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Reflection group

In mathematics, particularly in group theory and geometry, a reflection group is a discrete group generated by a finite set of reflections in a Euclidean space, where each reflection is an isometry that fixes a hyperplane pointwise and negates vectors orthogonal to it.^[1] These groups arise as symmetry groups of geometric objects, such as regular polytopes or root systems, and can be finite or infinite depending on whether the reflections generate a bounded fundamental domain.^[2] Reflection groups are closely related to Coxeter groups, which provide an abstract algebraic framework where the generators satisfy relations of the form (s_i s_j)^{m_{ij}} = 1 for i \neq j, with m_{ij} determining the angle between reflection hyperplanes, and each s_i^2 = 1.^[1] Finite reflection groups preserve a positive definite bilinear form and act faithfully on the associated root system, a finite set of vectors closed under the group action and consisting of pairs \{\alpha, -\alpha\}.^[3] Examples include the dihedral group of order $2n, symmetries of a regular n-gon generated by two reflections at angle \pi/n, and the symmetric group S_n, which acts as reflections on the roots \epsilon_i - \epsilon_j in \mathbb{R}^n.^[1] The finite irreducible reflection groups over the reals are completely classified into four infinite families—A_n (n ≥ 1), B_n = C_n (n ≥ 2), D_n (n ≥ 4), I_2(p) (p ≥ 3)—and six exceptional types: E_6, E_7, E_8, F_4, G_2, H_3, H_4, corresponding to the Coxeter-Dynkin diagrams of those types.^[3] Among these, the crystallographic ones—those with root lengths differing by integer factors and angles yielding integer m_{ij}—are precisely the types A_n, B_n, D_n, E_6, E_7, E_8, F_4, G_2, which underpin the Weyl groups of semisimple Lie algebras.^[3] Infinite reflection groups, such as affine Weyl groups, extend these structures to tilings of Euclidean space and play roles in representation theory and combinatorics.^[2]

Definition and Fundamentals

Reflections in Euclidean Space

In Euclidean space, a reflection is defined as an orthogonal linear transformation that fixes every point on a hyperplane pointwise while negating vectors in the direction orthogonal to that hyperplane.^[4] This geometric action mirrors points across the hyperplane, preserving distances between points since orthogonal transformations maintain the Euclidean norm, and preserving angles up to sign due to the orientation-reversing nature of the reflection.^[5]^[6] Algebraically, given a hyperplane defined by its normal vector \alpha \neq 0, the reflection s_\alpha acting on a vector v is given by the formula

s_\alpha(v) = v - 2 \frac{v \cdot \alpha}{\alpha \cdot \alpha} \alpha.

This expression, known as the Householder reflection formula, projects v onto the direction of \alpha and subtracts twice that projection from v, effectively reflecting it across the hyperplane perpendicular to \alpha.^[7]^[8] Reflections possess key properties: they are involutions, satisfying s_\alpha^2 = \mathrm{id}, meaning applying the reflection twice returns every vector to itself; they have determinant -1, distinguishing them from rotations (which have determinant $1); and they serve as the fundamental building blocks for the [orthogonal group](/page/Orthogonal_group), as every [orthogonal transformation](/page/Orthogonal_transformation) can be expressed as a [composition](/page/Composition) of at most nreflections inn$-dimensional space, per the Cartan–Dieudonné theorem.^[9]^[6]^[10]

Generating Reflection Groups

A reflection group G is a discrete group of isometries of the Euclidean space V generated by a finite set of reflections \{ s_i \}, where each reflection s_i is an involution fixing a hyperplane pointwise and negating the direction normal to it.^[11]^[12] The discreteness condition ensures that G acts properly discontinuously on V, meaning every point in V has a neighborhood that intersects only finitely many G-translates of itself, with all stabilizers G_x being finite subgroups.^[13] For linear reflection groups (where the hyperplanes pass through the origin), G embeds as a discrete subgroup of the compact Lie group O(V), and thus must be finite.^[11] A basic theorem states that G is finitely generated precisely when the generating reflections correspond to a finite collection of reflecting hyperplanes, as the relations among these reflections suffice to describe the entire group structure.^[12] The union of these hyperplanes divides V into connected components called chambers, and the fundamental domain for the G-action is a simplicial cone bounded by portions of these hyperplanes, serving as a strict fundamental region where the group acts simply transitively on its images.^[11] By the orbit-stabilizer theorem applied to the G-action on the set of chambers, the orbits partition V minus the hyperplanes into equivalent chambers, with the stabilizer of a chamber being trivial and the orbit size equaling the order of G, reflecting the faithful tiling of space by these domains.^[13]^[11]

