Orthogonal group

In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F) given by

$\mathrm{O}(n,F) = \{ Q \in \mathrm{GL}(n,F) \mid Q^T Q = Q Q^T = I \}$

where Q^T is the transpose of Q. The classical orthogonal group over the real numbers is usually just written O(n).

More generally the orthogonal group of a non-singular quadratic form over F is the group of linear operators preserving the form (the above group O(n, F) is then the orthogonal group of the sum-of-n-squares quadratic form). The Cartan–Dieudonné theorem describes the structure of the orthogonal group.

The special orthogonal group, SO(n,F), is the kernel of the Dickson invariant and usually has index 2 in O(n,F).^[1] When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n,F) is commonly defined to be the elements of O(n,F) with determinant 1. Each element in O(n,F) has determinant −1 or 1. Thus in characteristic 2, the determinant is always 1. By analogy with GL/SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO.

The derived subgroup Ω(n,F) of O(n,F) is an often studied object because when F is a finite field Ω(n,F) is often a finite simple group, a Chevalley group of type D_n. By the Jordan–Hölder Theorem, the finite simple groups are building blocks of finite groups much as the prime numbers are to the integers. The classification of finite simple groups is believed to be complete making Ω(n,F) one of many groups central to the study of group theory.

Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix.

Group theory

Group theory

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product

Finite groups and classification of finite simple groups
Cyclic group Z_n Symmetric group, S_n Dihedral group, D_n Alternating group A_n Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄ Conway groups Co₁, Co₂, Co₃ Janko groups J₁, J₂, J₃, J₄ Fischer groups F₂₂, F₂₃, F₂₄ Baby Monster group B Monster group M

Discrete groups and lattices
The integers, Z Lattice (group) Modular groups, PSL(2,Z) and SL(2,Z)

Topological groups and Lie groups
Solenoid (mathematics) Circle group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G₂ F₄ E₆ E₇ E₈ Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

Lie groups

Classical groups
General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)

Simple Lie groups
List of simple Lie groups Infinite simple Lie groups: A_n, B_n, C_n, D_n, Exceptional simple Lie groups: G₂ F₄ E₆ E₇ E₈

Other Lie groups
Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group

Lie algebras
Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry

Structure of semi-simple Lie groups
Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra

Representation theory
Representation of a Lie group Representation of a Lie algebra

Lie groups in Physics
Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

1 Over the real number field
2 Over the complex number field
3 Topology
- 3.1 Low dimensional
- 3.2 Homotopy groups
4 Over finite fields
5 The Dickson invariant
6 Orthogonal groups of characteristic 2
7 The spinor norm
8 Galois cohomology and orthogonal groups
9 Related groups
10 Applications to string theory
11 Principal homogeneous space: Stiefel manifold
12 See also
13 Notes
14 References
15 External links

Over the real number field

Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n,R) has two connected components, with SO(n,R) being the identity component, i.e., the connected component containing the identity matrix.

The real orthogonal and real special orthogonal groups have the following geometric interpretations

O(n,R) is a subgroup of the Euclidean group E(n), the group of isometries of Rⁿ; it contains those that leave the origin fixed – $O(n,\mathbf{R}) = E(n) \cap GL(n,\mathbf{R}).$ It is the symmetry group of the sphere (n = 3) or hypersphere and all objects with spherical symmetry, if the origin is chosen at the center.

SO(n,R) is a subgroup of E⁺(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – $SO(n,\mathbf{R}) = E^+(n) \cap GL(n,\mathbf{R}) = E(n) \cap GL^+(n,\mathbf{R}).$ It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{ I, −I } is a normal subgroup and even a characteristic subgroup of O(n,R), and, if n is even, also of SO(n,R). If n is odd, O(n,R) is the direct product of SO(n,R) and { I, −I }. The cyclic group of k-fold rotations C_k is for every positive integer k a normal subgroup of O(2,R) and SO(2,R).

Relative to suitable orthogonal bases, the isometries are of the form:

$\begin{bmatrix} \begin{matrix}R_1 & & \\ & \ddots & \\ & & R_k\end{matrix} & 0 \\ 0 & \begin{matrix}\pm 1 & & \\ & \ddots & \\ & & \pm 1\end{matrix} \\ \end{bmatrix}$

where the matrices R₁,...,R_k are 2-by-2 rotation matrices. The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,^{[note 1]} and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map $v \mapsto -v$ ), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

The symmetry group of a circle is O(2,R), also called Dih (S¹), where S¹ denotes the multiplicative group of complex numbers of absolute value 1.

SO(2,R) is isomorphic (as a Lie group) to the circle S¹ (circle group). This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix

$\begin{bmatrix}\cos(\phi)&-\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}.$

The group SO(3,R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering. See rotation group and the general formula for a 3 × 3 rotation matrix in terms of the axis and the angle.

