Dihedral Reflection Subgroups

• Dihedral reflection subgroups are rank-two groups generated by two distinct reflections, isomorphic to the dihedral Coxeter group I₂(m) with order 2m when finite.
• Their structure is determined by the order of the product of two reflections, playing a vital role in understanding combinatorial, geometric, and tiling properties in Coxeter theory.
• A combinatorial framework using palindromic reduced expressions and reflection factorizations underpins the construction and uniqueness of maximal dihedral subgroups with applications in poset theory and crystallography.

A dihedral reflection subgroup is a rank-two reflection subgroup generated by two distinct reflections in a Coxeter group. Such subgroups play a fundamental role in Coxeter theory, root system geometry, and applications to combinatorics and tiling, and underlie the algebraic and combinatorial structure of reflection-based groups. This article provides a comprehensive account of the theory, structure, classification, and applications of dihedral reflection subgroups, emphasizing the modern combinatorial approach and their real and abstract incarnations.

1. Structural Framework: Coxeter Groups and Reflection Subgroups

Let $(W,S)$ be a Coxeter system, with $W$ a group generated by a (possibly infinite) set $S=\{s_i\}$ subject to relations $s^2=1$ for all $s\in S$, and $(ss')^{m_{s,s'}}=1$ for $s\ne s'$, where $m_{s,s'}\in\{2,3,\ldots,\infty\}$.

A reflection in $W$ is any element of the form $w s w^{-1}$ with $w\in W, s\in S$. The set of all reflections is denoted $T=\bigcup_{w\in W} w S w^{-1}$.

A reflection subgroup is any subgroup generated by a subset of $T$. By the Dyer–Deodhar–Hée theorem, any such subgroup inherits a Coxeter structure via its canonical set of simple reflections.

A dihedral reflection subgroup is a rank-two reflection subgroup, i.e., a subgroup generated by two distinct reflections $t, t'\in T$. Each such subgroup is isomorphic to the dihedral Coxeter group of type $I_2(m)$, with $m$ being the order of $tt'$ in $W$, and has order $2m$ when $m<\infty$ (Gobet, 2023, Douglass et al., 2011).

2. Maximal Dihedral Reflection Subgroups: Existence and Uniqueness

Given any two distinct reflections $t, t'\in T$, Dyer’s theorem asserts the existence and uniqueness of a maximal dihedral reflection subgroup $D_{t,t'}\le W$ containing both $t$ and $t'$ (Gobet, 2023). Formally, $D_{t,t'}$ is defined as

$$D_{t,t'} = \langle R_{tt'} \rangle, \quad R_{tt'} = \left\{ r\in T \mid \exists\, r'\in T\!: \ tt'=r\,r' \right\}$$

where $tt'$ is the product of $t$ and $t'$.

The existence of a unique maximal such subgroup is established combinatorially, without recourse to root system theory, via properties of reduced words and exchange conditions in Coxeter groups. This subgroup contains all reflections expressible through $tt'$-factorizations and admits a Coxeter presentation: $$\langle t, t' \mid t^2=(t')^2=1,\, (tt')^{m_{t,t'}}=1\rangle \cong I_2(m_{t,t'})$$ with $m_{t,t'}$ the order of $tt'$ in $W$ (possibly $\infty$) (Gobet, 2023).

3. Classification, Conjugacy, and Types

The isomorphism type of a dihedral reflection subgroup is dictated by the order $m$ of $tt'$. For reflections $s_\alpha, s_\beta$ in a finite root system, $$m_{\alpha\beta}=\frac{\pi}{\arccos\left(-(\alpha,\beta)/\|\alpha\|\|\beta\|\right)}$$. Classification in finite Coxeter groups aligns dihedral reflection subgroups with rank-2 root subsystems (Douglass et al., 2011):

• For simply-laced types $A_n, D_n, E_6, E_7, E_8$, only $I_2(2)$ and $I_2(3)$ occur.
• In $B_n$ and $C_n$: $I_2(2)$, $I_2(3)$, $I_2(4)$, $I_2(6)$.
• Exceptional types such as $F_4$, $H_3$, and $H_4$ feature higher $m$ values, linked to special angles in the corresponding root systems.
• In all types except $E_8$, there is a canonical assignment between dihedral reflection subgroups and the conjugacy class of their Coxeter elements, which is injective up to conjugacy (Douglass et al., 2011).

The maximal dihedral subgroup $D_{t,t'}$ is generated by the two given reflections and captures all elements whose reflection factorizations intersect with $tt'$.

