Editing Character theory (section)

=="Twisted" dimension==
One may interpret the character of a representation as the "twisted" [[dimension (vector space)|dimension of a vector space]].<ref name="Gannon">{{Harv|Gannon|2006}}</ref> Treating the character as a function of the elements of the group {{math|''χ''(''g'')}}, its value at the [[Identity element|identity]] is the dimension of the space, since {{math|''χ''(1) {{=}} Tr(''ρ''(1)) {{=}} Tr(''I<sub>V</sub>'') {{=}} dim(''V'')}}. Accordingly, one can view the other values of the character as "twisted" dimensions.{{clarify|date=June 2011|reason=recursive definition}}

One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of [[monstrous moonshine]]: the [[j-invariant|{{mvar|j}}-invariant]] is the [[graded dimension]] of an infinite-dimensional graded representation of the [[Monster group]], and replacing the dimension with the character gives the [[McKay–Thompson series]] for each element of the Monster group.<ref name="Gannon" />