Examples of Finite Reflection Groups

Two Dimensions

In two dimensions, finite reflection groups acting irreducibly on the Euclidean plane \mathbb{R}^2 are precisely the dihedral groups D_n for n \geq 3, which are the symmetry groups of regular n-gons.^[14] These groups are generated by n reflections across lines passing through a vertex of the regular n-gon and its center (or through midpoints of opposite sides for even n), with the reflections' axes intersecting at the center and separated by angles of \pi/n.^[15] Equivalently, D_n can be generated by just two adjacent reflections, whose composition yields a rotation by $2\pi/n.^[14] The order of D_n is $2n, consisting of n rotations and n reflections.^[15] It admits the presentation \langle r, s \mid r^n = s^2 = (rs)^2 = 1 \rangle, where r is a rotation by $2\pi/n and s is a reflection.^[14] Geometrically, the action of D_n on \mathbb{R}^2 is such that a fundamental domain is a sector of angle \pi/n bounded by two adjacent reflection axes; the group's orbit of this domain tiles the plane with $2n congruent sectors, covering the full $2\pi around the center without overlap.^[14] These groups are the Coxeter groups of type I_2(n).^[15] All irreducible finite reflection groups in two dimensions are dihedral; reducible cases include direct products with cyclic or trivial groups acting on orthogonal lines, but the focus here is on the irreducible dihedral examples.^[14]

Three Dimensions

In three-dimensional Euclidean space, the finite reflection groups correspond to the full symmetry groups of the Platonic solids, generated by reflections across planes that bound the solid or its dual. These groups act faithfully on \mathbb{R}^3 and are classified as the irreducible Coxeter groups of types A_3, B_3, and H_3. Unlike in two dimensions, where all finite reflection groups are dihedral, the three-dimensional cases introduce more complex polyhedral symmetries, with the groups serving as the Coxeter groups associated with these regular polyhedra. The tetrahedral reflection group, of Coxeter type A_3, has order 24 and is generated by reflections across the six planes that pass through an edge and the midpoint of the opposite edge of a regular tetrahedron. This group is isomorphic to the symmetric group S_4, acting as all even and odd permutations of the four vertices of the tetrahedron. It preserves the tetrahedron and its dual, which is itself, and includes orientation-reversing isometries such as improper rotations combined with reflections.^[11] The octahedral reflection group, of Coxeter type B_3, has order 48 and arises as the symmetry group of the regular octahedron or its dual, the cube. It is generated by reflections across the nine planes that bisect the edges of the octahedron: three coordinate planes and six diagonal planes at 45 degrees to the axes. This group is isomorphic to the hyperoctahedral group of signed permutations on three elements, S_4 \times \mathbb{Z}/2\mathbb{Z}, and includes all symmetries that map the octahedron to itself, such as reflections through faces, edges, and vertices.^[16] The icosahedral reflection group, of Coxeter type H_3, has order 120 and is the full symmetry group of the regular icosahedron or its dual, the dodecahedron. It is generated by reflections across the 15 planes that pass through an edge and the midpoints of opposite edges of the icosahedron. This group is isomorphic to A_5 \times \mathbb{Z}/2\mathbb{Z}, where A_5 is the alternating group on five elements, reflecting the fivefold rotational symmetries around vertices. Unlike the other two, H_3 is non-crystallographic, meaning its root system does not correspond to a Lie algebra over the rationals.^[11] Geometrically, each of these reflection groups acts on the 2-sphere (the unit sphere in \mathbb{R}^3) by orthogonal transformations, with the fundamental domain being a spherical triangle whose angles are determined by the Coxeter diagram's labels (such as \pi/2, \pi/3, \pi/5 for H_3). Tiling the sphere with 24, 48, or 120 copies of this triangle, respectively, yields the orbit of the group action, illustrating how reflections generate the full symmetry. These groups also appear as Weyl groups in the classification of semisimple Lie algebras, providing a bridge to representation theory.^[16]

