Group ring over splitting field is direct sum of matrix rings for each irreducible representation

Statement

Suppose $G$ is a finite group and $K$ is a splitting field for $G$. Suppose the irreducible representations of $G$ are $\varphi_1, \varphi_2, \dots, \varphi_r$ and the degrees of irreducible representations are $d_1,d_2,\dots,d_r$ respectively. Then, the group ring $K[G]$ is a semisimple Artinian ring and is expressible as a direct sum of matrix rings over $K$ as follows:

$K[G] \cong M_{d_1}(K) \oplus M_{d_2}(K) \oplus \dots \oplus M_{d_r}(K)$

where $M_n(K)$ is the ring of $n \times n$ matrices over $K$. Note that each $M_n(K)$ 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 $K$ is not a splitting field, it is true that if the characteristic of $K$ does not divide the order of $G$, $K[G]$ 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 $K$ in its center. If the division rings that we need to use are not fields, then we have the Schur index phenomenon.

Alternative formulations

• Sum of squares of degrees of irreducible representations equals order of group
• Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree