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Sortable Projections and Finite Cambrian Lattices — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Canonical Roots, Signs, and Faithful Reflections
- Chains, Antichains, Sperner and Dilworth
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Coxeter Euler Forms and Sortable Chamber Cones
- Coxeter Presentations, Exchange, and Reduced Word Theorems
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Coxeter Diagrams and Complete Classification
- Finite Fields and Cyclotomic Extensions
- Finite Lattice Projections and Coxeter Chain Labels
- Finite Reflection Arrangements and Spherical Coxeter Complexes
- Foundations of the Real Numbers for Analysis
- Free Groups and Presentations
- Further Trigonometric Identities and Inverse Functions
- Graphs, Walks and Connectivity
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Incidence Algebras and Möbius Inversion
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Parabolic Subgroups and Double Coset Geometry
- Partitions of Unity and Paracompactness
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Real Forms and Reflection Geometry
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Sine, Cosine, and the Definition of Pi
- Sortable Projections and Finite Cambrian Lattices
- Splitting Fields
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Fundamental Theorem of Finite Abelian Groups
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The ZFC Axioms and the Basic Set Constructions
- Tits Cones, Chambers, and Parabolic Stabilizers
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Order, Inversions, and Lattice Operations
2 · Summary
This companion is a dependency leaf. Its computations use the theory on sortable-projections-and-finite-cambrian-lattices and that page’s prerequisite closure; no theory page may depend on a supplier homed here.
The c-sortable subset of A3 for c = s1s2s3, a three-element fiber, and the upper endpoint map lists the sortable elements for in type , computes all projection fibers for that orientation, and gives complete sorting and projection tables for the second orientation . It exhibits a three-element interval fiber and checks a meet/join preservation example directly.
The c-Cambrian quotient of S3 for both orientations: fibers, endpoints and meet/join preservation computes the two quotients for and . It records both sortable sets and fiber partitions, the right weak-order meet/join tables, the upper endpoints, and a finite check of meet and join preservation for all pairs in the first orientation.
These examples verify finite calculations in their stated types. General closure, homomorphism and interval claims are proved on the theory page.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
The c-sortable subset of A3 for c = s1s2s3, a three-element fiber, and the upper endpoint map
Statement
Let be the Coxeter system of type with and diagram . Use the standard model on with , right-to-left composition, and one-line notation; the type- identification and are as in Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4) and The finite symmetric group , one-line notation, and cycle notation. Let , , and . Write for the sortable projection and for the upper projection of The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0.
(i) The -sortable subset. An element of is -sortable (c-sortable elements, forced and unforced skips, skip roots, and the chamber cone (1)) if and only if its -sorting word has weakly decreasing block sequence. There are exactly such elements: where . The remaining ten elements are and each has a strict failure of weak decrease in its block sequence.
(ii) The fibers of . The nontrivial fibers are and the remaining eight fibers are singletons . In particular, the fiber has three elements; by The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (3), every fiber is the closed interval , and
(iii) The endpoint maps. On the fiber of , is constantly and is constantly . The other nontrivial upper endpoints are The map is order-preserving and idempotent, with and , by The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (2)-(3).
(iv) Meet and join preservation. For all , and by Sortable elements form a sublattice and the c-Cambrian quotient is its lattice-homomorphic image (4). For the three-element fibers of and ,
(v) A second orientation. For , the fiber of is whereas . The fiber partition depends on the Coxeter element, although the number of sortable elements is for both orientations.
Facts & Assumptions
Given: The type- Coxeter system, its standard permutation realization, the two displayed Coxeter elements and , the periodic words, and the right weak order.
Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4): for type , is isomorphic to with and is permutation inversion number.
The finite symmetric group , one-line notation, and cycle notation: permutation products use , so the right factor acts first; one-line notation records values in argument order.
Coxeter elements, the oriented Euler form, the skew form, and the periodic word (3): has a fixed sequence of positions; the sorting word is the lexicographically earliest reduced subword, and its block sequence records the letters selected between dividers.
The greedy scan computes the c-sorting word; commutation, conjugation and rank-two alignment (1): scanning positions in order and selecting a letter exactly when it is a left descent of the current remainder produces the -sorting word.
c-sortable elements, forced and unforced skips, skip roots, and the chamber cone (1): is -sortable exactly when its sorting-word block sequence is weakly decreasing under inclusion.
