Theorem. Wedderburn blocks over an algebraically closed field [fgap-001L]
Theorem. Wedderburn blocks over an algebraically closed field [fgap-001L]
Let \(G\) be finite and \(k\) an algebraically closed field whose characteristic does not divide \(|G|\). There are positive integers \(d_C\), indexed by the conjugacy classes \(C\) of \(G\), such that \[ k[G]\cong _{k\text {-alg}} \prod _{C\in \operatorname {Conj}(G)}M_{d_C}(k), \qquad \sum _C d_C^2=|G|. \] The class labels are an indexing choice: the statement does not canonically pair a particular conjugacy class with a particular irreducible module.
Maschke semisimplicity followed by Artin--Wedderburn gives the stated product. Lux and Pahlings state the results in [lux2010representations, thms. 1.5.5--1.5.6, pp. 56--57].
The dimension identity can also be read from the regular representation, where an irreducible module occurs with multiplicity equal to its dimension; compare [james2001representations, thms. 11.9 and 11.12, pp. 100--101].