Group ring

The group ring of a group is an -monoid constructed from the corresponding free module , such that the product of and coïncide. As such, it is a specialization of the monoid ring.

Construction as maps

Let be a group, and be a ring. The group ring may be identified with the set of maps of finite-support , with the convolution and conjugate operations defined below,¹ where we identify with . The convolution operation is defined by

Derivation

Convolution is defined by extending by linearity, so

which yields the definition given above. Note the similarity to the everyday Convolution operation.

If is an Involutive ring, the conjugate is defined by

Hilbert space

If , then the group ring can be made into a Hilbert space with some inner product, usually taken from those listed below.

Inner products

Two possible inner products on a complex group ring are

which has as an Orthonormal basis; or alternatively the renormalised

which has as a unit vector. This normalisation is used for orthogonality of irreps. In these notes I will try to stay consistent with distinguishing these two inner products as above.

Properties

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

1996, Representations of finite and compact groups, §II.3 ↩