In terms of algebraic topology, for n > 2 the fundamental group of SO(n,R) is cyclic of order 2, and the spinor group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the spinor group Spin(2) is the unique 2-fold cover).

Even and odd dimension

The structure of the orthogonal group differs in certain respects between even and odd dimensions – for example, $-I$ (reflection through the origin) is orientation-preserving in even dimension, but orientation-reversing in odd dimension. When this distinction wishes to be emphasized, the groups are generally denoted O(2k) and O(2k+1), reserving n for the dimension of the space ( $n=2k$ or $n=2k+1$ ). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is $B_r = so_{2r+1},$ while in even dimension the Lie algebra is $D_r = so_{2r}.$

Lie algebra

The Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skew-symmetric real n-by-n matrices, with the Lie bracket given by the commutator. This Lie algebra is often denoted by o(n,R) or by so(n,R), and called the orthogonal Lie algebra or special orthogonal Lie algebra. These Lie algebras are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension $B_r = so_{2r+1},$ while in even dimension $D_r = so_{2r}.$

More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2-blades) $v \wedge w$ . The correspondence is given by the map $v \wedge w \mapsto v^* \otimes w - w^* \otimes v,$ where $v^*$ is the covector dual to the vector v; in coordinates these are exactly the elementary skew-symmetric matrices.

This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras $so_{p,q}.$

The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called spin representation, which are important in physics.

3D isometries that leave the origin fixed

Isometries of R³ that leave the origin fixed, forming the group O(3,R), can be categorized as:

SO(3,R):
- identity
- rotation about an axis through the origin by an angle not equal to 180°
- rotation about an axis through the origin by an angle of 180°
the same with inversion in the origin (x is mapped to −x), i.e. respectively:
- inversion in the origin
- rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis
- reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.

See also the similar overview including translations.

Conformal group

Being isometries (preserving distances), orthogonal transforms also preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (Side-Side-Side) congruence of triangles and AAA (Angle-Angle-Angle) similarity of triangles. The group of conformal linear maps of Rⁿ is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: $\operatorname{CO}(2k+1) = \operatorname{O}(2k+1) \times \mathbf{R}$ , while if n is even, these subgroups intersect in $\pm 1$ , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: $\operatorname{CO}(2k) = \operatorname{O}(2k) \times \mathbf{R}^+$ .

Similarly one can define CSO(n); note that this is always : $\operatorname{CSO}(n)�:= \operatorname{CO}(n) \cap \operatorname{GL}_+(n) = \operatorname{SO}(n) \times \mathbf{R}^+$ .

Over the complex number field

Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n − 1)/2 over C (which means the dimension over R is twice that). O(n,C) has two connected components, and SO(n,C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n,C) is not simply connected. For n > 2 the fundamental group of SO(n,C) is cyclic of order 2 whereas the fundamental group of SO(2,C) is infinite cyclic.

The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator.

Topology

Low dimensional

The low dimensional (real) orthogonal groups are familiar spaces:

$\begin{align} \mbox{O}(1) &= \left\{\pm 1\right\} = S^0\\ \mbox{SO}(1) &= \left\{1\right\} = *\\ \mbox{SO}(2) &= S^1\\ \mbox{SO}(3) &= \mathbf{RP}^3 \end{align}$

The group $\mbox{SO}(4)$ is double covered by $\mbox{SU}(2)\times \mbox{SU}(2)=S^3\times S^3$ .

Here Sⁿ denotes the n-dimensional sphere, RPⁿ the n-dimensional real projective space, and SU(n) the special unitary group of degree n.

Homotopy groups

The homotopy groups of the orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute.

However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions

$\mbox{O}(0) \subset \mbox{O}(1)\subset \mbox{O}(2)\subset\cdots\subset O = \bigcup_{k=0}^\infty \mbox{O}(k)$

(as the inclusions are all closed inclusions, hence cofibrations, this can also be interpreted as a union).

$S^n$ is a homogeneous space for $\mbox{O}(n+1)$ , and one has the following fiber bundle:

$\mbox{O}(n) \to \mbox{O}(n+1) \to S^n,$ ,

which can be understood as "The orthogonal group $\mbox{O}(n+1)$ acts transitively on the unit sphere $S^n$ , and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower". The map $\mbox{O}(n) \to \mbox{O}(n+1)$ is the natural inclusion.

Thus the inclusion $\mbox{O}(n) \to \mbox{O}(n+1)$ is (n − 1)-connected, so the homotopy groups stabilize, and $\pi_k(O) = \pi_k(\mbox{O}(n))$ for $n > k + 1$ : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

Via Bott periodicity, $\Omega^8 O \simeq O$ , thus the homotopy groups of O are 8-fold periodic, meaning $\pi_{k+8} O = \pi_k O$ , and one need only compute the lower 8 homotopy groups to compute them all.

