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

  1. write down all the characters (i.e., 1-dimensional representations) of A=C_\ell^n,
  2. describe the action of S_n on A^*,
  3. for each \mu\in [A^*], compute the stabilizer (S_n)_{\mu},
  4. describe all irreducible representations of each (S_n)_{\mu},
  5. 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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