One could argue that representation theory is a branch of mathematics devoted to translating group theory into linear algebra. Informally, a representation of an abstract group G is a homomorphism from G to a group of matrices. The name comes from the fact that the above group of matrices is a concrete representative for the isomorphism class of G. Then matrices correspond to linear transformations of vector spaces, and therefore G may be viewed as a collection of transformations of Euclidean space. If you have ever thought of a cyclic group of order n as the group of rotations in the plane that preserve a regular n-gon centered at the origin, you have actually thought about a representation of the cyclic group! Historically, this is actually how groups were born: in Felix Klein's 1884 Lectures on the Icosahedron (Klein, 1956), one may see many concepts of modern group theory arising via the study of groups of symmetries of shapes, i.e. groups of linear transformations of two- or three-dimensional space which preserve a given shape.

Since then, representation theory has evolved into a vast, far-reaching and quite sophisticated area of mathematics. Here we wish to give an essential introduction to some of the ideas that are important in our next translation of the Hurwitz problem. Our goal is to come as efficiently as possible to understand the class algebra of the symmetric group Sd. For this reason we choose to make most of our exposition specific to the symmetric group Sd, and state without proof many facts where we feel the proof would not be especially relevant to our Hurwitz story. The reader interested in finding proofs and filling in more details may consult Dummit and Foote (2004, Chapter 18) or Fulton and Harris (1991, Part 1).

The Group Ring and the Group Algebra

A natural step in trying to convert group theory into linear algebra information is to construct a vector space that “knows a lot” about the group. We construct a ring that encodes the group operation as its multiplication, and then enlarge coefficients to also have a vector space structure.