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:
“Animals are stylized characters in a kind of old saga—stylized because even the most acute of them have little leeway as they play out their parts.”
—Edward Hoagland (b. 1932)
“For our vanity is such that we hold our own characters immutable, and we are slow to acknowledge that they have changed, even for the better.”
—E.M. (Edward Morgan)
“Children pay little attention to their parent’s teachings, but reproduce their characters faithfully.”
—Mason Cooley (b. 1927)