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:

    “The business of a novelist is, in my opinion, to create characters first and foremost, and then to set them in the snarl of the human currents of his time, so that there results an accurate permanent record of a phase of human history.”
    —John Dos Passos (1896–1970)

    “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)

    “The major men
    That is different. They are characters beyond
    Reality, composed thereof. They are
    The fictive man created out of men.
    They are men but artificial men.”
    —Wallace Stevens (1879–1955)