Splitting fields of representations of generalized symmetric groups, 2

In general, there are three types of (complex) representations of a finite group G. (A good reference for all this is Serre’s well-known book, Linear representations of finite groups.)

Let \rho:H\rightarrow Aut(W) be an n-dimensional irreducible representation of a finite group G on a complex vector space W. Let \chi denote the character of \rho.

Exactly one of the following possibilities must hold:

  • One of the values of the character \chi is not real. Such representations will be called complex (or type 1 or unitary).
  • All the values of \chi are real and \rho is realizable by a representation over a real vector space. Such representations will be called real (or type 2 or orthogonal).
  • All the values of \chi are real but \rho is not realizable by a representation over a real vector space. Such representations will be called quaternionic (or type 3 or symplectic).

Proposition (Frobenius-Schur): Let \rho:H\rightarrow Aut(W) be an irreducible representation of a finite group G on a complex vector space W with character $\chi$. Then

{1\over |G|} \sum_{g\in G}\chi(g^2)= \left\{ \begin{array}{cc} 0,&\rho\ {\rm complex},\\ 1,&\rho\ {\rm real},\\ -1,&\rho\ {\rm quaternionic}. \end{array} \right.

This quantity is sometimes called the Frobenius-Schur indicator of \rho.

It can be shown that if \rho \rho'\cong \rho are equivalent representations then \rho and \rho' have the same type.

In the next post we will examine the types that the irreducible representations of semi-direct product G=C_\ell^n\, >\!\!\lhd \, S_n fall into.

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