Splitting implies characters separate conjugacy classes - Revision history

Vipul at 02:20, 14 July 2011

2011-07-14T02:20:07Z

← Older revision		Revision as of 02:20, 14 July 2011
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	==Statement==		==Statement==

	For a [[finite group]] <math>G</math>, any [[fact about::splitting field]] <math>k</math>~~is a [[fact about::character-separating field]]~~: given any two distinct [[conjugacy class]]es, there is a finite-dimensional representation <math>\varphi</math> of <math>G</math> over <math>k</math> such that the character of <math>\varphi</math> takes different values on the two conjugacy classes.		For a [[finite group]] <math>G</math>, any [[fact about::splitting field]] <math>k</math> has the following property: given any two distinct [[fact about::conjugacy class]]es of <math>G</math>, there is a finite-dimensional representation <math>\varphi</math> of <math>G</math> over <math>k</math> such that the character of <math>\varphi</math> takes different values on the two conjugacy classes.

	==Related facts==		==Related facts==
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	===Converse===		===Converse===

	The converse is not true, ~~i.e., [[character-separating not implies splitting]]. In~~ fact, [[characters span class functions not implies splitting]] (this is due to the [[Schur index]] phenomenon).		The converse is not true, in fact: [[characters span class functions not implies splitting]] (this is due to the [[Schur index]] phenomenon).

			On the other hand, see [[characters span class functions iff they separate conjugacy classes iff field contains field generated by character values]] for alternative characterizations of a field where the characters separate the conjugacy classes.

	==Facts used==		==Facts used==

Vipul: moved Splitting implies character-separating to Splitting implies characters separate conjugacy classes

2011-07-14T02:16:50Z

moved Splitting implies character-separating to Splitting implies characters separate conjugacy classes

← Older revision	Revision as of 02:16, 14 July 2011
(No difference)

Vipul at 20:17, 13 July 2011

2011-07-13T20:17:43Z

← Older revision		Revision as of 20:17, 13 July 2011
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	===Converse===		===Converse===

	The converse is not true, i.e., [[character-separating not implies splitting]]. ~~However~~, ~~it ''is'' true that~~ [[characters span class functions implies splitting]].		The converse is not true, i.e., [[character-separating not implies splitting]]. In fact, [[characters span class functions not implies splitting]] (this is due to the [[Schur index]] phenomenon).

	==Facts used==		==Facts used==

Vipul: Created page with "==Statement== For a finite group $G$ , any fact about::splitting field $k$ is a fact about::character-separating field: given any two distinct [[..."

2011-07-13T20:16:02Z

Created page with "==Statement== For a finite group <math>G</math>, any fact about::splitting field <math>k</math>is a fact about::character-separating field: given any two distinct [[..."

New page

==Statement==

For a [[finite group]] <math>G</math>, any [[fact about::splitting field]] <math>k</math>is a [[fact about::character-separating field]]: given any two distinct [[conjugacy class]]es, there is a finite-dimensional representation <math>\varphi</math> of <math>G</math> over <math>k</math> such that the character of <math>\varphi</math> takes different values on the two conjugacy classes.

==Related facts==

===Corollaries===

* [[Splitting implies class-separating]]: Given any two distinct conjugacy classes, we can find a finite-dimensional representation where the images are not conjugate as linear transformations.

===Converse===

The converse is not true, i.e., [[character-separating not implies splitting]]. However, it ''is'' true that [[characters span class functions implies splitting]].

==Facts used==

# [[uses::Splitting implies characters span class functions]]

==Proof==

The proof follows directly from Fact (1). If it were true that there were two distinct conjugacy classes on which all characters were equal, then the span of the space of characters would not include any class function taking different values on the two conjugacy classes, and in particular would not include the indicator function for any one of the conjugacy classes. Hence, the span would not be all class functions.

Splitting implies characters separate conjugacy classes - Revision history

Vipul at 02:20, 14 July 2011

Vipul: moved Splitting implies character-separating to Splitting implies characters separate conjugacy classes

Vipul at 20:17, 13 July 2011

Vipul: Created page with "==Statement== For a finite group G, any fact about::splitting field kis a fact about::character-separating field: given any two distinct [[..."

Vipul: Created page with "==Statement== For a finite group $G$ , any fact about::splitting field $k$ is a fact about::character-separating field: given any two distinct [[..."