Representation Theory of Finite Groups - Other Formulations

Other Formulations

A representation ρ: G → GL(n,C) defines a group action of G on the vector space Cn. Moreover this action completely determines ρ. Hence to specify a representation it is enough to specify how it acts on its representing vector space.

Alternatively, the action of a group G on a complex vector space V induces a left action of group algebra C on the vector space V, and vice-versa. Hence representations are equivalent to left C-modules.

The group algebra C is a |G|-dimensional algebra over the complex numbers, on which G acts. (See Peter–Weyl for the case of compact groups.) In fact C is a representation for G×G. More specifically, if g₁ and g₂ are elements of G and h is an element of C corresponding to the element h of G,

(g₁,g₂)=g₁ h g₂-1.

C can also be considered as a representation of G in three different ways:

• Conjugation: g = g h g−1
• As a left action: g = g h (a regular representation)
• As a right action: g = h g−1 (also);

these are all to be 'found' inside the G×G action.