  In this investigation of character tables of finite groups we study basic
sets and associated representation theoretic data for complementary sets of
conjugacy classes. For the symmetric groups we find unexpected properties of
characters on restricted sets of conjugacy classes, like beautiful
combinatorial determinant formulae for submatrices of the character table and
Cartan matrices with respect to basic sets; we observe that similar phenomena
occur for the transition matrices between power sum symmetric functions to
bounded partitions and the $k$-Schur functions introduced by Lapointe and
Morse. Arithmetic properties of the numbers occurring in this context are
studied via generating functions.
