Class-separating field: Difference between revisions

Latest revision as of 16:49, 24 June 2008

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Class-separating field, all facts related to Class-separating field) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

Symbol-free definition

A field is termed class-separating for a group if it satisfies the following equivalent conditions:

Given any two conjugacy classes in the group, there exists a finite-dimensional linear representation of the group over the field such that the images of the conjugacy classes, are not conjugate in the general linear group.
Given two elements of the group whose images are conjugate in the general linear group for every finite-dimensional linear representations, the two elements must be conjugate in the group.
No two distinct conjugacy classes can be locally conjugate.

Definition with symbols

A field $k$ is termed class-separating for a group $G$ if it satisfies the following equivalent conditions:

Given any two conjugacy classes $c_{1}$ and $c_{2}$ , there exists a finite-dimensional linear representation $ρ : G \to G L (V)$ where $V$ is a finite-dimensional $k$ -vector space, such that $ρ (c_{1})$ and $ρ (c_{2})$ are not in the same conjugacy class in $G L (V)$ .
Given two elements $g$ and $h$ in $G$ such that $ρ (g)$ and $ρ (h)$ are conjugate in $G L (V)$ for every finite-dimensional linear representation $ρ$ of $G$ , we can conclude that $g$ and $h$ are conjugate elements inside $G$ .

Definition in terms of the conjugacy class-representation duality

Further information: conjugacy class-representation duality

Let $L (c, ρ)$ denote the conjugacy class in $G L (V)$ of the image of the conjugacy class $c$ of $G$ under the representation $ρ$ . Then, $k$ is class-separating for $G$ if and only if $L (c_{1}, ρ) = L (c_{2}, ρ)$ implies that $c_{1} = c_{2}$ .

Relation with other properties

Stronger properties

Character-separating field

Related properties

Facts

For a finite group, a sufficiently large field is a field of characteristic zero or relatively prime to the order of the group, which contains all the $m^{t h}$ roots of unity where $m$ is the exponent of the group.

It turns out that any sufficiently large field is character-separating, and hence also class-separating.

@@ Line 1: / Line 1: @@
-{{termrelatedto|linear representation theory}}
+{{term related to|linear representation theory}}
 {{group-parametrized field property}}
@@ Line 9: / Line 9: @@
 ===Symbol-free definition===
-A [[field]] is termed '''class-separating''' for a [[group]] if given any two conjugacy classes in the group, there exists a [[linear representation]] of the group over the field such that the images of the conjugacy classes, are not conjugate in the  general linear group.
+A [[field]] is termed '''class-separating''' for a [[group]] if it satisfies the following equivalent conditions:
-Another way of saying this is that no two distinct conjugacy classes can be [[locally conjugate conjugacy classes|locally conjugate]].
+# Given any two [[conjugacy class]]es in the group, there exists a [[finite-dimensional linear representation]] of the group over the field such that the images of the conjugacy classes, are not conjugate in the  general linear group.
+# Given two elements of the group whose images are conjugate in the general linear group for every finite-dimensional linear representations, the two elements must be conjugate in the group.
+# No two distinct conjugacy classes can be [[locally conjugate conjugacy classes|locally conjugate]].
 ===Definition with symbols===
-A field <math>k</math> is termed '''class-separating''' for a group <math>G</math> if given any two conjugacy classes <math>c_1</math> and <math>c_2</math>, there exists a representation <math>\rho:G \to GL(V)</math> where <math>V</math> is a <math>k</math>-vector space, such that <math>\rho(c_1)</math> and <math>\rho(c_2)</math> are not in the same conjugacy class in <math>GL(V)</math>.
+A field <math>k</math> is termed '''class-separating''' for a group <math>G</math> if it satisfies the following equivalent conditions:
-===Definition in terms of the L-notation===
+# Given any two conjugacy classes <math>c_1</math> and <math>c_2</math>, there exists a [[finite-dimensional linear representation]] <math>\rho:G \to GL(V)</math> where <math>V</math> is a finite-dimensional  <math>k</math>-vector space, such that <math>\rho(c_1)</math> and <math>\rho(c_2)</math> are not in the same conjugacy class in <math>GL(V)</math>.
+# Given two elements <math>g</math> and <math>h</math> in <math>G</math> such that <math>\rho(g)</math> and <math>\rho(h)</math> are conjugate in <math>GL(V)</math> for every [[finite-dimensional linear representation]] <math>\rho</math> of <math>G</math>, we can conclude that <math>g</math> and <math>h</math> are [[conjugate elements]] inside <math>G</math>.
+===Definition in terms of the conjugacy class-representation duality===
 {{further|[[conjugacy class-representation duality]]}}