Alphabeta MathArticle items · 05 — The inversion formula $|N(w)|=\ell(w)$, the root-reflection dictionary and strong exchange
Presents 27 items from the article and its immediate prerequisites, covering published library prerequisites and dependency level 0.
1 · Prerequisites
- The geometric inversion set N(w) of an element of a Coxeter group
- The real Coxeter form, its radical, reflections, and form-preserving maps
- Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order
- The canonical reflection homomorphism, roots, reflections, and the positive cone
- Descent of the reflection representation, unit root norms, and conjugation of reflections
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness
- The principle of mathematical induction
- Group and abelian group
- Monoid homomorphism and group homomorphism
- The rank-two half-space alternative and the chamber-length induction (Pₙ), (Qₙ)
- The dual action, chambers, faces, and root hyperplanes
- The dual action, the faces, and the rank-two chamber tiling
- Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action
- Linear functionals and the algebraic dual V^*=L(V,F)
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- The Tits cone, its interior, and the negative-root set of a functional
- The cardinality | A| of a finite set
- Continuity of a map of topological spaces at a point and globally
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- Basis and subbasis for a topology, and the topology generated by a family of sets
- A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Based loops and the fundamental group
- Paths, path-connected spaces and path components
- Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- For continuous maps into a convex subset of ℝⁿ, the straight-line formula defines a continuous homotopy
- Deg:π₁(S¹,[0])→(ℤ,+) is a group homomorphism
- Two based circle loops are path-homotopic if and only if they have equal degree
- deg(ωₙ)=n for every integer n
- Group isomorphisms, automorphisms and the set Aut(G)
- The integers form a commutative ring
- The geometric representation on the simple-root basis over a common splitting field, and the root set
- The natural numbers ℕ (von Neumann)
- Linear map between vector spaces over the same field
- Linear subspace of a vector space
- The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
- Left and right cosets gH and Hg of a subgroup
- The well-ordering principle
- The finite symmetric group Sₙ, one-line notation, and cycle notation
- Inversions, inversion number, the sign sgn(σ)=(-1)^inv(σ), and even and odd permutations
- Adjacent transpositions generate the finite symmetric group Sₙ
- Cycles with disjoint supports commute
- Loops over a two-set path-connected open cover factor through the covering sets
- Antipodal complements cover Sⁿ by simply connected sets with path-connected overlap for n≥2
- Simply connected topological spaces
- The kernel and image of a 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
- One-step subgroup test: a nonempty H ⊆ G is a subgroup iff gh⁻¹ ∈ H for all g, h ∈ H; the identity and the inverses of H are then those of G
- Normal subgroup: invariance under conjugation
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"
- A continuous image of a connected space is connected, and connectedness is a topological property
- A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Connected components, quasicomponents, and totally disconnected spaces
- 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
- Lifts of maps, paths, and homotopies through a covering map
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- 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
- Every complete ordered field is Archimedean
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- The circle as S¹=ℝ/ℤ with basepoint [0]
- ℝ/ℤ is compact and path-connected
- For a quotient map q : X → Y, a map out of Y is continuous iff its composite with q is; a continuous map on X constant on the fibres of q factors uniquely through q; and a composite of quotient maps is a quotient map
- The derivatives of sine and cosine are cosine and minus sine
- A function differentiable at c is continuous at c
- A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
- The zero sets of sine and cosine and the least positive common period 2 pi
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- Pi as twice the smallest positive zero of cosine
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology
- Distinct points of a metric space have disjoint balls around them
- Group extensions, sections, complements, and split extensions
- Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product
- First isomorphism theorem for groups: G/ker f congimf
- A group homomorphism is injective if and only if its kernel is trivial
- Linear hyperplane
- A convex subset of ℝᵐ contains every line segment between two of its points
- Refinements, locally finite families, point-finite families, and star refinements
- Affine subspaces as translates x+U of linear subspaces
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- The quotient vector space V/W and its canonical projection
- 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
- Vector space over a field
- Invertible linear maps, linear isomorphisms, and inverse linear maps
- Projective morphisms before Proj
- Derived category of an abelian category
- Coherent module sheaves
- Coherent sheaves on a locally Noetherian scheme
- Affine quasi-coherent sheaves are modules
- Compatible local sheaves glue uniquely up to unique isomorphism
- Finite modules over Noetherian rings are Noetherian
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Every quotient and every localisation of a Noetherian ring is Noetherian
- The derived category inherits a triangulated structure
- Exact functor between triangulated categories
- Natural isomorphism
- The Axiom of Choice
- Projective dimension of an object
- Ext via a projective resolution of the first variable
- Under the Axiom of Choice, every module admits a projective resolution
