Finite Coxeter Groups

• Finite Coxeter groups are groups defined by involutive reflections subject to specific order relations, with properties encoded in Coxeter matrices and Dynkin diagrams.
• They are classified into types such as A, B, D, I and exceptional families, providing a unified combinatorial and geometric framework for understanding symmetric and reflection groups.
• Research integrates geometric realizations, reflection subgroup analysis, and representation theory, leveraging homological invariants and reduced word combinatorics for deeper structural insights.

Finite Coxeter groups are groups generated by involutive reflections that satisfy specific order relations encoded by a Coxeter matrix or diagram. These groups play a central role in algebra, geometry, Lie theory, combinatorics, and the theory of algebraic groups. Their classification is intimately linked to the ADE–Dynkin diagrams and underpins the structure theory of semisimple Lie algebras, algebraic groups, and reflection groups. The subject incorporates a blend of geometric, combinatorial, and representation-theoretic methodologies, making finite Coxeter groups a cornerstone in modern mathematics.

1. Definitions, Presentations, and Root System Realization

A finite Coxeter group is defined via a presentation by generators and relations: $W = \langle S \mid (ss')^{m(s,s')}=1 \rangle,$ where $S$ is a finite set of involutive generators ($m(s,s)=1$), $m(s,s')=m(s',s)\geq 2$ for $s\neq s'$, and $m(s,s')=\infty$ encodes the absence of relation between $s$ and $s'$. The Coxeter matrix $M=(m(s,s'))$ is symmetric. The associated Coxeter graph (Dynkin diagram in the finite case) has one node for each generator and edges labeled when $m(s,s')\geq 4$.

Every finite Coxeter group can be realized as a finite real reflection group: there exists a Euclidean space $V$ with inner product $B$ and a set of roots $\{\alpha_s\}$ such that $s$ acts as the orthogonal reflection in the hyperplane orthogonal to $\alpha_s$. The associated bilinear form is given by

$$B(\alpha_s, \alpha_{s'}) = -\cos\left(\dfrac{\pi}{m(s,s')}\right).$$

The finiteness criterion for the Coxeter group is that the associated Gram matrix is positive definite (Singla, 15 Sep 2025).

A central theorem states: an irreducible finite Coxeter group is determined by a connected Dynkin diagram of the ADE, BC, F, H, or I type (constituting the Dynkin$^+$ family). Conversely, every connected Dynkin$^+$ diagram produces a finite Coxeter group.

2. Classification and Combinatorial Structure

Finite Coxeter groups are classified as follows:

Type	Structure Summary	Parameters
$A_n$	Symmetric group $S_{n+1}$	$n\geq 1$
$B_n$	Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$	$n\geq 2$
$D_n$	Even-signed permutation group (index 2 in $B_n$)	$n\geq 4$
$I_2(m)$	Dihedral group of order $2m$	$m\geq 3$
$E_6,\,E_7,\,E_8$	Exceptional, via Dynkin diagrams	—
$F_4,\,H_3,\,H_4,\,G_2$	Exceptional, non-crystallographic	—

A finite Coxeter group is a direct product of irreducibles. The associated root system allows for a combinatorial (Dynkin diagram) and geometric (reflection) perspective on subgroups, conjugacy classes, and parabolic subgroups (Singla, 15 Sep 2025, Douglass et al., 2011).

Reflection subgroups are classified by parabolic and maximal rank subdiagrams; in type $A_n$ these correspond to partitions, in $B_n$ to structures modeled by triple-partitions, and in $D_n$ by double-partitions and further restrictions to ensure maximal rank. Exceptional groups are analyzed directly via computations, often using the Borel–de Siebenthal algorithm and extended Dynkin diagrams (Douglass et al., 2011).

3. Conjugacy, Involutions, and Reflection Subgroups

Conjugacy classes in finite Coxeter groups are determined by their action on the root system and the structure of parabolic subgroups. Involutions, especially those represented as products of mutually commuting reflections, play a foundational role. The degree of an involution is the codimension of its fixed point space.

A standard construction identifies conjugacy classes of involutions with certain subsets of the Coxeter diagram (parabolic subgroups with longest elements), modulo diagram automorphisms. An important symmetry is that multiplication by the longest element $w_0$ often interchanges the +1 and –1 eigenspaces, providing duality between conjugacy classes of involutions (Zibrowius, 2022).

Explicit classification of reflection subgroups is achieved by relating each to combinatorial invariants associated with the Coxeter diagram. The map $\gamma$ sending a reflection subgroup to the conjugacy class of its Coxeter element is injective if and only if the set of all reflections forms a single conjugacy class in $W$, except for one exceptional case in $E_8$. In type $A_n$ this map is a bijection; in $B_n$ it is surjective but not injective; in $D_n$ it is injective but not surjective (Douglass et al., 2011).

Cubes (abelian subgroups generated by mutually commuting reflections) are central to many cohomological invariants and appear in applications to Sylow theory and fusion in finite Coxeter groups (Serre, 2020).

4. Presentations, Generation Methods, and Homological Invariants

Symmetric generation constructs the standard presentations for Weyl groups of types $A_n$, $D_n$, and $E_n$ by leveraging actions of symmetric groups on points, pairs, or triples, respectively. These presentations are realized as quotients of semidirect products of free products of $\mathbb{Z}/2\mathbb{Z}$ by symmetric groups, modulo succinct order-3 relators derived from the combinatorics of the diagrams: $$W(A_n) \cong 2^{*1}: S_n / (t_1(12))^3=1,\qquad W(D_n) \cong 2^{*2}: S_n / (t_{12}(23))^3=1,$$ and analogously for $E_6$, $E_7$, $E_8$ (Fairbairn, 2011).

