Locally conjugate conjugacy classes
This term is related to: linear representation theory
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Two conjugacy classes in a group are said to be locally conjugate over a field if for any indecomposable linear representation (or equivalently, for any linear representation) over the field, the images of the conjugacy classes in the general linear group, are in the same conjugacy class of the general linear group.
Definition with symbols
Let be a group and a field. Two conjugacy classes of are termed locally conjugate if for any indecomposable linear representation of over , and for elements , the elements and are conjugate in . In other words, there exists such that:
Definition using the L-map
where is the union of the set of conjugacy classes in for all vector spaces over .
Then, we say that two conjugacy classes and are locally conjugate with respect to if for any :
Locally conjugate representations
Further information: locally conjugate representations
These are representations that send each conjugacy class in the group to within the same conjugacy class in the general linear group.
Character-conjugate conjugacy classes
Further information: character-conjugate conjugacy classes
Further information: class-separating field
A class-separating field is a field in which no two distinct conjugacy classes are locally conjugate.
In a sufficiently large field
A sufficiently large field for a finite group is a field which contains all the roots of unity, where is the exponent of the group. It turns out that in a sufficiently large field, no two distinct conjugacy classes are locally conjugate. This can also be rewritten as: any sufficiently large field is class-separating.
In other fields of coprime characteristic
Take a finite group and a field whose characteristic does not divide the order of the group. Then, if the field is not sufficiently large, there may be multiple conjugacy classes that are locally conjugate to each other. In general, the following is true. Let be the smallest sufficiently large field containing the field for a finite group . Then includes automorphisms that raise the primitive roots of unity to some exponent. So can be identified with a subgroup of the multiplicative group of the ring of integers mod where is the exponent of .
Now it turns out that and are locally conjugate if and only if there exists a in this multiplicative group such that elements of can be expressed as powers of elements of .
- Suppose we take the field of real numbers and a finite group whose exponent is bigger than 2. Then the smallest sufficiently large field containing the reals is the complex numbers. The Galois group in this case is simply the subgroup of comprising the elements and (the automorphism is complex conjugation).
Thus two conjugacy classes in the group are locally conjugate over real numbers if and only if they are inverses of each other, viz one goes to the other via the inverse map.