- Ext is hom in the derived category
- Enough injective sheaves of modules
- Bounded below complexes admit injective replacements
- Derived hom in the bounded setting
- Cohomology of derived hom is ext
- Sheaf cohomology as right derived global sections
- Krull dimension of a nonzero ring
- Wemyss, A lockdown survey on cDV singularities · Definition 1.1, printed p.1; §2.1, printed p.9; §3.3, printed p.15; §4.1 and Definition 4.2, printed p.16; opening paragraph of §2, printed p.9, and §2.2, printed p.11; §1.2, first geometric-overview paragraph, printed p.2
- Hirano–Wemyss, Faithful actions from hyperplane arrangements · §1.3, printed p.3; Setup 4.1, printed p.8, and §4.1, printed p.9
- Miles Reid, The Du Val singularities Aₙ, Dₙ, E₆, E₇, E₈ · §1.5 and Table 1, printed p.5; Theorem 2.1(2), printed p.6; Exercise 4, printed p.11
- The Stacks Project, Lemma 10.31.2 (0306) · Lemma 10.31.2, statement
- Wemyss, Flops and clusters in the homological minimal model programme · §1.3, printed pp.4--5; §1.6, p.9; §2.2, Setups 2.8--2.9 and Notation 2.10 and Lemma 2.12, pp.12--13; §5.1, Definition 5.1, p.29; Remark 5.16 and Notation 5.17, printed p.35
- Van den Bergh, Three-dimensional flops and non-commutative rings · arXiv:math/0207170v1: §3.1, opening hypotheses, p.5; §3.2, definition of V, p.7; Corollary 3.2.8, Lemma 3.2.9, Proposition 3.2.10 and Corollary 3.2.11, pp.9--10; Lemma 3.5.2, Proposition 3.5.4, Theorems 3.5.5--3.5.6, pp.13--15; §3.4 opening hypotheses and Lemma 3.4.3, pp.11--12; Lemma 3.5.2, equation (3.8) and proof, pp.13--14; equation (3.10) immediately before Proposition 3.5.4, p.14
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Covariant functor, identity functor, composite functor, and contravariant functor
- Isomorphism, groupoid, and connected category
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Invertible matrices and the general linear group GLₙ(F)
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- The change-of-basis matrix P_C leftarrowB=[id_V]_B^C
- The reals form a totally ordered field
- Lifting criterion for maps from path-connected locally path-connected spaces
- Existence and uniqueness of homotopy lifts through a covering map
- Higher homotopy group by based cubes
- Higher homotopy groups are functorial and based homotopy invariant
- Eilenberg--Mac Lane space
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- CW complex with closure finiteness and weak topology
- Higher homotopy basepoint transport and moving homotopies
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Disconnected diagrams, direct products, and comparison of invariant forms
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- Coxeter diagrams: edges, labels, components and finite type
- 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
- Subgroup
- The external semidirect product N⋊_α H
- The inverse of a bijective group homomorphism is a group homomorphism
- Injection, surjection, bijection
- An action of a group H on a group N by automorphisms
- Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- The right and left weak orders, intervals, covers, and meets and joins of subsets
- The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc
- Kernel and image of a linear map
- The tensor and direct-sum models of complexification are canonically complex-linearly isomorphic
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- 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
- span(S) is exactly the set of linear combinations of finite lists of elements of S, and span(∅) = {0_V}
- Coset equality, well-defined quotient operations, and the canonical projection with kernel W
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
TheoremLibrary prerequisite
The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange
TheoremLibrary prerequisite
The root-length criterion and faithfulness of the canonical reflection representation
TheoremLibrary prerequisite
Root sign coherence and the action of simple reflections on positive roots
TheoremLibrary prerequisite
The finite-negativity criterion, the reduction step, and convexity of the Tits cone
TheoremLibrary prerequisite
For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
TheoremLibrary prerequisite
Loop classes form the group π₁(X,x₀) under concatenation
TheoremLibrary prerequisite
Deg:π₁(ℝ/ℤ,[0])→(ℤ,+) is an isomorphism
TheoremLibrary prerequisite
Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups
TheoremLibrary prerequisite
Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
TheoremLibrary prerequisite
Sⁿ is simply connected for every n≥2
TheoremLibrary prerequisite
The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
TheoremLibrary prerequisite
Every path-connected space is connected, and every path component lies inside a component
TheoremLibrary prerequisite
Existence and uniqueness of path lifts through a covering map
TheoremLibrary prerequisite
[t]↦(cos 2π t,sin 2π t) is a homeomorphism from ℝ/ℤ to the unit circle
TheoremLibrary prerequisite
Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent
DefinitionDependency level 0
Locally finite arrangements and relative face terminology
DefinitionDependency level 0
Complete-local smooth threefold flops and their slice data
DefinitionDependency level 0
Real Coxeter representation and contragredient notation
DefinitionDependency level 0
Set-sized groupoids, functors and composition conversion
LemmaDependency level 0
Positive degree and atoms of the presented Artin monoid
LemmaDependency level 0
Nonnegative inverse matrices preserve extremal basis rays
PropositionDependency level 0
Coverings preserve the higher-homotopy obstruction to K(pi,1)
PropositionDependency level 0
Finite parabolic longest elements and disconnected factors
PropositionDependency level 0
Normalizer quotients, kernels and conditional surjectivity
PropositionDependency level 0
Parabolic root subsystems and unique coset minima
PropositionDependency level 0
Reduced-word lifts and the prefix-order convention
PropositionDependency level 0