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Schur index divides degree of irreducible representation

This article states a result of the form that one natural number divides another. Specifically, the (Schur index of irreducible representation) of a/an/the (irreducible linear representation) divides the (degree of a linear representation) of a/an/the (irreducible linear representation).
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Suppose G is a finite group, \varphi is an Irreducible linear representation (?) over a splitting field for G, \chi is the character of \varphi, and m(\chi) is the Schur index (?) of \chi. Then, m(\chi) divides the degree of \varphi (which is the same as the degree of \chi). In particular, this bounds the possible values of the Schur index for irreducible representations in terms of the Degrees of irreducible representations (?).

(Note that instead of requiring \varphi to be irreducible over a splitting field, we could instead require \varphi to be absolutely irreducible over any field whose characteristic does not divide the order of G).

In particular:

  • The Schur index of a linear character is 1 (this is obvious even otherwise).
  • The Schur index of an irreducible character of prime degree is either 1 or equal to that prime.

Related facts


This statement can be combined with results about the degrees of irreducible representations. In particular:

Other related facts