Class-determining field - Revision history

Vipul at 13:56, 21 July 2011

2011-07-21T13:56:43Z

← Older revision		Revision as of 13:56, 21 July 2011
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	A [[field]] is said to be a '''class-determining field''' for a [[group]] if any [[finite-dimensional linear representation]] is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.		A [[field]] is said to be a '''class-determining field''' for a [[group]] if any [[finite-dimensional linear representation]] is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.

	In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the [[general linear group]].		In other words, for any two distinct (i.e. [[fact about::equivalent linear representations\|inequivalent]]) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the [[general linear group]].

	Equivalently, no two inequivalent linear representations are [[locally conjugate representation\|locally conjugate]].		Equivalently, no two inequivalent linear representations are [[locally conjugate representation\|locally conjugate]].
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	===Definition with symbols===		===Definition with symbols===

	A [[field]] <math>k</math> is termed a '''class-determining field''' for a [[group]] <math>G</math> if for any two [[finite-dimensional linear representation]]s <math>\rho_1,\rho_2:G \to GL(V)</math>, there exists <math>g \in G</math> such that <math>\rho_1(g)</math> and <math>\rho_2(g)</math> are not conjugate.		A [[field]] <math>k</math> is termed a '''class-determining field''' for a [[group]] <math>G</math> if for any two [[fact about::equivalent linear representations\|inequivalent]] [[finite-dimensional linear representation]]s <math>\rho_1,\rho_2:G \to GL(V)</math>, there exists <math>g \in G</math> such that <math>\rho_1(g)</math> and <math>\rho_2(g)</math> are not conjugate.

	~~==Relation with other properties==~~

	~~===Stronger properties===~~

	* [[Character-determining field]]

	==Facts==		==Facts==

	For a finite group, any field of ~~coprime characteristic~~ is ~~character~~-determining~~, and hence also~~ class-determining.		* [[Non-modular implies class-determining]]: For a [[finite group]], any field whose characteristic does not divide the order of the group is a class-determining field.
			* [[Cyclic implies every field is class-determining]]
			* [[Elementary abelian of prime-square order implies corresponding prime field is not class-determining]]

Vipul at 16:45, 24 June 2008

2008-06-24T16:45:30Z

← Older revision		Revision as of 16:45, 24 June 2008
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	===Symbol-free definition===		===Symbol-free definition===

	A [[field]] is said to be a '''class-determining field''' for a [[group]] if any representation is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.		A [[field]] is said to be a '''class-determining field''' for a [[group]] if any [[finite-dimensional linear representation]] is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.

	In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the [[general linear group]].		In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the [[general linear group]].
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	===Definition with symbols===		===Definition with symbols===

	A [[field]] <math>k</math> is termed a '''class-determining field''' for a [[group]] <math>G</math> if for any two finite-dimensional linear ~~representations~~ <math>\rho_1,\rho_2:G \to GL(V)</math>, there exists <math>g \in G</math> such that <math>\rho_1(g)</math> and <math>\rho_2(g)</math> are not conjugate.		A [[field]] <math>k</math> is termed a '''class-determining field''' for a [[group]] <math>G</math> if for any two [[finite-dimensional linear representation]]s <math>\rho_1,\rho_2:G \to GL(V)</math>, there exists <math>g \in G</math> such that <math>\rho_1(g)</math> and <math>\rho_2(g)</math> are not conjugate.

	==Relation with other properties==		==Relation with other properties==

Vipul: 3 revisions

2008-05-07T23:18:03Z

3 revisions

← Older revision	Revision as of 23:18, 7 May 2008
(No difference)

Vipul at 21:10, 15 January 2008

2008-01-15T21:10:45Z

← Older revision		Revision as of 21:10, 15 January 2008
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	{{termrelatedto\|linear representation theory}}		{{termrelatedto\|linear representation theory}}

			{{group-parametrized field property}}

	==Definition==		==Definition==

Vipul at 16:08, 13 September 2007

2007-09-13T16:08:08Z

← Older revision		Revision as of 16:08, 13 September 2007
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	==Facts==		==Facts==

	For a finite group, any field of coprime characteristic is character-determining, ~~but not~~ class-determining.		For a finite group, any field of coprime characteristic is character-determining, and hence also class-determining.

Vipul at 16:07, 13 September 2007

2007-09-13T16:07:59Z

New page

{{termrelatedto|linear representation theory}}

==Definition==

===Symbol-free definition===

A [[field]] is said to be a '''class-determining field''' for a [[group]] if any representation is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.

In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the [[general linear group]].

Equivalently, no two inequivalent linear representations are [[locally conjugate representation|locally conjugate]].

===Definition with symbols===

A [[field]] <math>k</math> is termed a '''class-determining field''' for a [[group]] <math>G</math> if for any two finite-dimensional linear representations <math>\rho_1,\rho_2:G \to GL(V)</math>, there exists <math>g \in G</math> such that <math>\rho_1(g)</math> and <math>\rho_2(g)</math> are not conjugate.

==Relation with other properties==

===Stronger properties===

* [[Character-determining field]]

==Facts==

For a finite group, any field of coprime characteristic is character-determining, but not class-determining.