Reflection Length in Coxeter Groups

• Reflection length is a group-theoretic invariant that measures the minimal number of reflections required to express an element in a Coxeter group.
• In affine Coxeter groups, the invariant decomposes into a translation part and a finite subgroup part, yielding a uniform upper bound of 2n, where n is the dimension.
• The study highlights a dichotomy where spherical and affine groups exhibit bounded reflection length, contrasting with the unbounded complexity in more general infinite Coxeter groups.

Reflection length is a group-theoretic and geometric invariant defined in the context of Coxeter groups and their generalizations, quantifying the minimal number of reflections required to express a given group element. The function encapsulates subtle structural, combinatorial, and geometric properties, distinguishing between finite, affine, and more general infinite (non-affine) Coxeter groups by the presence or failure of uniform bounds on this length. In the affine case, reflection length admits strong metric, combinatorial, and geometric interpretations, and its boundedness reflects the profound rigidity and regularity of these groups.

1. Definitions and Fundamental Properties of Reflection Length

Let $(W,S)$ be a Coxeter system with minimal generating set $S$. The set of all reflections is defined by

$R = \{ w s w^{-1}\mid s\in S,\, w\in W \}.$

The reflection length $\ell_R(w)$ of an element $w\in W$ is the minimal integer $k$ such that $w$ can be written as a product of $k$ elements of $R$: $w = r_1 r_2 \cdots r_k, \quad r_i \in R.$ This induces a bi-invariant word metric on $W$, which is naturally symmetric and satisfies the triangle inequality. In Coxeter groups, especially in the context of geometric representations (for example, as groups generated by Euclidean (affine) or hyperbolic reflections), these reflections correspond geometrically to actual symmetries of the space, and $\ell_R(w)$ counts the minimal number of such symmetries needed to transport a fundamental chamber to its image under $w$.

2. Affine Coxeter Groups: Structure and Computation of Reflection Length

An affine Coxeter group $W$ acting faithfully and cocompactly on $\mathbb{R}^n$ arises from a crystallographic root system $\Phi$ and the lattice $L$ of translations by coroots. Each reflection is associated to an affine hyperplane of the form

$$H_{a,i} = \{ x\in V : \langle x, a\rangle = i \},\quad a\in\Phi,\ i\in \mathbb{Z},$$

where $r_{a,i}$ denotes reflection across $H_{a,i}$. The group structure splits as

$w = t_x w_0,$

where $t_x\in T$ is translation by $x\in L$ and $w_0\in W_0$ is an element of the spherical (finite) Coxeter subgroup generated by reflections fixing the origin.

For a translation $t_x$, the paper defines two notions of dimension:

• Real dimension: minimal $k$ such that $x$ lies in a $k$-dimensional subspace spanned by coroots.
• Integral dimension: minimal $k$ such that $x$ is an integral linear combination of $k$ coroots.

A translation $t_x$ with integral dimension $k$ satisfies

$\ell_R(t_x) = 2k,$

and more generally, for $w = t_x w_0$ with $k = \text{integral dimension}(x) \leq n$,

$k \leq \ell_R(w) \leq k + n \leq 2n.$

For $k=n$ (the maximal integral dimension), $\ell_R(w) = 2n$ is achieved, demonstrating the sharpness of the uniform upper bound.

3. Uniform Upper Bound and its Optimality

The central theorem established is that for any affine Coxeter group $W$ acting faithfully and cocompactly on $\mathbb{R}^n$, the function $\ell_R$ admits a sharp uniform upper bound: $\ell_R(w) \leq 2n \quad \forall w\in W,$ with equality realized for maximal-dimensional translations. The proof constructs any $w = t_x w_0$ as a product of two elements:

• $u$ (moving the origin to $x$) with $\ell_R(u) = k$,
• $v$ (an element fixing the origin) in $W_0$, with $\ell_R(v) \leq n$.

By the triangle inequality, $\ell_R(w) \leq k + n \leq 2n$. Combinatorial independence arguments establish $2n$ as a lower bound for elements with full integral dimension, demonstrating optimality.

4. Metric, Combinatorial, and Geometric Interpretations

For finite (spherical) Coxeter groups, reflection length coincides with the codimension of the fixed space: $$\ell_R(w) = \operatorname{codim} \operatorname{Fix}(w)$$. In affine groups, the reflection length is governed by the interplay of translation parts and their dimensionality, together with the (finite) reflection behavior of the spherical subgroup. The normal form $w = t_x w_0$ provides a framework for decomposing reflection length computation into translation (dimension-based) and finite group (permutation or root structure) parts.

The combinatorial structure is further illuminated in the context of symmetric and affine symmetric groups, where, for a permutation $\pi$, $\ell_R(\pi)$ equals $n$ minus the number of cycles, tying reflection length directly to classical permutation statistics.

5. Beyond Affine: Comparison with Other Coxeter Groups

The work conjectures that only spherical and affine Coxeter groups admit a uniform upper bound for reflection length. For other (e.g., hyperbolic or free) infinite Coxeter groups, explicit examples demonstrate $\ell_R$ is unbounded. For instance, in the free Coxeter group on three generators, the $n$th power of the product of standard generators has reflection length $n+2$, growing without bound.

This dichotomy anchors the boundedness phenomenon in the rigid structure of spherical/affine Coxeter groups, while infinite non-affine Coxeter groups—their geometry more complex—do not constrain the complexity of elements as measured by reflection length.

6. Key Formulas and Computational Implications

Central formulas, collecting the main technical content, include:

Description	Formula
Definition of reflections	$R = \{ w s w^{-1} \mid s \in S,\, w \in W \}$
Reflection length (factorization)	$$\ell_R(w) = \min \{ k \mid w = r_1 \cdots r_k,\; r_i \in R \}$$
Affine normal form	$w = t_x w_0$ with $t_x \in T$, $w_0\in W_0$
Upper bound for affines	$k \leq \ell_R(w) \leq k+n \leq 2n$
Maximal translation case	$\ell_R(t_x) = 2n$ for $x$ of full integral dimension
Conjecture for bounding reflection length	"Spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound"

These provide direct computational schemes: any element decomposed as a translation part and finite Coxeter group part yields immediate bounds and, in the maximal translation case, exact values.

7. Broader Implications and Open Questions

The techniques—bridging root-theoretic, combinatorial, and geometric arguments—demonstrate that reflection length is a metric closely reflecting the internal rigidity or flexibility of the group. The rigid, bounded behavior in affine Coxeter groups stands in contrast with unbounded growth in more general infinite Coxeter groups.

The conjecture, supported by concrete examples and structural arguments, focuses subsequent research on understanding the spectrum of possible reflection length functions across Coxeter groups, their geometric group theory, and algorithmic implementations for factorization and word metrics. The sharp $2n$ bound and its proof also influence algorithmic strategies for decomposing elements in affine groups and understanding automorphism group structure, factorization diameters, and geometric navigation in Coxeter group actions.