I.3.5 Group Representation

Group representations describe abstract groups in terms of linear transformations of vector spaces. They can be used to represent group elements as matrices, so that the group operation can be represented by matrix multiplication, which allows many group-theoretic problems to be reduced to problems in linear algebra. A representation means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation.

A representation of a group is an action of on a vector space by invertible linear maps. Therefore, representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important, depending on the type of space. One must also consider the type of field over which the vector space is defined. The most important fields are complex, real and finite fields. In general, algebraically closed fields are easier to handle than non-algebraically closed ones. The characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the order of the group.

There are various representation theories of groups, depending on the kind of group being represented, though the basic definitions and concepts are similar. For example, if the field of scalars of the vector space has some characteristic that divides the order of the group, then this is called modular representation theory. Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. The resulting theory is a central part of harmonic analysis.

Many important Lie groups are compact, so the results of compact representation theory apply to them, but other techniques specific to Lie groups are also used. For example, most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.

On the other hand, linear algebraic groups, or more generally affine group schemes, are the analogues of Lie groups, but over more general fields than just  or . Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different and much less well understood. Therefore, the analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak topology causes many technical complications.

The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using somead hoctechniques.

Group representations help understanding their properties by studying how do they act on different spaces, so they are very important in physics because they describe how the symmetry group of a physical system affects the solutions of equations describing that system.