Higher Dimensions

Finite reflection groups in dimensions n \geq 4 are classified into irreducible and reducible types, with the irreducible ones falling into infinite classical families and a finite number of exceptional cases. The classical series consist of types A_{n-1}, B_n = C_n, and D_n (for n \geq 4), while the exceptional irreducible groups of rank at least 4 are E_6, E_7, E_8, F_4, and H_4. The group of type A_{n-1} is isomorphic to the symmetric group S_n, realized geometrically as the group generated by reflections across the hyperplanes x_i = x_j (for i < j) in the (n-1)-dimensional subspace of \mathbb{R}^n orthogonal to the all-ones vector; its order is n!, and it has $2n(n-1) roots corresponding to the differences e_i - e_j. The types B_n and C_n coincide and form the hyperoctahedral group of signed permutations, acting on \mathbb{R}^n via reflections across the coordinate hyperplanes x_i = 0 and the diagonals x_i = \pm x_j (for i < j); the order is $2^n n!. The group D_n is the index-2 subgroup of B_n consisting of even signed permutations, generated by reflections across x_i = \pm x_j but excluding odd numbers of sign flips; its order is $2^{n-1} n!. The exceptional groups include F_4 and H_4 in dimension 4, E_6 in dimension 6, E_7 in dimension 7, and E_8 in dimension 8. The orders are 1152 for F_4, 14400 for H_4, 51840 for E_6, 2903040 for E_7, and 696729600 for E_8. Geometrically, F_4 arises as the symmetry group of the 24-cell, while H_4 is the symmetry group of the 120-cell and 600-cell. All irreducible finite reflection groups of rank n act faithfully and irreducibly on \mathbb{R}^n, with a fundamental chamber that is an n-simplex bounded by the reflecting hyperplanes of the simple reflections. The full set of reflecting hyperplanes partitions \mathbb{R}^n into |W| chambers, where |W| is the group order, and the group acts transitively on them. Reducible finite reflection groups are direct products of irreducible ones acting on the orthogonal direct sum of their representation spaces; for example, the symmetry group of a rectangular box in \mathbb{R}^n is B_1 \times \cdots \times B_1 (n factors).

Connection to Coxeter and Weyl Groups

As Coxeter Groups

A real reflection group admits an abstract presentation as a Coxeter group, capturing its structure through a finite generating set of reflections and their relations. Specifically, a Coxeter system is a pair (W, S), where W is a group generated by a finite set S of involutions (corresponding to reflections), satisfying the relations s_i^2 = 1 for all s_i \in S and (s_i s_j)^{m_{ij}} = 1 for i \neq j, with m_{ii} = 1 and m_{ij} \geq 2 (or \infty if the order is infinite).^[17] In this presentation, the elements of S are the simple reflections whose fixed hyperplanes bound a fundamental chamber of the group action.^[17] The relations in the Coxeter system are conveniently encoded in a Coxeter diagram, a graph with vertices labeled by the elements of S. Vertices s_i and s_j are connected by an edge if m_{ij} \geq 3; the edge is unlabeled if m_{ij}=3, labeled by the integer m_{ij} if m_{ij} \geq 4, and labeled \infty if m_{ij} = \infty; if m_{ij} = 2, there is no edge.^[17] Finite Coxeter diagrams (without \infty labels) classify the possible finite reflection groups up to isomorphism, with the diagram's connectedness reflecting the group's irreducibility.^[17] For instance, the dihedral group arises from a diagram with two vertices connected by an edge labeled n, corresponding to m_{12} = n.^[17] A fundamental theorem establishes the equivalence between geometric and abstract structures: a discrete reflection group acting on Euclidean space is a Coxeter group if and only if it is generated by reflections whose mirrors meet at dihedral angles \pi / m_{ij} for integers m_{ij} \geq 2.^[12] Here, the exponents m_{ij} directly determine the braid relations in the presentation, linking the group's geometry to its algebraic relations.^[12] Regarding finiteness, the spherical Coxeter groups—those for which the associated symmetric bilinear form on the vector space with basis S (defined by \langle s_i, s_j \rangle = -\cos(\pi / m_{ij})) is positive definite—are precisely the finite real reflection groups.^[17] This criterion ensures the group action is cocompact on a sphere, distinguishing finite cases from their infinite counterparts like affine groups.^[17]

