User:TakuyaMurata/Frobenius formula

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 $s^{\lambda }$, distinguished symmetric polynomials parametrized by partitions $\lambda$. The properties that we need to use are

1. Schur polynomials are an integral basis for the ring of symmetric functions.
2. (Cauchy formula) $\prod {1 \over 1-x_{i}y_{i}}=\sum _{\lambda }s^{\lambda }(x)s^{\lambda }(y).$
3. For $\Delta =\prod _{i<j}(x_{i}-x_{j})$, we have $\Delta \cdot s^{\lambda }$ is a polynomial such that

By Property 1., for each symmetric polynomial P, we can write

$P=\sum _{\lambda }\omega _{\lambda }(P)s^{\lambda }$

for the integers $\omega _{\lambda }(P)$. First we establish the following:

1. For a symmetric polynomial P,

   the coefficient of $x^{\lambda }$ in P is $\sum _{\mu }K_{\mu \lambda }\omega _{\mu }(P)$

   for some unique integers $K_{\mu \lambda }$ (called the Kostka numbers).
2. For a symmetric polynomial P, $\omega _{\lambda }(P)$ is the coefficient of $x_{1}^{\ell _{1}}\cdot {\dots }\cdot x_{k}^{\ell _{k}}$ in $\Delta \cdot P$.
3. Writing $P^{\mu }=P_{1}^{i_{1}}\cdot {\dots }\cdot P_{n}^{i_{n}}$ ($i_{j}$ = the number of j in $\mu$) and viewing $\omega _{\lambda }$ as a function $C(\mu )\mapsto \omega _{\lambda }(P^{\mu })$, $\omega _{\lambda }$ are orthonormal with respect to the inner product on the space of class functions on $S_{n}$.

The proof is now completed by descending induction on partitions $\lambda$, as follows. Let $S_{\lambda }=S_{\lambda _{1}}\times \dots \times S_{\lambda _{k}}$ be the subgroup of $S_{n}$ (so-called the Young subgroup), $U_{\lambda }=\operatorname {Ind} _{S_{\lambda }}^{S_{n}}(1)$ the representation induced from the trivial representation and $\psi _{\lambda }$ its character. The basic case is not hard to see; thus, assume that for all $\mu >\lambda$, $\chi _{\mu }=\omega _{\mu }$ ($\omega _{\mu }$ is viewed as a class function as above). The Mackey formula for an induced character says

$\psi _{\lambda }(C(\mu ))={[S_{d}:S_{\lambda }] \over \#(C(\mu ))}\#(C(\mu ))\cap S_{\lambda }).$

...

Hence,

$\psi _{\lambda }=\omega _{\lambda }+\sum _{\nu >\lambda }K_{\nu \lambda }\chi _{\nu }$.

By the linear independence of characters, this is possible only when $\omega _{\lambda }=\chi _{\lambda }$.