Template:Character table facts to check against: Difference between revisions

From Groupprops
No edit summary
No edit summary
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{quotation|'''FACTS TO CHECK AGAINST FOR CHARACTERS OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD''':<br>'''Orthogonality relations''': [[Character orthogonality theorem]] <nowiki>|</nowiki> [[Column orthogonality theorem]] <nowiki>|</nowiki> [[Splitting implies characters span class functions]] <br>'''Separation results''' (basically says any two rows distinct, any two columns distinct): [[Character determines representation in characteristic zero]] <br>'''Numerical facts''': [[Characters are cyclotomic integers]] <nowiki>|</nowiki> [[Size-degree-weighted characters are algebraic integers]] <nowiki>|</nowiki> [[Irreducible character of degree greater than one takes value zero on some conjugacy class]] <nowiki>|</nowiki> [[Zero-or-scalar lemma]]}}
{{quotation|'''FACTS TO CHECK AGAINST (for [[character of a linear representation|characters]] of [[irreducible linear representation]]s over a [[splitting field]])''':<br>'''Orthogonality relations''': [[Character orthogonality theorem]] <nowiki>|</nowiki> [[Column orthogonality theorem]] <br>'''Separation results''' (basically says rows independent, columns independent): [[Splitting implies characters form a basis for space of class functions]]<nowiki>|</nowiki>[[Character determines representation in characteristic zero]] <br>'''Numerical facts''': [[Characters are cyclotomic integers]] <nowiki>|</nowiki> [[Size-degree-weighted characters are algebraic integers]]<br>'''Character value facts''': [[Irreducible character of degree greater than one takes value zero on some conjugacy class]]<nowiki>|</nowiki> [[Conjugacy class of more than average size has character value zero for some irreducible character]] <nowiki>|</nowiki> [[Zero-or-scalar lemma]]}}

Latest revision as of 05:01, 9 May 2013

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Template:Character_table_facts_to_check_against&oldid=46568"