Weyl Groups and Root Systems

In the context of finite reflection groups, Weyl groups arise as the symmetry groups associated with root systems, which are finite sets of vectors in a Euclidean space satisfying specific geometric properties. A root system Φ is a finite subset of a real Euclidean vector space V, spanning V, such that for each root α ∈ Φ, its reflection s_α(v) = v - 2(α, v)/(α, α) α maps Φ to itself, and the only scalar multiples of α in Φ are ±α. The Weyl group W(Φ) is the finite subgroup of the orthogonal group O(V) generated by these reflections {s_α | α ∈ Φ}, acting faithfully on V. A key distinguishing feature of Weyl groups among reflection groups is the crystallographic condition: the reflections s_α preserve the root lattice ℤΦ = {∑ n_α α | n_α ∈ ℤ, α ∈ Φ}, ensuring that the group has a natural integral structure compatible with lattice symmetries in crystallography. This condition implies that the Cartan integers ⟨α, β⟩ = 2(α, β)/(β, β) are integers for all α, β ∈ Φ, leading to an integer-valued Cartan matrix A = (a_{ij}) for a choice of simple roots {α_1, ..., α_n} ⊂ Φ, where a_{ij} = ⟨α_i, α_j⟩ and a_{ii} = 2. The diagonal entries reflect possible differing root lengths, with off-diagonal entries a_{ij} ∈ {-3, -2, -1, 0} determining the angles between simple roots.^[18] The irreducible Weyl groups are classified up to isomorphism by their associated Dynkin diagrams, which encode the Cartan matrix via nodes for simple roots and bonds between adjacent nodes: single, double, or triple bonds indicating |a_{ij}|=1, 2, or 3, respectively, with directed arrows pointing from longer to shorter roots when lengths differ.^[18] They are classified into four infinite families—A_n (n ≥ 1), B_n (n ≥ 2), C_n (n ≥ 2), D_n (n ≥ 4)—and five exceptional types: E_6, E_7, E_8, F_4, and G_2.^[19] Among these, the simply-laced types—A_n, D_n, E_6, E_7, E_8—have all roots of equal length, corresponding to Dynkin diagrams without directed edges and underlying the classical Lie algebras of types ADE. The Weyl group W(Φ) preserves a partial order on roots induced by a choice of positive roots Φ^+, with simple roots forming a basis for this cone. Elements w ∈ W(Φ) act by permuting roots while preserving Φ^+, and the length function l(w) with respect to the generating set of simple reflections S = {s_{α_i}} equals the number of inversions, i.e., the cardinality of {α ∈ Φ^+ | w(α) ∈ -Φ^+}. This length function governs the Bruhat order on W(Φ) and plays a central role in applications to representation theory and Lie algebras, where W(Φ) is the Weyl group of the semisimple Lie algebra with root system Φ.^[20]

Reflection Groups over Finite Fields

Definition and Basic Properties

In the context of linear algebra over finite fields, reflection groups are studied as subgroups of the general linear group acting on a vector space. Let V be an n-dimensional vector space over the finite field \mathbb{F}_q, where q is a power of a prime, and let GL(V) denote the group of invertible linear endomorphisms of V. Assuming characteristic not 2, a reflection is defined as an element g \in GL(V) such that the fixed-point subspace \Fix(g) = \{ v \in V \mid g v = v \} has codimension 1 in V, and g acts as multiplication by -1 on the one-dimensional quotient space V / \Fix(g). This ensures that g fixes a hyperplane pointwise while acting non-trivially on the quotient, generalizing the notion of reflections to this setting.^[21]^[22] A reflection group G is a subgroup of GL(V) generated by a set of such reflections. These groups are finite and act semisimply on V, meaning the natural module V is completely reducible as a representation of G. Reflections within G are precisely the elements acting as -1 on a codimension-1 subspace. The natural action of G on V, known as the reflection representation, decomposes into a direct sum of irreducible G-modules. This decomposition highlights the modular representation theory over \mathbb{F}_q, where the structure mirrors aspects of the characteristic-zero case but accounts for the finite nature of the point set in V. In analogy to reflection groups over the real numbers—where reflections are isometries fixing hyperplanes (mirrors)—the finite-field version acts linearly on the finite set of points in the affine space associated to V, with hyperplanes serving as analogous mirrors but over discrete point sets.

