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

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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