Group ring

Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations

Consider unitary irreps for . Then there exists a unitary isomorphism from the Group ring to the direct sum of matrix algebras

defined by with1 #m/thm/rep

and likewise

which is unitary from to the following inner product

and a homomorphism in the sense that

Proof

To verify the given inverse, note that by orthogonality of irreps, form an orthonormal basis with respect to the inner product

and hence it is a linear bijection. Since

it is unitary. From Convolution of matrix representations, it follows that

hence preserves the algebra operations.

The isomorphism is denoted in such a way to evoke the Fourier transform due to similar properties. This may be viewed as a special case of the Wedderburn–Artin theorem.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 1996, Representations of finite and compact groups, §III.1, pp. 38–39 ↩