Chapter 8. Complements
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 7 definitions, 16 theorems, 1 proposition, 2 corollaries, 2 examples, 7 constructions, 1 convention and 4 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: chapter-wide Coxeter-system setting
Unless otherwise stated, (W,S) denotes an arbitrary Coxeter system throughout this survey chapter.
Construction: Tits elementary simplifications for words
Let F be the free group on symbols in bijection with S and F^+ the positive word monoid mapping onto W. Elementary simplifications either apply one braid relation without changing word length or delete an adjacent equal pair; E(u) is the finite effectively computable set of words reachable from u by such simplifications.
Theorem: Tits solution of the word problem
For positive words u,v, their images in W are equal iff E(u)∩E(v) is nonempty. In particular a word represents 1 iff its simplification set contains the empty word.
Theorem/recorded result: conjugacy problem for extra-large Coxeter groups
The source records Appel–Schupp’s algorithm solving the conjugacy problem for extra-large Coxeter groups, where every finite m(s,s′) for distinct generators is greater than 4.
Definition: reflection subgroup
Let T be the set of all reflections of W. A reflection subgroup is a subgroup W′≤W generated by a subset of T.
Construction: canonical generators of a reflection subgroup
For w∈W put N(w)={t∈T | ℓ(tw)<ℓ(w)}. If W′ is a reflection subgroup, define S′ as the reflections t′∈W′ for which N(t′)∩W′={t′}.
Theorem: reflection subgroups are Coxeter groups
If W′ is generated by reflections and S′ is the canonical set defined from N, then (W′,S′) is a Coxeter system.
Definition: the (−1)-condition and W-equivalence for parabolic subsets
For I⊆S, say I satisfies the (−1)-condition if W_I is finite and its longest element w_I acts as −1 on the geometric subspace V_I. Let J be the family of such I; I,J∈J are W-equivalent when some w∈W carries their simple-root sets to one another.
Theorem: Richardson classification of involutions
Every involution of W is conjugate to w_I for some I satisfying the (−1)-condition; for I,J in that family, w_I and w_J are conjugate iff I and J are W-equivalent.
Theorem/recorded result: canonical orthogonal reflection decomposition of involutions
The source records Deodhar’s result that every involution has a unique totally orthogonal set of roots whose commuting reflections multiply to that involution.
Definition: Coxeter element of an arbitrary Coxeter system
For an enumeration S={s_1,…,s_n}, the product w=s_1⋯s_n is a Coxeter element.
Proposition/result: conjugacy of Coxeter elements for tree Coxeter graphs
If the Coxeter graph is a tree, Coxeter elements obtained from different enumerations of S form a single conjugacy class; this can fail when the graph has circuits.
Construction/result: Howlett matrix for a Coxeter element
In the geometric representation, let A be the Gram matrix of B and write 2A=U+U^t with U upper triangular unipotent according to the chosen ordering. The Coxeter element has matrix −U{-1}Ut.
Theorem: spectral characterization of infinite and affine Coxeter groups
W is infinite iff its Coxeter element has infinite order, iff the geometric-representation matrix has a real eigenvalue >1. W is of affine type iff 1 is an eigenvalue and all other eigenvalues have absolute value 1.
Definition/result: Möbius function and inversion on a locally finite poset
For a poset in which every lower interval is finite, the Möbius function μ(x,y) is uniquely characterized by Σ_{x≤z≤y}μ(x,z)=δ_{x,y}; it gives the usual Möbius inversion formula for abelian-group-valued functions.
Theorem: Möbius function of Bruhat order
For every Coxeter group and x≤w in Bruhat order, μ(x,w)=(−1)^{ℓ(x)+ℓ(w)}.
Corollary: Eulerian Bruhat intervals and rank-two diamonds
Every nontrivial Bruhat interval has equally many even- and odd-length elements. In particular, if ℓ(w)−ℓ(x)=2 there are exactly two z with x<z<w.
Theorem/recorded result: Möbius function on parabolic quotient representatives
For minimal coset representatives W^I with induced Bruhat order, Deodhar’s formula gives μI(x,w)=(−1){ℓ(x)+ℓ(w)} when the full interval [x,w] lies in W^I, and μ^I(x,w)=0 otherwise.
Definition: Bruhat graph
The Bruhat graph has vertex set W and a directed edge tw→w whenever t is a reflection and ℓ(tw)<ℓ(w), without requiring the length difference to be one.
Theorem: finiteness of fixed-length Bruhat interval types
For finite Coxeter groups, only finitely many poset-isomorphism types of Bruhat intervals of any fixed length occur.
