Splitting fields of representations of generalized symmetric groups, 4

First a technical definition.

Let $A=C_\ell^n$ . Let $\eta_k(z)=z^k$ , for $z\in C_\ell$ and $1\leq k\leq \ell-1$ . For $\eta\in C_\ell^*$ , let $\mu\otimes \eta =(\mu_1\eta,\mu_2\eta,...,\mu_n\eta)$ where $\mu=(\mu_1,\mu_2,...,\mu_n)$ . This defines an action of $C_\ell^*$ on $A^*$ and hence on the set of equivalence classes of $G$ , $G^*$ . We call two representations $\theta_{\mu,\rho}$ , $\theta_{\mu',\rho'}$ $C_\ell^*$ -equivalent, and write

$\theta_{\mu,\rho}\sim_\ell \theta_{\mu',\rho'},$

if $\rho=\rho'$ and $\mu'=\mu\otimes \eta$ for some $\eta\in C_\ell^*$ . Similarly, we call two characters $\mu$ , $\mu'$ of $C_\ell^n$ $C_\ell^*$ -equivalent, and write

$\mu\sim_\ell \mu',$

if $\mu'=\mu\otimes \eta$ for some $\eta\in C_\ell^*$ .

For example, Let $\ell =9$ , $n=3$ and $\mu=(\eta_2,\eta_5,\eta_8)$ . Then $\mu\sim \mu\otimes\eta_3$ .

Let $\theta_{\mu,\rho}$ be as in the previous post. Note that

$\theta_{\mu\otimes \eta,\rho} = \theta_{\mu,\rho}\otimes \eta ,$

for $\eta\in C_\ell^*$ . Therefore, the matrix representations of two $C_\ell^*$ -equivalent representations differ only
by a character.

Let $G=C_\ell^n\, >\!\!\lhd \, S_n$ .

The results in the above section tells us how to construct all the irreducible representations of $G$ . We must

write down all the characters (i.e., 1-dimensional representations) of $A=C_\ell^n$ ,
describe the action of $S_n$ on $A^*$ ,
for each $\mu\in [A^*]$ , compute the stabilizer $(S_n)_{\mu}$ ,
describe all irreducible representations of each $(S_n)_{\mu}$ ,
write down the formula for the character of $\theta_{\mu,\rho}$ .

Write $\mu\in [A^*]$ as $\mu=(\mu_1,...,\mu_n)$ , where each component is a character of the cyclic group $C_\ell$ , $\mu_j\in C_\ell^*$ . Let $\mu'_1,...,\mu'_r$ denote all the distinct characters which occur in $\mu$ , so

$\{\mu'_1,...,\mu'_r\}=\{ \mu_1,...,\mu_n\}.$

Let $n_1$ denote the number of $\mu'_1$ ‘s in $\mu$ , $n_2$ denote the number of $\mu'_2$ ‘s in $\mu$ , …, $n_r$ denote the number of $\mu'_r$ ‘s in $\mu$ . Then $n=n_1+...+n_r$ . Call this the partition associated to $\mu$ .

If two characters $\mu=(\mu_1,...,\mu_n)$ , $\mu'=(\mu'_1,...,\mu'_n)$ belong to the same class in $[(C_\ell^n)^*]$ , under the $S_n$ -equivalence relation, then their associated partitions are equal.

The Frobenius formula for the character of an induced representation gives the following character formula. Let $\chi$ denote the character of $\theta_{\mu,\rho}$ . Then

$\chi(\vec{v},p)=\sum_{g\in S_n/(S_n)_\mu} \chi^o_\rho(gpg^{-1})\mu^g(\vec{v}),$

for all $\vec{v}\in C_\ell^n$ and $p\in S_n$ . In particular, if $p=1$ then

$\chi(\vec{v},1)=({\rm dim}\ \rho)\sum_{g\in S_n/(S_n)_\mu} \mu^g(\vec{v}).$

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Splitting fields of representations of generalized symmetric groups, 4

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