Group ring

Regular representation

The (left) Regular representation for a group is both a group representation of a group and a ∗-representation of its complex group ring carried by the group ring itself and defined using the group ring’s convolution operation. For and

and thus for and

Proof these are representations

If we prove that is a ∗-representation it follows that is a unitary representation. Properties 1, 2, and 4 follow from properties of the ∗-algebra (distributivity, associativity, monoid identity), hence all that is left to prove is that for any . Using as defined in the Zettel for Group ring

as required.

The right regular representation is defined the same way using right multiplication.

Matrix

If group elements are identified with indices for a matrix then for each

i.e. , so each is basically the group table.

Properties

• The regular representation contains all irreducible representations
• Its character is times the indicator function .

---

tidy | en | sembr