Key Examples and Constructions

One prominent class of reflection groups over finite fields \mathbb{F}_q (with q odd) consists of the monomial groups generated by reflections that permute a basis and change signs of coordinates. These groups are isomorphic to the wreath product \mathbb{Z}_2 \wr S_n acting faithfully on the standard module \mathbb{F}_q^n, where the generators are the diagonal matrices with a single -1 entry and the identity elsewhere (for sign changes) combined with permutation matrices. This construction parallels the hyperoctahedral group B_n in the classical setting and arises as irreducible reflection groups of Coxeter type B_n reduced modulo an odd prime.^[21]^[22] In reductive algebraic groups defined over \mathbb{F}_q, Borel subgroups and more generally parabolic subgroups provide settings where reflection subgroups emerge naturally. For instance, in Chevalley groups such as SL_n(\mathbb{F}_q), the Borel subgroup of upper triangular matrices contains the Weyl subgroup generated by reflections corresponding to simple roots, acting on the Lie algebra or standard module. Parabolic subgroups, as stabilizers of flags in the associated Tits building, often contain maximal reflection subgroups corresponding to the Weyl group, yielding finite groups generated by reflections of Coxeter type. These constructions are irreducible when the ambient representation is, and their structure is determined by the underlying Dynkin diagram reduced over \mathbb{F}_q.^[23]^[24] Exceptional examples of reflection groups over finite fields include finite analogs of higher-rank Weyl groups like those of type E_8 in low dimensions and small q. For q=5, the group H_{3,5} \cong O_1(3,5) acts irreducibly on a 3-dimensional module over \mathbb{F}_5 as a reflection group of rank 3, with diagram [3,5], distinct from classical types due to the characteristic. Similarly, the Weyl subgroup of the Suzuki group Sz(8) \cong ^2B_2(8), which is dihedral, admits a reflection representation reflecting its twisted Chevalley structure; the Weyl group of the Ree group ^2G_2(3) likewise realizes as a rank-2 dihedral reflection group on a 2-dimensional module over \mathbb{F}_3. These cases highlight non-crystallographic or twisted types viable only in positive characteristic.^[22]^[25] Reflection groups in Chevalley groups G(\mathbb{F}_q) can be constructed systematically via the BN-pair structure, where the normalizer N_G(T) of a maximal split torus T contains the reflections, and the quotient W = N_G(T)/T is the finite Coxeter group generated by images of root reflections. This yields the Weyl group as a reflection subgroup of GL(\Phi, \mathbb{F}_q) in the root space representation, with |W| independent of q but the embedding sensitive to the field; for example, in type A_{n-1}, W \cong S_n acts on \mathbb{F}_q^{n-1} via the permutation representation modulo the trace. Such constructions ensure the group is generated by the simple reflections s_i satisfying the Coxeter relations, and subgroups thereof form parabolic reflection groups.

Infinite Reflection Groups

Affine Reflection Groups

Affine reflection groups, also known as affine Weyl groups, are infinite discrete subgroups of the affine isometry group of Euclidean space that extend the finite Weyl groups associated with root systems of semisimple Lie algebras. For a finite irreducible root system \Phi in a Euclidean vector space V, the corresponding affine Weyl group \tilde{W} is defined as the semidirect product \tilde{W} = W \ltimes \Lambda, where W is the finite Weyl group generated by reflections in the hyperplanes perpendicular to the roots in \Phi, and \Lambda is the coroot lattice spanned by the coroots \{\alpha^\vee \mid \alpha \in \Phi\}.^[26] This structure arises naturally in the context of affine Lie algebras and captures the symmetries of crystallographic arrangements in higher dimensions.^[27] The generators of \tilde{W} consist of the finite reflections s_\alpha for \alpha \in \Phi, together with affine reflections s_{\alpha + k\delta} for \alpha \in \Phi, integers k \in \mathbb{Z}, and \delta the basic imaginary root, which introduces a periodic extension to the root system. These affine reflections act on the space V by isometries that fix affine hyperplanes of the form \{x \in V \mid \langle x, \alpha \rangle = k\}, combining linear reflections with translations by elements of the coroot lattice. The action of \tilde{W} on V is faithful and discrete, tiling the space with copies of a fundamental domain known as the alcove—a bounded open simplex contained within the fundamental chamber of the finite Weyl group W, specifically the set of points x \in V satisfying $0 < \langle x, \alpha_i \rangle < 1 for all simple roots \alpha_i.^[27] The closure of this alcove serves as a fundamental domain for the action, ensuring that \tilde{W} acts cocompactly on V.^[26] As Coxeter groups, affine reflection groups admit a presentation via an affine Coxeter system (\tilde{W}, \tilde{S}), where the generating set \tilde{S} includes the simple reflections of W plus one additional affine reflection, corresponding to the highest root. The associated Coxeter diagram is obtained by adjoining a new node to the finite Dynkin diagram of \Phi, connected according to the Cartan integers, resulting in an infinite group despite all braid relations having finite order. This infinite order stems from the unbounded translations inherent in the semidirect product, distinguishing affine groups from their finite counterparts. The affine root system \tilde{\Phi} = \Phi \cup \{\alpha + k\delta \mid \alpha \in \Phi, k \in \mathbb{Z}\} is preserved by \tilde{W}, providing a geometric realization of the extended symmetries. For example, in type A_1, the affine Weyl group is generated by reflections across lines at integer positions on the real line, tiling it with intervals of length 1.^[28]