Theorem/recorded result: interval poset determines the induced Bruhat graph
For a closed Bruhat interval Z, its poset isomorphism type determines the isomorphism type of the full subgraph of the Bruhat graph on Z.
Recorded conjecture: interval invariance of R- and Kazhdan–Lusztig polynomials
The source records Dyer’s conjecture, prompted by Kazhdan–Lusztig, that R_{x,w} and hence P_{x,w} depend only on the poset-isomorphism type of the Bruhat interval [x,w].
Theorem: lexicographic shellability of Bruhat intervals
Every closed interval [x,w] in the Bruhat order of a Coxeter group is lexicographically shellable.
Construction/result: canonical increasing chain label in a Bruhat interval
Fixing a reduced word for w labels each maximal chain in [x,w] by the successive omitted generator positions. There is a unique maximal chain with increasing label, and its label is lexicographically first among all maximal-chain labels.
Corollary: topological and Cohen–Macaulay consequences of shellability
Shellability recovers the Bruhat Möbius formula; passing to the order complex gives the standard topological formulation, and the associated Stanley–Reisner ring is Cohen–Macaulay.
Definition: automorphism of the Bruhat poset
A Bruhat-order automorphism is a bijection θ:W→W such that x≤w iff θ(x)≤θ(w). Such an automorphism fixes 1 and preserves length.
Construction: inversion and Coxeter-graph automorphisms of Bruhat order
The inversion map w↦w^{-1} is a Bruhat-poset automorphism. Every automorphism of the labelled Coxeter graph induces a group automorphism preserving Bruhat order.
Example: Bruhat automorphisms in rank two
For a dihedral Coxeter group, elements of each nonzero intermediate length may be interchanged independently; in the finite order-2m case the Bruhat automorphism group is (Z/2Z)^{m−1}.
Theorem: classification of Bruhat-order automorphisms in irreducible rank at least three
If |S|≥3 and (W,S) is irreducible, every automorphism of Bruhat order is induced by a Coxeter-graph automorphism or is inversion composed with one.
Construction/comparison: factorization input for affine Poincaré series
For a finite Weyl group W with degrees d_i and exponents m_i=d_i−1, identify the distinguished coset representatives of W inside its affine Weyl group W_a with the coroot lattice; Bott’s loop-space cell decomposition supplies the Poincaré series of this quotient.
Theorem: Bott formula for affine-Weyl Poincaré series
If W has exponents m_1,…,m_n, then the affine Weyl group satisfies W_a(t)=W(t)/∏_{i=1}n(1−t{m_i}).
Examples/comparison: irreducible representations of dihedral and symmetric groups
Over C, irreducible dihedral representations have degree 1 or 2. Irreducible S_n-representations are parametrized by partitions of n, as are conjugacy classes; all irreducible S_n-representations can be realized over Q.
Theorem/recorded result: integrality of Weyl-group characters
The source records that all irreducible characters of finite Weyl groups are integer-valued, in contrast with general noncrystallographic finite reflection groups.
Theorem: rational realizability of irreducible Weyl-group representations
Every irreducible representation of a finite Weyl group can be realized over Q.
Comparison: Springer and Kazhdan–Lusztig constructions of Weyl-group representations
The source compares Springer’s rational cohomological construction with Kazhdan–Lusztig cell representations; for symmetric groups the cell construction yields a canonical Q-basis for each irreducible representation.
Definition: Schur multiplier
The Schur multiplier is taken as H²(G,C) of factor-set classes under pointwise multiplication. The source also records the homological convention H_2(G,Z) and the natural identification H²(G,C^)≅Hom(H_2(G,Z),C*).
Construction: Coxeter-graph invariants for the Schur multiplier
Let n_1 be the number of Coxeter-graph vertices; n_2 the number of finite-labelled edges with label at least 3; n_3 the number of equivalence classes of nonadjacent vertex pairs under the odd-edge relation; and n_4 the number of components after deleting even- or ∞-labelled edges.
Theorem: Howlett formula for the Schur multiplier
For any Coxeter group W, its Schur multiplier is an elementary abelian 2-group of rank n_2+n_3+n_4−n_1.
Comparison/result: crystallographic Coxeter groups as Kac–Moody Weyl groups
The source records that the Weyl groups attached to Kac–Moody Lie algebras may be arbitrary crystallographic Coxeter groups, extending the finite and affine Weyl-group settings.
Comparison/result: Schubert geometry and Bruhat/Kazhdan–Lusztig data
Schubert varieties are closures of Bruhat cells indexed by the Weyl group; cell-closure inclusion is governed by Bruhat order. The source highlights the Kazhdan–Lusztig connection between Schubert-variety geometry and representation theory as the geometric thread linking the chapter’s constructions.