Representation Ring - Characters

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 the other characters in the book there is, likewise, little to say. The most endearing one is obviously the old Captain Maksim Maksimich, stolid, gruff, naively poetical, matter-of- fact, simple-hearted, and completely neurotic.”
    —Vladimir Nabokov (1899–1977)

    “Though they be mad and dead as nails,
    Heads of the characters hammer through daisies;
    Break in the sun till the sun breaks down,
    And death shall have no dominion.”
    —Dylan Thomas (1914–1953)

    “Children pay little attention to their parent’s teachings, but reproduce their characters faithfully.”
    —Mason Cooley (b. 1927)