Hyperbolic and Other Infinite Cases

Hyperbolic Coxeter groups are infinite discrete groups generated by reflections acting on hyperbolic space \mathbb{H}^n, where the fundamental domain is a hyperbolic simplex bounded by hyperplanes corresponding to the reflecting hyperplanes, or mirrors.^[29] These groups arise from Coxeter systems whose associated bilinear form has indefinite signature, leading to actions on spaces of constant negative curvature.^[29] A representative example in two dimensions is the hyperbolic triangle group (2,3,\infty), generated by reflections across the sides of an ideal triangle in \mathbb{H}^2 with angles \pi/2, \pi/3, and $0.^[30] In higher dimensions, such groups correspond to Coxeter diagrams where at least one pair of nodes is unconnected (indicating m_{ij} = \infty), ensuring the group is infinite and acts discretely on \mathbb{H}^n.^[29] These groups act freely and properly on the complement of their mirrors in \mathbb{H}^n, with the mirrors forming a tessellation of the space.^[29] The volume of the fundamental domain is finite when the Coxeter polyhedron is ideal, meaning its vertices lie on the boundary at infinity, as in the case of crystallographic reflection groups with finite-volume polytopes.^[31] Other infinite cases include Euclidean reflection groups beyond the irreducible parabolic (affine) ones, such as reducible groups acting on Euclidean space \mathbb{E}^n via reflections, which can produce infinite-volume fundamental domains like infinite simplices or strips. For example, the infinite dihedral group generated by reflections across two parallel lines in \mathbb{E}^2 has an infinite strip as its fundamental domain.^[28] Lorentzian reflection groups operate in indefinite quadratic spaces, such as Minkowski space, where the bilinear form has signature (n,1), generating discrete subgroups that may not preserve a positive definite metric but still tile the space with simplices.^[32]

Generalizations Beyond Real Euclidean Spaces

Complex Reflection Groups

A pseudo-reflection in the general linear group GL(n, ℂ) is a non-identity element that fixes a hyperplane pointwise and acts as multiplication by a primitive m-th root of unity (for some m > 1) on the one-dimensional quotient space orthogonal to that hyperplane.^[33] This generalizes the notion of a real reflection, which corresponds to the case m=2 where the action is multiplication by -1 on the normal line.^[33] A complex reflection group is a finite subgroup G of GL(n, ℂ) that is generated by pseudo-reflections and acts faithfully and linearly on the complex vector space ℂ^n.^[33] Such groups extend the classical real reflection groups to the complex setting, preserving key structural properties like the existence of a polynomial ring of invariants.^[33] By the Shephard-Todd-Chevalley theorem, G is a complex reflection group if and only if the ring of invariants ℂ[ℂ^n]^G is a polynomial algebra generated by n algebraically independent homogeneous polynomials.^[33] The irreducible complex reflection groups were fully classified by Shephard and Todd in 1954, up to conjugacy in GL(n, ℂ).^[33] The classification consists of an infinite family of groups, denoted G(m, p, n) where m ≥ 1, p divides m, and n ≥ 2 is the dimension (with the order of G(m, p, n) given by m^n n! / p), alongside 34 primitive exceptional groups labeled G_4 through G_{37}.^[33] For example, the group G(6, 3, 2) is an imprimitive group of order 24 acting on ℂ^2, while G_4 is a primitive example of order 24.^[33] The well-generated cases, including complexifications of real Coxeter groups, are embedded within this classification; for instance, the symmetric groups correspond to G(1,1,n), dihedral groups to G(m,m,2), and hyperoctahedral groups to G(m,m,n).^[33] A complex reflection group G is the complexification of a real reflection group if it is conjugate in GL(n, ℂ) to a subgroup of real matrices, in which case its reflections include real ones and its structure mirrors finite Coxeter groups.^[33] More generally, the order of G equals the product of the degrees d_1, ..., d_n of its basic invariant polynomials, where the degrees satisfy d_i = m_i + 1 and the m_i are the exponents of G (non-negative integers such that the coinvariant algebra has Poincaré series ∏ (1 + t^{m_i + 1}) / (1 - t^{m_i + 1})).^[33] For instance, in G(m, p, n), the degrees are m, 2m, ..., (n-1)m, and (m n)/p.^[33] These degrees provide a complete invariant for the group up to isomorphism in the irreducible case.^[33]

