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:
“Thus we may define the real as that whose characters are independent of what anybody may think them to be.”
—Charles Sanders Peirce (1839–1914)
“I have often noticed that after I had bestowed on the characters of my novels some treasured item of my past, it would pine away in the artificial world where I had so abruptly placed it.”
—Vladimir Nabokov (1899–1977)
“Do you set down your name in the scroll of youth, that are written down old with all the characters of age?”
—William Shakespeare (1564–1616)