Homological invariants of Coxeter groups, particularly the low-dimensional integral homology groups $H_2(W;\mathbb{Z})$ and $H_3(W;\mathbb{Z})$, are computed in terms of the diagrams. The second homology can be decomposed as

$$H_2(W;\mathbb{Z}) \cong H_0(D_{\bullet\bullet};\mathbb{Z}_2) \oplus \mathbb{Z}_2[E(D_{\text{even}})] \oplus H_1(D_{\text{odd}};\mathbb{Z}_2),$$

providing diagrammatic invariants distinguishing non-isomorphic groups. The full structural formula for $H_3(W;\mathbb{Z})$ depends on cyclic and acyclic components of the diagrams and incorporates nontrivial extensions reflecting intricate relations among the generating set and their parities (Boyd, 2018).

Cohomological invariants with coefficients in $\mathbb{Z}/2\mathbb{Z}$ are fully described for classical types, notably $D_n$, using restriction to maximal abelian (reflection-generated) subgroups and explicit trace forms, yielding free bases for the invariant modules (Ducoat, 2011, Serre, 2020).

5. Combinatorics, Reduced Words, and Representation Theory

Reduced word combinatorics is fundamental in the paper of Coxeter groups—each group element admits decompositions as products of minimal length, subject to the braid and commutation relations. In type $A_n$, reduced words correspond to permutations and are intricately connected to Young tableaux and equivalence classes under elementary moves.

Enumerative and structural properties of special classes of elements, such as fully commutative or cyclically fully commutative elements, are studied via generating functions. In all finite Coxeter types, the generating functions for such elements by length are rational; e.g., for $A_{n-1}$: $$A_{n-1}^{\mathrm{(CFC)}}(q) = (2q+1)A_{n-2}^{\mathrm{(CFC)}}(q) - qA_{n-3}^{\mathrm{(CFC)}}(q)$$ with associated rational generating functions, revealing deep combinatorial regularities (Pétréolle, 2014).

Explicit connections between reduced words and geometric tilings are realized by Elnitsky's bijections and their generalizations: for each finite Coxeter group $W$ equipped with a suitable embedding into $\mathrm{Sym}(n)$, there is a bijection between certain tilings of polygons and equivalence classes of reduced words, thereby relating algebraic properties of $W$ with polygonal geometry (Nicolaides et al., 2021).

Representation theory of finite Coxeter groups is rich and explicitly realized for the infinite families. For $A_n\cong S_{n+1}$, irreducibles are indexed by partitions and constructed by Specht modules, with dimensions given by the hook-length formula. For $B_n=(\mathbb{Z}/2\mathbb{Z})^n\rtimes S_n$, irreducibles arise via the Wigner–Mackey method and pairwise partition parametrization. In $D_n$, representations are induced from $B_n$ by restriction and analyzing the splitting and non-splitting cases (Singla, 15 Sep 2025).

The notion of perfect models for representations relates to the existence of Gelfand models realized via induced characters associated with special subgroups (quasiparabolic centralizers of involutions), complete classification depending on the type, with uniqueness up to explicit equivalences outside several low-rank cases (Marberg et al., 2022).

6. Structural, Model-Theoretic, and Isomorphism Aspects

Finite Coxeter groups possess a high degree of rigidity: most have a unique realization up to isomorphism of the generating diagrams (complete graphs). The "Coxeter galaxy" is a ranked simplicial complex whose vertices are complete Coxeter graphs; finite groups often correspond to isolated points, reflecting this rigidity. Connections between different presentations (via "blow-ups", "diagram twists") provide a structural perspective on the isomorphism problem (Rego et al., 2022).

Model theory contributes by fully characterizing when finite-rank Coxeter groups are superstable (if all components are spherical or affine) and when the theory is controlled by the combinatorics of the diagrams, especially in the even and right-angled cases. In these contexts, key sets such as the reflections are first-order definable, supporting robust classification and transfer of properties (Muhlherr et al., 2020).

For specific combinatorial and geometric invariants (length functions, reflection length, excess), one finds remarkable symmetries: in each conjugacy class, elements of both minimal and maximal length can be factored as products of two involutions with zero excess, connecting to questions in length statistics and minimal expressions (Hart et al., 2015).

7. Broader Applications and Open Directions

Finite Coxeter groups underpin fundamental theory in algebraic groups, Lie algebras, and algebraic geometry. Their subgroups and element classes correspond to important subvarieties in flag varieties (Schubert varieties), with deep connections to representation theory and spherical varieties (Balogh et al., 2021).

Research directions include:

• Deeper understanding of tensor product decompositions (for example, the Saxl conjecture for symmetric groups),
• Structure and invariants of exceptional and non-crystallographic groups,
• Algorithmic aspects of the word and conjugacy problems for interval and Artin groups,
• Categorification, geometric group theory, and applications to cohomological invariants.

The paper of reflection representations and their generalizations (parameterized by characters of graphical homology) opens new connections to cell representations and Hecke algebras (Hu, 2023).

In all, finite Coxeter groups are among the best-understood and most structurally rich classes of groups, with a categorical synthesis of geometric, combinatorial, and representation-theoretic methodologies forming the foundation for their modern theory.