Coxeter Element Overview

• Coxeter elements are defined as the product of all simple reflections in a Coxeter system, serving as pivotal regular elements in both finite and complex reflection groups.
• They play a critical role in spectral theory and conjugacy classifications, linking eigenvalue data directly to exponents and the Coxeter number.
• Their combinatorial structure underpins noncrossing partitions, primitive factorizations, and applications in Schubert geometry and representation theory.

A Coxeter element is a fundamental object in the theory of Coxeter and complex reflection groups, playing a central role in geometric, combinatorial, and representation-theoretic aspects of Lie theory, singularity theory, and algebraic combinatorics. Its definition, algebraic structure, spectral theory, and connection to noncrossing partitions, monodromy, and primitive factorizations constitute a rich area of research with deep consequences for the understanding of finite and infinite reflection groups.

1. Definition and Characterization

Let $(W, S)$ be a Coxeter system with $W$ a group generated by involutive simple reflections $S = \{s_1, \ldots, s_n\}$, subject to braid relations determined by the Coxeter matrix. A Coxeter element is any product of all simple reflections in some order: $$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$ The set of all such products, each involving every simple reflection exactly once, exhausts all Coxeter elements; different orderings may yield the same or distinct group elements. In the classical finite real case, Coxeter elements correspond to regular elements whose eigenvalues and spectral data encode the exponents and Coxeter number of $W$ (Hodges et al., 2020).

For irreducible, well-generated complex reflection groups $W \subset \mathrm{GL}(V)$, with degrees $1 \leq d_1 \leq \dots \leq d_n$ of basic polynomial invariants and Coxeter number $h = d_n$, a Coxeter element is precisely a regular element of order $h$, i.e., an element that acts on $V$ with a regular eigenvector outside the reflecting hyperplanes, for an eigenvalue $W$0 (Reiner et al., 2014, Douvropoulos, 2018).

2. Conjugacy, Automorphisms, and Enumeration

Conjugacy and Automorphisms

In finite irreducible $W$1, all Coxeter elements form a single conjugacy class, except in types where the Dynkin diagram admits a nontrivial automorphism. Eriksson and Eriksson proved that two Coxeter elements are conjugate if and only if their reduced words are related by a sequence of rotations (moving the first letter to the end) and legal commutations (swapping commuting generators) (Eriksson et al., 2013). This combinatorial criterion is tightly linked to acyclic orientations and chip-firing games on the Coxeter graph.

For complex reflection groups, the set of Coxeter elements forms a single orbit under the action of reflection automorphisms: group automorphisms that permute the set of all reflections. The Galois group of the field of definition $W$2 acts simply transitively on the set of conjugacy classes of Coxeter elements, with the number of such classes given by $W$3, where $W$4 counts exponents coprime to $W$5 (Reiner et al., 2014).

Enumeration

The total number of Coxeter elements in $W$6 is $W$7, where $W$8 is the order of any Coxeter element (the Coxeter number). For example, in type $W$9 ($S = \{s_1, \ldots, s_n\}$0), there are $S = \{s_1, \ldots, s_n\}$1 Coxeter elements, each an $S = \{s_1, \ldots, s_n\}$2-cycle in the symmetric group (Hodges et al., 2020).

Type	$S = \{s_1, \ldots, s_n\}$3	$S = \{s_1, \ldots, s_n\}$4	$S = \{s_1, \ldots, s_n\}$5 Coxeter elements
$S = \{s_1, \ldots, s_n\}$6	$S = \{s_1, \ldots, s_n\}$7	$S = \{s_1, \ldots, s_n\}$8	$S = \{s_1, \ldots, s_n\}$9
$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$0	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$1	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$2	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$3
$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$4	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$5	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$6	$$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$7

3. Algebraic and Spectral Properties

Coxeter elements are regular: in the reflection representation, their eigenvalues are $$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$8, with the $$c = s_{\pi(1)} s_{\pi(2)} \cdots s_{\pi(n)},\quad \pi \in S_n$$9 the exponents of the group. The centralizer $W$0 is the cyclic group generated by $W$1 itself, i.e., $W$2, which has order $W$3 when $W$4 is finite (Hollenbach et al., 2019). This rigidity is foundational for the study of regular conjugacy classes and for the structural properties of absolute order, representations, and monodromy.

In the context of complex groups and their duals, the Coxeter element is realized as the value of the dual cocharacter $W$5 (associated to half the sum of positive roots) at $W$6, thus connecting Coxeter theory to the Langlands dual group. Kostant’s theorem states that the character value of an irreducible representation at the Coxeter element is always in $W$7, nonzero precisely when the corresponding weight lies in the Weyl group orbit of a specific regular semisimple torsion element in the dual torus (Prasad, 2014).

