Note: This draft page is used to work out the derivation of the formula and will be merged back to
Frobenius formula.
Derivation
The proof here relies on some basic facts about
Schur polynomials, distinguished symmetric polynomials parametrized by partitions . The properties that we need to use are
For a symmetric polynomial P, is the coefficient of in .
Writing ( = the number of j in ) and viewing as a function , are orthonormal with respect to the inner product on the space of class functions on .
The proof is now completed by descending induction on partitions , as follows. Let be the subgroup of (so-called the Young subgroup), the
representation induced from the trivial representation and its character. The basic case is not hard to see; thus, assume that for all , ( is viewed as a class function as above). The
Mackey formula for an induced character says