This post shall list some properties of the Schur index in the case where is a generalized symmetric group and is either the reals or rationals.

Let , for , .

**Theorem:** Let . Let , for some , and let . Let

denotes the character of .

- Suppose that one of the following conditions holds:
- and divides in , or
- ,

Then .

- Suppose that one of the following conditions holds:
- , , and is not solvable, or
- and .

Then .

This theorem is proven in this paper. Benard has shown that , for all as in the above theorem.

Since the Schur index over of any irreducible character of a generalized symmetric group is equal to , each such character is associated to a representation all of whose matrix coefficients belong to the splitting field .

What is the splitting field , for ?

This will be addressed in the next post.