The recursive initial-letter sortable projection: if is initial in , then when , and when .
The right and left weak orders, intervals, covers, and meets and joins of subsets (1) and Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics (2): iff with additive length; in finite every pair has a meet and join.
The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (2)-(3): is the upper endpoint of each -fiber, every fiber is , and the stated monotonicity, idempotence and composites hold.
Sortable elements form a sublattice and the c-Cambrian quotient is its lattice-homomorphic image (4): preserves meet and join in finite weak order.
Proof
Use and right-to-left composition, as in [F1]-[F2]. In each table, a position list is the set selected by the greedy scan of , and a block code such as means the consecutive subsets . The scan is the sorting word by [F4], so the displayed letters multiply to and determine its sortable status by [F5].
For , the first twelve scan records are : .
The remaining twelve scan records for are : .
The weakly decreasing rows in the c-sorting tables above are exactly . The other ten rows fail respectively at , , , , , , , in the last blocks of , in the last blocks of , and . Since the tables contain 24 distinct reduced words and , this proves (i).
For , the first twelve scan records are : .
The remaining twelve scan records for are : .
The weakly decreasing rows in the c'-sorting tables above are exactly . The ten other rows fail at , , , , , , , , , and , respectively; hence also has 14 sortable elements.
Write for a descent branch of [F6] at initial letter and for an ascent branch whose -prefix is ; after rotate the word to , and after restrict it by deleting the initial . Recursing through all inputs gives these image fibers (the indicated traces end at the base ): . The fibers are disjoint and their sizes sum to , so the table is exhaustive and proves (ii).
In right weak order the nontrivial fiber chains are , , , , , and ; successive quotients are simple generators with additive length. By [F8] each top is of the fiber image, giving exactly the endpoint values in (iii), and each displayed fiber is the interval between its bottom and top.
Put and . To move from position to position in requires the adjacent swaps in that order, so this is its unique reduced word; is the longest element on positions and has the two reduced words and . Their right weak-order prefixes intersect in , so . The upper interval of is : has right ascents ; only ; only ; only ; only ; and none, and every upper element is reached by a sequence of such simple right ascents. For these six , the data are . By [F7], exactly when , so the common upper bounds are exactly . Since the latter two are and , the least common upper bound is .
The projection table gives , , , and . The same results are and (use the reduced word for ). This verifies the two sample equalities in (iv); the universal identities there are [F9].
The complete projection recursion table, in the same notation, is . Its disjoint preimages cover all inputs, and the -row is exactly the five-element fiber in (v).
The -sorting tables show sortable elements, matching the in the -sorting tables; the different sizes of the displayed -fibers show the partitions differ. Every calculation is finite, with no selection from an arbitrary family, so no Choice is used.
The c-Cambrian quotient of S3 for both orientations: fibers, endpoints and meet/join preservation
Statement
Let be the Coxeter system of type : presented by , with right weak order , length , longest element and left descent sets . The and -sorting-word conventions are as in Coxeter elements, the oriented Euler form, the skew form, and the periodic word. The six elements of are
| reduced word | ||||||
| 0 | 1 | 1 | 2 | 2 | 3 | |
(i) The quotient for . With , the -sorting words of the six elements are the empty word, , , , and in the positions of ; the corresponding block sets are . Hence the -sortable elements (weakly decreasing block sequence) are and the unique non-sortable element is . The fibers of are where ; the nontrivial fiber is the interval with lower endpoint and upper endpoint (The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (2)–(3)).
(ii) The quotient for . Exchanging and : the -sortable elements are , the unique non--sortable element is , , and the unique nontrivial fiber is . Thus the two orientations of the diagram give the same fiber-size profile , and in both cases the non-sortable element is the one whose two letters occur in the order opposite to the Coxeter element, its projection being the shorter of the two rank-two chains.
(iii) Meet and join preservation. The join and meet tables of the right weak order on (rows and columns in the order ) are
With the values , , , , , , the two tables give, for each of the pairs, and ; for example and . This is the rank-two case of Sortable elements form a sublattice and the c-Cambrian quotient is its lattice-homomorphic image (4).
