Schur index

The Schur index of a central simple algebra over a field (cf. Central simple algebra) is the degree of the division algebra such that is a full matrix algebra over .

Let be a finite group, a field and the algebraic closure of . Let be an irreducible -module with character (cf. Irreducible module). Let be obtained from by adjoining the values , . The Schur index of the module , , (or the Schur index of the character ) is the minimal degree of a field extending such that descends to , i.e. such that there is an -module for which .

For a finite field the Schur index is always , [a1].

A basic result on the Schur index is that for each -module the multiplicity of in is a multiple of .

A field is a splitting field for a finite group if each irreducible -module is absolutely irreducible, i.e. if is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [a1] that if is the exponent of a finite group (i.e. is the smallest integer such that for all ), then is a splitting field for .

The set of classes of central simple algebras over which occur as components of a group algebra for some finite group is a subgroup of the Brauer group of , and is known as the Schur subgroup of . Cf. [a4] for results on the structure of .

References

[a1]	R. Brauer, "On the representation of a group of order in the field of -th roots of unity" Amer. J. Math. , 67 (1945) pp. 461–471
[a2]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §90, §41
[a3]	B. Huppert, "Finite groups" , 2 , Springer (1982) pp. §1
[a4]	T. Yamada, "The Schur subgroup of the Brauer group" , Springer (1974)