Group ring over splitting field is direct sum of matrix rings for each irreducible representation
Statement
Suppose is a finite group and
is a splitting field for
. Suppose the irreducible representations of
are
and the degrees of irreducible representations are
respectively. Then, the group ring
is a semisimple Artinian ring and is expressible as a direct sum of matrix rings over
as follows:
where is the ring of
matrices over
. Note that each
is a simple ring and therefore corresponds to a minimal two-sided ideal in the decomposition.
Related facts
Over non-splitting fields
Further information: Group ring of finite group over field of characteristic not dividing its order is semisimple Artinian
Even if is not a splitting field, it is true that if the characteristic of
does not divide the order of
,
is a semisimple Artinian ring. By the Artin-Wedderburn theorem, it splits as a direct sum of matrix rings over division rings (in an essentially unique fashion), where each division ring has
in its center. If the division rings that we need to use are not fields, then we have the Schur index phenomenon.