$\begin{align} \pi_0 O &= \mathbf Z/2\\ \pi_1 O &= \mathbf Z/2\\ \pi_2 O &= 0\\ \pi_3 O &= \mathbf Z\\ \pi_4 O &= 0\\ \pi_5 O &= 0\\ \pi_6 O &= 0\\ \pi_7 O &= \mathbf Z\\ \end{align}$

Relation to KO-theory

Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: $\pi_k O = \pi_{k+1} BO$ .

Setting $KO = BO \times \mathbf Z = \Omega^{-1} O \times \mathbf Z$ (to make $\pi_0$ fit into the periodicity), one obtains:

$\begin{align} \pi_0 KO &= \mathbf Z\\ \pi_1 KO &= \mathbf Z/2\\ \pi_2 KO &= \mathbf Z/2\\ \pi_3 KO &= 0\\ \pi_4 KO &= \mathbf Z\\ \pi_5 KO &= 0\\ \pi_6 KO &= 0\\ \pi_7 KO &= 0\\ \end{align}$

Computation and Interpretation of homotopy groups

Low-dimensional groups

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

$\pi_0(O) = \pi_0(\mbox{O}(1)) = \mathbf Z/2$ from orientation-preserving/reversing (this class survives to $\mbox{O}(2)$ and hence stably)

$\mbox{SO}(3) = \mathbf{RP}^3 = S^3/(\mathbf Z/2)$ yields

$\pi_1(O) = \pi_1(\mbox{SO}(3)) = \mathbf Z/2$ , which is spin
$\pi_2(O) = \pi_2(\mbox{SO}(3)) = 0$ , which surjects onto $\pi_2(\mbox{SO}(4))$ ; this latter thus vanishes

Lie groups

From general facts about Lie groups, $\pi_2 G$ always vanishes, and $\pi_3 G$ is free (free abelian).

Vector bundles

From the vector bundle point of view, $\pi_0(KO)$ is vector bundles over $S^0$ , which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so

$\pi_0(KO) = \mathbf Z$ is dimension

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. Using $\pi_0$ , O and O/U have two components, $KO = BO \times \mathbf Z$ and $KSp = BSp \times \mathbf Z$ have $\mathbf Z$ components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:^[2]

$\pi_0(KO) = \mathbf Z$ is dimension
$\pi_1(KO) = \mathbf Z/2$ is orientation
$\pi_2(KO) = \mathbf Z/2$ is spin
$\pi_4(KO) = \mathbf Z$ is topological quantum field theory

Let $F = \mathbf R, \mathbf C, \mathbf H, \mathbf O$ , and let $L_F$ be the tautological line bundle over the projective line $\mathbf{FP}^1$ , and $[L_F]$ its class in K-theory. Noting that $\mathbf{RP}^1 = S^1, \mathbf{CP}^1 = S^2, \mathbf{HP}^1 = S^4, \mathbf{OP}^1 = S^8$ , these yield vector bundles over the corresponding spheres, and

$\pi_1(KO)$ is generated by $[L_{\mathbf R}]$
$\pi_2(KO)$ is generated by $[L_{\mathbf C}]$
$\pi_4(KO)$ is generated by $[L_{\mathbf H}]$
$\pi_8(KO)$ is generated by $[L_{\mathbf O}]$

From the point of view of symplectic geometry, $\pi_0(KO) \cong \pi_8(KO) = \mathbf Z$ can be interpreted as the Maslov index, thinking of it as the fundamental group of the stable Lagrangian Grassmannian $\pi_1(U/O),$ as $U/O \simeq \Omega^7(KO),$ so $\pi_1(U/O)=\pi_{1+7}(KO).$

Over finite fields

Orthogonal groups can also be defined over finite fields $\mathbf{F}_q$ , where $q$ is a power of a prime $p$ . When defined over such fields, they come in two types in even dimension: $O^+(2n, q)$ and $O^-(2n, q)$ ; and one type in odd dimension: $O(2n+1, q)$ .

If $V$ is the vector space on which the orthogonal group $G$ acts, it can be written as a direct orthogonal sum as follows:

$V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W$ ,

where $L_i$ are hyperbolic lines and $W$ contains no singular vectors. If $W = 0$ , then $G$ is of plus type. If $W = <w>$ then $G$ has odd dimension. If $W$ has dimension 2, $G$ is of minus type.

In the special case where n = 1, $O^\epsilon(2, q)$ is a dihedral group of order $2(q - \epsilon)$ .

