group representation

Imagine you have an abstract group — say the symmetries of a triangle — and you want to do calculations with it. The cleanest way is to turn each symmetry into a concrete matrix so that composing symmetries becomes multiplying matrices. A representation is exactly such a faithful-to-the-structure dictionary: it lets you replace abstract group elements with linear maps you can actually compute with.

Precisely, a representation of a group G on a vector space V over a field k is a group homomorphism rho from G to GL(V), the group of invertible linear maps on V. Equivalently it is an action of G on V by linear transformations: each g gives a map rho(g) with rho(g)rho(h) = rho(gh) and rho(e) = the identity. The dimension of V is called the degree (or dimension) of the representation. When V has finite dimension n and we fix a basis, rho becomes a homomorphism G -> GL(n, k), a family of n-by-n invertible matrices.

A representation is the same data as a module over the group algebra k[G]: the action of the formal sum sum a_g g on a vector v is sum a_g rho(g)v. This translation is not a mere notation change — it lets the whole machinery of module theory (submodules, quotients, direct sums, semisimplicity) act on representations. Two representations are called equivalent (isomorphic) when there is an invertible linear map intertwining the two actions, i.e. a change of basis carrying one matrix family to the other.