# Schur index

The Schur index of a central simple algebra $A$ over a field $K$( cf. Central simple algebra) is the degree of the division algebra $D$ such that $A$ is a full matrix algebra $M _ {n} ( D)$ over $D$.

Let $G$ be a finite group, $K$ a field and $\overline{K}\;$ the algebraic closure of $K$. Let $V$ be an irreducible $\overline{K}\; [ G]$- module with character $\rho$( cf. Irreducible module). Let $K( \rho )$ be obtained from $K$ by adjoining the values $\rho ( g)$, $g \in G$. The Schur index of the module $V$, $m _ {K} ( V)$, (or the Schur index of the character $\rho$) is the minimal degree of a field $S$ extending $K( \rho )$ such that $V$ descends to $S$, i.e. such that there is an $S[ G]$- module $W$ for which $V \simeq \overline{K}\; \otimes _ {S} W$.

For a finite field $K$ the Schur index is always $1$, [a1].

A basic result on the Schur index is that for each $K[ G]$- module $W$ the multiplicity of $V$ in $\overline{K}\; \otimes _ {K} W$ is a multiple of $m _ {K} ( V)$.

A field $S \subset \overline{K}\;$ is a splitting field for a finite group $G$ if each irreducible $S[ G]$- module is absolutely irreducible, i.e. if $\overline{K}\; \otimes _ {S} V$ is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [a1] that if $d$ is the exponent of a finite group $G$( i.e. $d$ is the smallest integer such that $g ^ {d} = e$ for all $g \in G$), then $\mathbf Q ( 1 ^ {1/d} )$ is a splitting field for $G$.

The set $S( K)$ of classes of central simple algebras over $K$ which occur as components of a group algebra $K[ G]$ for some finite group $G$ is a subgroup of the Brauer group $\mathop{\rm Br} ( K)$ of $K$, and is known as the Schur subgroup of $\mathop{\rm Br} ( K)$. Cf. [a4] for results on the structure of $S( K)$.

#### 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)
How to Cite This Entry:
Schur index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_index&oldid=48623