(iv) The upper projection. Here for and , so is not the identity but is order preserving and idempotent with and (The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (2)–(3)).
Facts & Assumptions
Given: The type- Coxeter presentation , the two orientations and , and the standard definitions of right weak order, sortable projection and upper projection.
Coxeter elements, the oriented Euler form, the skew form, and the periodic word: the -sorting word is the lexicographically earliest reduced subword, and its block sequence records the letter sets between successive dividers.
c-sortable elements, forced and unforced skips, skip roots, and the chamber cone (1): an element is -sortable exactly when its sorting-word block sequence is weakly decreasing under inclusion.
The recursive initial-letter sortable projection: at an initial letter , use when and on the parabolic prefix when ; the base value is .
The right and left weak orders, intervals, covers, and meets and joins of subsets (1): exactly when with .
The right and left weak orders, intervals, covers, and meets and joins of subsets (3): meets and joins are greatest lower and least upper bounds in the right weak order.
Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics (2): finite has a right-weak-order meet and join for every pair.
The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 (2)-(3): is order preserving and idempotent, its fibers agree with those of , and each fiber is the closed interval from to , with and .
Sortable elements form a sublattice and the c-Cambrian quotient is its lattice-homomorphic image (4): preserves binary meets and joins in the finite weak-order lattice.
The sortable projection kernel and the c-Cambrian quotient (1): exactly when ; the quotient order is the right weak order on the projection images.
Proof
Put and . Cancellation reduces every word to an alternating word; gives and , so every alternating word of length at least four shortens, while . Thus every element is represented by one of . The map , satisfies the presentation and sends these six forms to six distinct permutations, so they are exactly the elements of . Their lengths are respectively, and left multiplication gives , , , , , .
For , the lexicographically first reduced position sets in are , , , , , . Their block sequences are respectively , , , , , and . Hence exactly are sortable, while fails the inclusion test.
For , the first reduced position sets in are , , , , , . Their block sequences are , , , , , and . Thus exactly are -sortable and is the unique failure.
On a rank-one parabolic, and , and likewise and . Applying the initial-letter recursion at for and at for gives . For , the entries follow from , , , , and . For , they follow from , , , , and .
Reduced prefixes give the Hasse diagram and . These are the only covers: multiplying each of the six reduced forms on the right by or either cancels its last letter or gives the next displayed form. The four incomparable pairs are exactly one choice from and one from ; their only common lower bound is and their only common upper bound is . Comparable pairs have their lesser and greater elements as meet and join, respectively. This determines both operation tables in the statement and verifies them as greatest lower and least upper bounds.
By the kernel definition, the fibers for are , and those for are . Their sortable images inherit the right weak order as the quotient order, so these are the stated two quotient partitions.
The map fixes every element except , which it sends to . If neither operand is , their meet and join are also fixed: a join can equal only when both operands lie below it, and among such pairs without their join is or ; a meet can equal only when both operands lie above it, and among such pairs without the only possibility is , whose meet is . For an operand , the six cases give and ; applying to the first coordinate yields equality in every case. By symmetry of meet and join this checks all 36 pairs. These are the quotient operations from [F10] and the direct calculation is the rank-two instance of [F9].
With , the products for are respectively . Applying the opposite-orientation projection table and multiplying by gives in that order. In particular, . Reversing gives in the same input order, so .
The nontrivial fibers are the chains and , since and are length-additive right extensions; all other fibers are singletons. The endpoints in step 3.3 therefore give exactly and and singleton intervals. For , changes only to ; it preserves each cover in step 2.2, is idempotent, and the projection table gives and . The same checks with and establish the corresponding assertions for .
Both orientations have five sortable elements, but their nonsortable elements and nontrivial fibers are interchanged: for and for . Thus the fiber-size profile is in either orientation, while the quotient partitions differ. All claims follow from six explicitly listed elements and finite case checks, so no Choice is used.
Sources
- N. Reading, Sortable elements and Cambrian lattices, arXiv:math/0512339v1 (2005); Algebra Universalis 56 (2007) 35-56
- N. Reading and D. E. Speyer, Sortable elements in infinite Coxeter groups, arXiv:0803.2722v3 (2010); Trans. Amer. Math. Soc. 363 (2011) 699-761
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer 2005