Schur index of irreducible character

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Direct definition

Suppose G is a finite group, K is a splitting field for G, and \chi is the character of an irreducible linear representation \varphi of G over K. Suppose k is the subfield of K generated by the character values \chi(g), g \in G. The Schur index of \chi (also termed the Schur index of \varphi) is defined in the following equivalent ways:

  1. It is the smallest positive integer m such that there exists a degree m extension L of k such that \varphi can be realized over L, i.e., we can change basis so that all the matrix entries are from L. Note that it is not necessary that L be a subfield of K, but rather we need to work within a suitable larger field that contains both L and K to perform the necessary conjugation.
  2. It is the multiplicity of \varphi in any irreducible linear representation \alpha of G over k that has \varphi as one of its irreducible constituents over K.

The Schur index of a character \chi is often denoted m(\chi).

Note that if the representation can be realized over the field generated by the character values for that representation, the Schur index is one.

Definition in terms of division rings




From the facts above, it is clear that to get an example of an irreducible character/representation with Schur index greater than 1, we should not look at odd-order p-groups. Here are the smallest examples: