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{{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]]}}
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