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:
“Of all the characters I have known, perhaps Walden wears best, and best preserves its purity. Many men have been likened to it, but few deserve that honor. Though the woodchoppers have laid bare first this shore and then that, and the Irish have built their sties by it, and the railroad has infringed on its border, and the ice-men have skimmed it once, it is itself unchanged, the same water which my youthful eyes fell on; all the change is in me.”
—Henry David Thoreau (1817–1862)
“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)
“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)