Group extensions, sections, complements, and split extensions
- The inverse of a bijective group homomorphism is a group homomorphism
- Left and right cosets gH and Hg of a subgroup
- Injection, surjection, bijection
- An action of a group H on a group N by automorphisms
- Real Coxeter representation and contragredient notation
- 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
- Linear subspace of a vector space
- 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)
- Locally finite arrangements and relative face terminology
- A convex subset of ℝᵐ contains every line segment between two of its points
- 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
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
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
Bounded weak-order joins and finite parabolic criteria
Lemma
Nonnegative inverse matrices preserve extremal basis rays
Proposition
Normalizer quotients, kernels and conditional surjectivity
Proposition
Root signs, inversions and descent translation
Proposition