Pseudo-Reflection Groups and Further Extensions

Pseudo-reflection groups provide a unifying framework for reflection groups across various fields and structures, extending the classical real and complex cases. In a finite-dimensional vector space V over a field k of characteristic zero, a pseudo-reflection is a non-identity element g \in \mathrm{GL}(V) of finite order whose fixed subspace V^g = \{v \in V \mid gv = v\} has codimension one in V. A pseudo-reflection group is then a finite subgroup G \leq \mathrm{GL}(V) generated by pseudo-reflections. Over the complex numbers \mathbb{C}, these groups are precisely the complex reflection groups, whose irreducible representations were classified by Shephard and Todd into the infinite family G(m,p,n) (with p dividing m)—which includes the symmetric groups S_n = G(1,1,n), the groups G(m,1,n), and G(m,m,n)—along with 34 exceptional primitive cases labeled G_4 to G_{37}.^[33] The cornerstone result for these groups is the Chevalley–Shephard–Todd theorem, which links their generation to the structure of invariant rings. For a finite subgroup G \leq \mathrm{GL}(V, \mathbb{C}), the ring of invariants \mathbb{C}[V]^G is a polynomial algebra if and only if [G](/page/G) is a pseudo-reflection group; Chevalley established the sufficiency for real reflection groups, Shephard and Todd proved the full equivalence over \mathbb{C}, and Serre extended the necessity (polynomial invariants imply generation by pseudo-reflections) to arbitrary fields of characteristic zero.^[33] This theorem highlights the polynomial nature of invariants, with degrees given by the Shephard-Todd numbers d_1, \dots, d_n satisfying |G| = \prod d_i, and has profound implications for singularity theory, as the quotient V/[G](/page/G) is then smooth.^[33] In positive characteristic p > 0, the theory requires adjustments due to non-semisimplicity of elements. Serre's necessity result persists: if k[V]^G is polynomial, then G is generated by pseudo-reflections, where pseudo-reflections are now defined similarly but without assuming diagonalizability. However, the converse fails; for example, the Weyl group of type A_2 in characteristic 3 has non-polynomial invariants despite being generated by reflections. Broer provided a modular analogue, proving that for an irreducible kG-module V, the action is coregular—meaning k[V]^G is generated by algebraically independent homogeneous elements and k[V] is a free k[V]^G-module—if and only if G is generated by pseudo-reflections and satisfies the direct summand property (i.e., k[V] is a direct summand of k[V]^H as k[V]^H-modules for certain subgroups H). Further extensions address actions over rings and schemes. Over Dedekind domains R (integral domains with all localizations at primes being principal ideal domains, such as rings of integers in number fields), Mundelius generalized the Chevalley–Shephard–Todd theorem: for a pseudo-reflection group G acting faithfully on a free R-module V of finite rank, the invariant ring R[V]^G is arithmetically Cohen–Macaulay and regular in codimension one, preserving key multiplicity and depth properties analogous to the polynomial case over fields.^[34] In the setting of finite linearly reductive group schemes (étale over fields of characteristic zero or flat over perfect fields), Satriano extended the theorem: if G acts faithfully on a vector space V and is generated by pseudo-reflection subgroup schemes (those with fixed locus of codimension one), then k[V]^G is polynomial, with the converse under a stability condition ensuring reducedness.^[35] These developments connect pseudo-reflection groups to arithmetic geometry, stack theory, and modular representation theory, with applications to quotient singularities and essential dimensions.^[34]^[35]

