Orthogonal group Totally Explained

Everything about Orthogonal Group totally explained

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 ť

mathrm)

The Dickson invariant

For orthogonal groups in even dimensions, 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. Over fields that are not of characteristic 2 it's 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 many authors define the special orthogonal group 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 . Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthgonal 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 ť 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's trivial, but it's often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that isn't 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. ť

1 
ightarrow mu_2 
ightarrow Pin_V 
ightarrow O_V 
ightarrow 1

Here μ₂ is the algebraic group of square roots of 1; over a field of characteristic not 2 it's 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.

Important subgroups

In physics, particular in the areas of Kaluza-Klein compacification, it's important to find out the subgroups of the orthogonal group. The main ones are: ť

O(n) supset O(n-1)

O(2n) supset SU(n)

O(2n) supset USp(n)

O(7) supset G_2

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

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)

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

Footnotes

Further Information

Get more info on 'Orthogonal Group'.

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