Alphabeta MathCoxeter Groups
Coxeter groups organize reflection-style presentations through generators of order two and prescribed orders of pairwise products. Symmetric groups give the first concrete family: permutations, cycles, transpositions and sign lead to adjacent generators and word length. Three published pairs develop symmetric-group structure, conjugacy and simplicity, and permutation statistics including weak and Bruhat orders.
The draft programme begins with general presentations, exchange and reduced words, then develops canonical roots, faithful reflection geometry, Tits cones, chambers and parabolic subgroups. Finite classification includes dihedral, crystallographic and noncrystallographic types. Spherical arrangements, invariant and coinvariant degrees, reflection length and bipartite root enumeration connect geometry to combinatorial structure. Affine alcoves and translations, growth, heaps, noncrossing partitions and sortable quotients extend the theory.
The geometric branch builds on the library’s existing metric, simplicial, CW and homotopy foundations. New supplier contracts cover the genuinely missing polyhedral and angular constructions, CAT comparison, quantitative short-loop arguments and the Davis complex. Published and draft suppliers throughout the library may be used, with their actual statuses and hypotheses retained.
General Group Theory keeps its existing pairs. Lie Theory, Braid Groups, Hopf Algebras & Hecke Algebras and Special Topics in Representation Theory retain their application homes. The category currently contains three published pairs and thirty draft pairs. The new scaffold has undergone an independent proof-design audit; its actual item proofs remain future authoring work.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Symmetric Groups and Word Foundations
3 pagesThe published pages develop concrete permutations, sign, conjugacy, inversion statistics and Bruhat order in symmetric groups. The draft general-Coxeter pair supplies presentations, exchange, reduced words and parabolic word control for the subsequent programme.
The symmetric group acts on its finite underlying set, and the orbit-partition theorem applies to the cyclic subgroup generated by a permutation.
4 definitions, 2 lemmas, 4 theorems, 4 corollariesExamples & counterexamples →Cycle decomposition, cycle type, permutation parity, the sign homomorphism, and the alternating group come from the preceding symmetric-group development.
1 definition, 4 lemmas, 7 theorems, 6 corollariesExamples & counterexamples →Finite symmetric groups, one-line notation, inversions, cycle decomposition, binomial coefficients, factorials, and the symbolic-method page provide the background for this development.
8 definitions, 2 lemmas, 6 theorems, 4 corollaries, 5 examples, 2 counterexamples, 1 false statement, 1 remarkExamples & counterexamples →
Part 2 · Reflection and Metric Foundations
0 pagesCanonical reflection geometry, roots, chambers and parabolics share this stage with the specific polyhedral and angular constructions needed later. Existing poset, metric, simplicial and topological proofs are reused. The new contracts supply the intrinsic metric and CAT comparison facts those general foundations do not establish.
Part 3 · Finite Geometry and Curvature Tools
0 pagesFinite classification includes crystallographic and noncrystallographic types. Bruhat labels and shelling, spherical arrangements, reflection length and bipartite root enumeration prepare invariant degrees. Quantitative loop shortening and the spherical metric-flag criterion supply the curvature branch. The Artin and Hecke interfaces retain their separate application homes.
Part 4 · Affine and Combinatorial Extensions
0 pagesWeak order, affine alcoves and affine classification develop the global geometry. Descents and growth have independent degree and length comparisons; heaps control commutation classes. The Davis coset realization and the Euler-form and sortable-cone constructions prepare the final geometric and combinatorial results.
Part 5 · Davis, Noncrossing and Cambrian Theory
0 pagesThe final branches prove the designed Davis CAT(0) and finite-subgroup statements, noncrossing lattice and complement constructions, and the sortable projection quotient. Every prerequisite remains explicit. These pages are drafts, and their actual item proofs remain future authoring work.