An Explicit Matrix Unit Realisation of the Wedderburn Decomposition

We construct explicit group-algebra elements corresponding to the elementary matrices in the Wedderburn decomposition of a finite group algebra. These elements yield an explicit algebra isomorphism that coincides with the classical Wedderburn isomorphism. As applications, we apply our construction to the symmetric group to compute the transition matrix between two natural bases of the centre and reinterpret the Frobenius characteristic map through the Wedderburn decomposition. Consequently, we show that the Schur basis is naturally compatible with the identity-matrix basis arising from the Wedderburn decomposition.