K-monoid

Group ring

The group ring 𝑅[𝐺] of a group 𝐺 is an [[R-monoid|𝑅-monoid]] constructed from the corresponding free module 𝑅(𝐺), such that the product of and coïncide. As such, it is a specialization of the monoid ring.

Construction as maps

Let 𝐺 be a group, and 𝑅 be a ring. The group ring 𝑅[𝐺] may be identified with the set of maps of finite-support 𝐺 →𝑅, with the convolution and conjugate operations defined below,¹ where we identify 𝑔 ∈𝐺 with 𝛿𝑔 :ℎ ↦[𝑔 =ℎ]. The convolution operation is defined by

(𝑎∗𝑏)(𝑥)=∑ℎ∈𝐺𝑎(𝑥ℎ−1)𝑏(ℎ)

Derivation

Convolution is defined by extending 𝛿𝑔 ∗𝛿ℎ =𝛿𝑔ℎ by linearity, so

(∑𝑔∈𝐺𝑎(𝑔)𝛿𝑔)∗(∑ℎ∈𝐺𝑏(𝑔)𝛿𝑔)=∑𝑔,ℎ∈𝐺𝑎(𝑔)𝑏(ℎ)𝛿𝑔ℎ=∑𝑥,ℎ∈𝐺𝑎(𝑥ℎ−1)𝑏(ℎ)𝛿𝑥

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

If 𝑅 is an Involutive ring, the conjugate is defined by

𝑎†(𝑔)=――――𝑎(𝑔−1)

Hilbert space

If 𝑅 =ℂ, 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

⟨𝑎|𝑏⟩=∑𝑔∈𝐺―――𝑎(𝑔)𝑏(𝑔)

which has {𝛿𝑥}𝑥∈𝐺 as an Orthonormal basis; or alternatively the renormalised

(𝑎|𝑏)=1|𝐺|∑𝑔∈𝐺―――𝑎(𝑔)𝑏(𝑔)

which has 𝑐1 :𝑔 ↦1 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

• ∗-representation of the complex group ring
• Regular group representation
• Ideal of the complex group ring
• Idempotent of the complex group ring
• Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations
• Centre of the group ring

---

tidy | en | SemBr

Footnotes

1. 1996, Representations of finite and compact groups, §II.3 ↩