4. Combinatorial Foundations and Constructions

The combinatorial approach to maximal dihedral reflection subgroups leverages reduced expressions and the exchange condition:

• Palindromic reduced expressions: Any reduced expression for a reflection $t$ yields another reduced expression by reversing and concatenating the halves.
• Two-reflection factorization: A product $w=s\,w'\,t$ with both $sw'$ and $w't$ reflections, implies $w=(st)^{n+1}$.
• Rank-two criterion: If every factorization of $w\neq 1$ as $s\,s'$ involves $s'\in T$, then $|S|=2$.
• Explicit construction: For $w=tt'$, the set $R_w$ defined as all $r\in T$ such that $w=rr'$ for some $r'\in T$ generates $W_w=\langle R_w\rangle$, which is a rank-two Coxeter group, i.e., dihedral (Gobet, 2023).

Uniqueness follows since any dihedral reflection subgroup containing $t,t'$ must contain $tt'$, and the appropriate $R_{tt'}$ exhausts all possible conjugates within such a subgroup.

5. Classical and Affine Examples

Symmetric and Hyperoctahedral Groups

For $W=S_{n+1}$ (type $A_n$), $T$ is the set of all transpositions. If $t=(i\,j)$ and $t'=(j\,k)$, $tt'$ is a $3$-cycle of order 3, and $D_{t,t'}\cong S_3 \cong I_2(3)$. In $B_n$, $I_2(m)$ for even $m$ occurs as subgroups generated by signed-coordinate reflections (Gobet, 2023, Douglass et al., 2011).

Affine Dihedral Subgroups

In affine extension, $\widetilde W(B_n)$, the affine dihedral subgroup $\widetilde I_2(h)$ with $h=2n$ is generated by two reflections and an affine reflection, acting on the projected Coxeter plane. The orbits under these groups describe the local dihedral symmetry observed in tilings and quasicrystals, including the Ammann–Beenker and Penrose arrangements. Detailed constructions project higher-dimensional cubic lattice Voronoi cells into the Coxeter plane, yielding overlapping $h$-gons tiled by rhombi, with the subgroup $\widetilde I_2(h)$ acting transitively on these configurations (Koca et al., 2023).

6. Dihedral Reflection Subgroups in Group Theory and Beyond

Beyond abstract Coxeter theory, dihedral reflection subgroups unify several classes of finite groups with involutive "reflection" automorphisms:

Group Family	Presentation	Order
Dihedral $D_{2n}$	$$\langle a, r \mid a^n=1, r^2=1, r a r = a^{-1}\rangle$$	$2n$
Dicyclic $Dic_n$	$$\langle a, r \mid a^n=1, r^2=a^{n/2}, r a r = a^{-1}\rangle$$, $n$ even	$2n$
Semidihedral $SD_{2^m}$	$$\langle a, s \mid a^{2^m}=1, s^2=1, s a s=a^{2^{m-1}-1}\rangle$$	$2^{m+1}$
Semiabelian $SA_{2^m}$	$$\langle a, s \mid a^{2^m}=1, s^2=1, s a s=a^{2^{m-1}+1}\rangle$$	$2^{m+1}$
Diquaternion $DQ_n$	$$DQ_n = Dic_n \rtimes \langle f \mid f^2=1, f a f=a^{-1}, f r f=r \rangle$$	$4n$

These families capture diverse generalizations of the dihedral group via distinct twists in the reflection (involution) action, and their subgroup lattices (especially the reflection-generated subgroups) classify the possible dihedral subgroups within more general group-theoretic contexts (Macauley, 2023).

7. Applications to Poset Theory and Quasicrystallography

Maximal dihedral reflection subgroups underpin the structure of generalized noncrossing partitions. Specifically, for any Coxeter group of rank three, the interval $[1, c]_T$ in the absolute order (with $c$ a Coxeter element) forms a lattice; this lattice property is established combinatorially using the uniqueness of maximal dihedral reflection subgroups. For intervals of absolute length three, $[u, v]_T$ is always a lattice, leading to the quasi-Garside property of the associated interval group $G([1,w]_T)$ (Gobet, 2023).

In mathematical crystallography, affine dihedral subgroups of higher-dimensional cubic lattices generate tilings with exact local dihedral symmetries—these structures form the basis for explaining aperiodic quasicrystal tilings such as those with 8- and 10-fold symmetry, by projection into the Coxeter plane, where the action of $\tilde I_2(h)$ ensures local and global symmetry of the resultant configurations (Koca et al., 2023).

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References

• (Gobet, 2023) Gobet, T.: "On maximal dihedral reflection subgroups and generalized noncrossing partitions", 2023.
• (Douglass et al., 2011) Douglass, J.M., Pfeiffer, G., Röhrle, G.: "On reflection subgroups of finite Coxeter groups", 2011.
• (Koca et al., 2023) Koca, M., Koca, N.O., and Koca, E.: "Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices $\mathbb{Z}^n$ and Quasicrystallography", 2023.
• (Macauley, 2023) Lee, J.: "Dihedralizing the quaternions", 2023.