We have the following formulas for the order of these groups, O(n,q) = { A in GL(n,q) : A·A^t=I }, when the characteristic is greater than two

$|O(2n+1,q)|=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i}).$

If −1 is a square in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).$

If −1 is a nonsquare in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).$

The Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to Z/2Z, and is 0 or 1 depending on whether an element is the product of an even or odd number of reflections. More concretely, the Dickson Invariant can be defined as $D(f)=rank (I-f) \mod 2$ where I is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives extra information. In characteristic 2, the special orthogonal group is defined to be the elements of Dickson invariant 0, rather than the elements of determinant 1.

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).

Orthogonal groups of characteristic 2

Over fields of characteristic 2 orthogonal groups often behave differently. This section lists some of the differences.

Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4 dimensional over the field with 2 elements and the Witt index is 2 (Grove 2002, Theorem 6.6 and 14.16). Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v+B(v,u)/Q(u)·u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v,u)/Q(u)·u.

The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2.

In odd dimensions 2n+1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.

In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm

The spinor norm is a homomorphism from an orthogonal group over a field F to

F^*/F^*2,

the multiplicative group of the field F up to square elements, that takes reflection in a vector of norm n to the image of n in F^*/F^*2.

For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

Galois cohomology and orthogonal groups

In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H¹, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.

$1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow O_V \rightarrow 1$

Here μ₂ is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H⁰(O_V), which is simply the group O_V(F) of F-valued points, to H¹(μ₂) is essentially the spinor norm, because H¹(μ₂) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from H¹ of the orthogonal group, to the H² of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.

Related groups

The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions $O(n) \subset U(n) \subset Sp(n)=USp(2n)$ and $USp(n) \subset U(n) \subset O(2n)$ are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, $U(n)/O(n)$ is the Lagrangian Grassmannian.

Lie subgroups

In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:

$O(n) \supset O(n-1)$ – preserves an axis

$O(2n) \supset U(n) \supset SU(n)$ – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.

$O(2n) \supset USp(n)$

$O(7) \supset G_2$

Lie supergroups

The orthogonal group O(n) is also an important subgroup of various Lie groups:

$U(n) \supset SU(n) \supset O(n)$

$USp(2n) \supset O(n)$

$G_2 \supset O(3)$

$F_4 \supset O(9)$

$E_6 \supset O(10)$

$E_7 \supset O(12)$

$E_8 \supset O(16)$

Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.^{[note 2]} These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:

Permutation matrices (the Coxeter group A_n)
Signed permutation matrices (the Coxeter group B_n); also equals the intersection of the orthogonal group with the integer matrices.^{[note 3]}

Covering and quotient groups

The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:

Two covering Pin groups, Pin₊(n) → O(n) and Pin₋(n) → O(n),
The quotient projective orthogonal group, O(n) → PO(n).

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

Spin group, Spin(n) → SO(n),
Projective special orthogonal group, SO(n) → PSO(n).

Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k+1) is a 1-to-1 cover, i.e., isomorphic to SO(2k+1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie algebra, $\mathfrak{so}_n(\mathbf{R})$ – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.^{[note 4]}

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

Applications to string theory

The group O(10) is of special importance in superstring theory because it is the symmetry group of 10 dimensional space-time.

Principal homogeneous space: Stiefel manifold

The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold $V_n(\mathbf{R}^n)$ of orthonormal bases (orthonormal n-frames).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given a orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds $V_k(\mathbf{R}^n)$ for $k < n$ of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.

Notes

↑ The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.
↑ Infinite subsets of a compact space have an accumulation point and are not discrete.
↑ $O(n) \cap GL(n,\mathbf{Z})$ equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.
↑ In odd dimension, SO(2k+1) $\cong$ PSO(2k+1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected.

References

↑ (Taylor 1992, page 160)
↑ John Baez "This Week's Finds in Mathematical Physics" week 105

Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, MR 1859189, ISBN 978-0-8218-2019-3
Taylor, Donald E. (1992), The Geometry of the Classical Groups, 9, Berlin: Heldermann Verlag, MR 1189139, ISBN 3-88538-009-9

Orthogonal group

Contents

Over the real number field

Even and odd dimension

Lie algebra

3D isometries that leave the origin fixed

Conformal group

Over the complex number field

Topology

Low dimensional

Homotopy groups

Relation to KO-theory

Computation and Interpretation of homotopy groups

Low-dimensional groups

Lie groups

Vector bundles

Loop spaces

Interpretation of homotopy groups

Over finite fields

The Dickson invariant

Orthogonal groups of characteristic 2

The spinor norm

Galois cohomology and orthogonal groups

Related groups

Lie subgroups

Lie supergroups

Discrete subgroups

Covering and quotient groups

Applications to string theory

Principal homogeneous space: Stiefel manifold

See also

Specific transforms

Specific groups

Related groups

Lists of groups

Notes

References

External links