III.77 Representations

A linear representation of a finite GROUP [I.3 §2.1] G is a way of associating a linear map T_g, from some VECTOR SPACE [I.3 §2.3] V to itself, with each element g of G. Of course, this association must reflect the group structure of G, so T_g T_h should equal T_gh, and if e is the identity of G, then T_e should be the identity map on V.

One useful aspect of linear representations is that the dimension of the vector space V may be considerably smaller than the size of G. If this is the case, then the representation packages the information about G in a particularly efficient way. For example, the ALTERNATING GROUP [III.68] A_5, which has sixty elements, is isomorphic to the group of rotational symmetries of an icosahedron, and can therefore be thought of as a group of transformations of ³ (or, equivalently, of 3 × 3 matrices).

A more fundamental reason for representations being useful is that every representation can be decomposed into building blocks known as irreducible representations. It turns out that a great deal of information about G can be deduced from a few basic facts about its irreducible representations.

These ideas can be generalized to infinite groups as well, and are particularly important in the case of LIE GROUPS [III.48 §1]. Since Lie groups have a differentiable structure, the representations of interest are those where the homomorphism g T_g reflects this structure (for example, by being differentiable).

Representations are discussed in much greater detail in REPRESENTATION THEORY [IV.9]. See also OPERATOR ALGEBRAS [IV.15 §2].