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Dec 5, 2017 · We have defined reflectiong groups to be groups generated by reflections in Euclidean space. However, looking at the examples in Section 2.4, it ...
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[9]
[PDF] Lecture 8: Examples of linear transformations
Feb 16, 2011 · Reflections have the property that they are their own inverse. If we combine a reflection with a dilation, we get a reflection-dilation.
[10]
[PDF] The Cartan–Dieudonné Theorem - CIS UPenn
The Cartan–Dieudonné Theorem. 7.1 Orthogonal Reflections. In this chapter the structure of the orthogonal group is studied in more depth. In particular, we ...
[11]
[PDF] COXETER GROUPS (Unfinished and comments are welcome)
In dimension n = 4 there are three additional regular polytopes, and all their symmetry groups are finite reflection groups [23], [4]. Reflection groups were ...
[12]
[PDF] Discrete Groups Generated by Reflections
Of the 230 space groups, the following seven are generated by reflections: ... the spherical or Euclidean space in which the given group operates, beginning.
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[PDF] REFLECTION GROUPS ON RIEMANNIAN MANIFOLDS
If G is a discrete subgroup ... Then there exists a unique orthogonal representation of G as a linear reflection group on an Euclidean space Rn such that the si ...
[14]
[PDF] Reflection Groups - Cornell Mathematics
Oct 19, 2016 · The Coxeter graph of the symmetry group of your polytope. • How certain Schläfli symbols determine tesselations instead of regular polytopes.
[15]
[PDF] Introduction to finite Coxeter groups and their representations - arXiv
Sep 15, 2025 · In two dimensions, the dihedral groups describe the symmetries of regular n-gons and it can be easily seen that dihedral groups are generated ...
[16]
[PDF] The Mathematics of Mirrors and Kaleidoscopes
Table 1. Finite reflection groups in three dimensions. The symmetry group B3 of the cube or octahedron consists of the signed permuta- tions of three objects ( ...
[17]
[PDF] Lectures on Coxeter groups
A finite Coxeter group is called a Weyl group if it has a reflection representa- tion over Q. These groups are particularly important in mathematics; they are.
[18]
https://www.springer.com/gp/book/9783540642422
[19]
[PDF] CRYSTALLOGRAPHIC ROOT SYSTEMS
Definition 3.1.3 (Weyl Group). Let W be a subgroup of GL(E) generated by re ... We also note that the Cartan matrix is the n ⇥ n matrix. 0. B. B. B. B. B. B.
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invariants of finite groups generated by pseudo-reflections in
with $lT_{n}(R)$ . LEMMA 2.2. Let $G\subseteqq GL_{n}(R)$ be a finite subgroup of monomial form generated ... Reflection groups over finite fields of ...<|control11|><|separator|>
[21]
(PDF) Finite linear groups generated by reflections - ResearchGate
Aug 7, 2025 · PDF | A complete classification is given of all finite irreducible linear groups generated by reflections over an arbitrary field of ...Missing: Zalesskii | Show results with:Zalesskii
[22]
Reflection groups and polytopes over finite fields, I | Request PDF
Aug 10, 2025 · In a series of three papers, Monson and Schulte studied families of crystallographic Coxeter groups with string diagrams [MS2, MS3, MS4]. In ...
[23]
Reflection groups and polytopes over finite fields, I - ScienceDirect
In many cases, the latter group will be the symmetry group of a finite regular polytope. In this paper, we investigate several such families of polytopes and ...
[24]
[PDF] On the simple groups of Suzuki and Ree
If r is a short root, so that rr = 1, then reflection in r is the map z 7→ −rzr, while if r is a long root, so that rr = p, it is the map z 7→ −rzr/p. The ...
[25]
[PDF] Affine Weyl Groups
I L(Φ∨)# =: bL(Φ) the weight lattice. Proposition. The affine Weyl group of Φ. Wa(Φ) := hsα,k | α ∈ Φ, k ∈ Zi ∼= L(Φ∨) : W(Φ) is the semidirect product of ...
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[PDF] Weyl groups, affine
The Weyl group of the root system is the subgroup of GL(V) generated by the reflections sa for a € . Note that W is a crystallographic finite reflection group ...
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[PDF] Reflection groups and Coxeter groups
Remark. Given a root system ☀ and corresponding reflection group W, define ☀' to be the set of unit vectors proportional to the vectors in §.
[28]
[PDF] Hyperbolic Coxeter Groups Gabor Moussong - OSU Math
An important class of discrete transformation groups is the class of reflection groups, or, in an abstract situation, Coxeter gruops. They occur naturally in a ...
[29]
[PDF] Complex Hyperbolic Triangle Groups - arXiv
For instance, the (2, 3, ∞)-reflection triangle group is commensurable to the classical modular group. The reflection triangle groups are rigid in Isom(H3), in ...
[30]
[PDF] Hyperbolic reflection groups
and "reflection group" means "group generated by reflections". ... If Κ Φ Q, or Κ = Q but / does not represent zero in Q, then a fundamental domain ...
[31]
[PDF] Coxeter groups, Lorentzian lattices, and K3 surfaces. - Berkeley Math
By taking Π to be Conway's Coxeter diagram of the reflection group of II1,25 we compute the automorphism groups of some Lorentzian lattices and K3 surfaces.Missing: indefinite metric
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[PDF] FINITE UNITARY REFLECTION GROUPS - UCSD Math
These groups are all known, and many of them have. Page 5. 278. G. C. SHEPHARD AND J. A. TODD been extensively studied. We shall list them here, together with ...Missing: classification | Show results with:classification
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[PDF] FINITE UNITARY REFLECTION GROUPS - UCSD Math
A reflection in unitary space is a congruent transformation of finite period that leaves invariant every point of a certain prime, and it is characterised by ...
[34]
Arithmetic invariants of pseudoreflection groups and regular graded ...
Nov 22, 2017 · Title:Arithmetic invariants of pseudoreflection groups and regular graded algebras. Authors:David Mundelius. View a PDF of the paper titled ...
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[PDF] Shephard--Todd theoremfor finite linearly reductive group schemes
We say that a subgroup scheme N of G is a pseudoreflection if V N has codimension 1 in V. We define the subgroup scheme generated by pseudoreflections to be the.