The character of a representation φ of a group G on a finite-dimensional vector space V over a field F is the trace of the representation φ (Serre 1977). In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher dimensional characters. The study of representations using characters is called "character theory" and one dimensional characters are also called "linear characters" within this context.
Read more about this topic: Character (mathematics)
Famous quotes containing the words character of and/or character:
“The slanders poured down like Niagara. If you take into consideration the setting—the war and the revolution—and the character of the accused—revolutionary leaders of millions who were conducting their party to the sovereign power—you can say without exaggeration that July 1917 was the month of the most gigantic slander in world history.”
—Leon Trotsky (1879–1940)
“Play builds the kind of free-and-easy, try-it-out, do-it-yourself character that our future needs. We must become more self-conscious and more explicit in our praise and reinforcement as children use unstructured play materials: “That’s good. You use your own ideas....” “That’s good. You did it your way....” “That’s good. You thought it all out yourself.””
—James L. Hymes, Jr. (20th century)