Characters
Any representation defines a character χ:G → C. Such a function is constant on conjugacy classes of G, a so-called class function; denote the ring of class functions by C(G). The homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from RF(G) → C(G) defined by Brauer characters is no longer injective.
For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).
Read more about this topic: Representation Ring
Famous quotes containing the word characters:
“Waxed-fleshed out-patients
Still vague from accidents,
And characters in long coats
Deep in the litter-baskets
All dodging the toad work
By being stupid or weak.”
—Philip Larkin (1922–1986)
“Socialist writers are made of sterner stuff than those who only let their characters steeplechase through trouble in order to come out first in the happy ending of moral uplift.”
—Christina Stead (1902–1983)
“Philosophy is written in this grand book—I mean the universe—
which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it.”
—Galileo Galilei (1564–1642)