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:
“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)
“I make it a kind of pious rule to go to every funeral to which I am invited, both as I wish to pay a proper respect to the dead, unless their characters have been bad, and as I would wish to have the funeral of my own near relations or of myself well attended.”
—James Boswell (1740–1795)
“Unresolved dissonances between the characters and dispositions of the parents continue to reverberate in the nature of the child and make up the history of its inner sufferings.”
—Friedrich Nietzsche (1844–1900)