4. Coxeter Elements and Root Systems

Action on Roots

The action of a Coxeter element $W$8 on the root system $W$9 of $W \subset \mathrm{GL}(V)$0 partitions the roots into $W \subset \mathrm{GL}(V)$1 orbits of size $W \subset \mathrm{GL}(V)$2 in the finite case; a natural transversal is given by certain nested sequences of simple reflections. In affine types, the orbit structure becomes more intricate: there are $W \subset \mathrm{GL}(V)$3 infinite orbits and, for ranks greater than two, countably many finite real $W \subset \mathrm{GL}(V)$4-orbits. These are completely classified in terms of certain rank-$W \subset \mathrm{GL}(V)$5 finite root subsystems and associated discrete parameters (Reading et al., 2018).

Infinite and Non-finite Rank Variants

For root systems of "half-infinite" types such as $W \subset \mathrm{GL}(V)$6 and $W \subset \mathrm{GL}(V)$7, the Coxeter operator is defined as the product of monodromy automorphisms about the critical values, and its spectrum becomes continuous over an interval rather than discrete, providing an infinite-dimensional, analytic extension of the finite-type spectral theory (Saito, 2012).

5. Noncrossing Partitions and Absolute Order

Coxeter elements index intervals in the absolute order on $W \subset \mathrm{GL}(V)$8: $W \subset \mathrm{GL}(V)$9 is defined by reflection length, and the noncrossing partition lattice $1 \leq d_1 \leq \dots \leq d_n$0 is the interval $1 \leq d_1 \leq \dots \leq d_n$1. The poset $1 \leq d_1 \leq \dots \leq d_n$2 is isomorphic under reflection automorphisms for different Coxeter elements; the associated combinatorial and geometric structures (EL-shellability, Catalan enumeration, self-duality) do not depend on the choice of $1 \leq d_1 \leq \dots \leq d_n$3 (Reiner et al., 2014, Douvropoulos, 2018).

Reduced reflection factorizations of $1 \leq d_1 \leq \dots \leq d_n$4 correspond bijectively to maximal chains in $1 \leq d_1 \leq \dots \leq d_n$5, connecting representation theory, braid groups, and algebraic geometry. The primitive factorization theory enumerates length-additive factorizations of $1 \leq d_1 \leq \dots \leq d_n$6 into parabolic Coxeter elements and reflections, with uniform formulas depending on reflection length and normalizer indices (Douvropoulos, 2018).

6. Geometry, Schubert Varieties, and Broader Contexts

Coxeter elements appear in the geometry of Schubert varieties: for a simple algebraic group $1 \leq d_1 \leq \dots \leq d_n$7 with Weyl group $1 \leq d_1 \leq \dots \leq d_n$8, each Schubert variety $1 \leq d_1 \leq \dots \leq d_n$9 in $h = d_n$0 is indexed by $h = d_n$1. Coxeter elements are maximally spherical elements, and $h = d_n$2 is $h = d_n$3-spherical. Generalizations allow for spherical elements indexed by more intricate subsets, with applications to the study of spherical Schubert geometry, Demazure modules, and key polynomials (Hodges et al., 2020).

In the broader categorical and monodromy contexts, Coxeter elements serve as Garside elements in braid groups, generate centralizers in Artin group theory, and underpin continous spectral decompositions in singularity and Hodge theory (Saito, 2012, Douvropoulos, 2018).

7. Generalizations and Regular Elements

Properties of Coxeter elements robustly extend to regular elements of arbitrary order in well-generated (and, in many cases, badly generated) complex reflection groups. Reflection automorphisms act transitively on regular elements of a given order, and the corresponding Galois-theoretic and combinatorial statements carry over with adjustments to degree and orbit counting determined by the exponents and regular numbers (Reiner et al., 2014).

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Key references:

• (Hodges et al., 2020) Hodges, Yong: Coxeter combinatorics and spherical Schubert geometry.
• (Reiner et al., 2014) Reiner, Ripoll, Stump: On non-conjugate Coxeter elements in well-generated reflection groups.
• (Prasad, 2014) Prasad: Half the sum of positive roots, the Coxeter element, and a theorem of Kostant.
• (Hollenbach et al., 2019) Wegener: The centralizer of a Coxeter element.
• (Reading et al., 2018) Hohlweg, Ripoll: The action of a Coxeter element on an affine root system.
• (Douvropoulos, 2018) Douvropoulos: Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element.
• (Saito, 2012) Saito: Coxeter elements for vanishing cycles of type $h = d_n$4 and $h = d_n$5.
• (Eriksson et al., 2013) Eriksson, Eriksson: Conjugacy of Coxeter elements.