Splitting fields of representations of generalized symmetric groups, 8

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In this post, we give an example.

Let G=C_3^8\, >\!\!\lhd \, S_8 and let

\pi = \theta_{\mu,\rho}=Ind_{G_\mu}^G(\mu\cdot \tilde{\rho}),

where \mu is a character of C_3^8 and \rho is an irreducible representation of its stabilizer in S_8, (S_8)_\mu.

The real representations \pi of G are the ones for which

  1. \mu is represented by a character of the form

    (1,1,1,1,1,1,\omega,\omega^2) \ {\rm or}\ (1,1,...,1),

    and \rho anything, or

  2. \mu is represented by a character of the form

    (1,1,1,1,\omega,\omega,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_4)^*\times (S_2)^*\times (S_2)^*,

    or

  3. \mu is represented by a character of the form

    (\omega,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_2,\pi_2)\in (S_4)^*\times (S_4)^*,

    or

  4. \mu is represented by a character of the form

    (1,1,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_2)^*\times (S_3)^*\times (S_3)^*.

The complex representations of G are: the representations
whose characters have at least one complex value. Such representations \pi = \theta_{\mu,\rho} are characterized by the fact that (\mu,\rho) is inequivalent to (\overline{\mu},\rho) under the obvious $S_8$-equivalence relation (which can be determined from the equivalence relation for representations in G^*).

The complex representations of G are the remaining representations not included in the above list.

There are no quaternionic representations of G.

The claims above follow from the fact that a representation
\theta_{\rho,\mu} is complex if and only if \mu is not self-dual.

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