Representation theory

The mathematical branch that connects groups and vector spaces is called representation theory. A representation of a finite group \(G\) on a finite-dimensional vector space \(V\) is a homomorphism \[\begin{equation} \rho: G \rightarrow \text{GL}(V): g \mapsto \rho(g) \end{equation}\] of the group \(G\) to the general linear group \(\text{GL}(V)\), such that every group element \(g\) is associated to an element of the general linear group. In other words, we associate every group element \(g\) with an \(n \times n\)-matrix \(\rho(g)\). The term homomorphism means that group structure is preserved: \[\begin{equation} \forall g_1, g_2 \in G: \rho(g_1 \cdot g_2) = \rho(g_1) \rho(g_2) \thinspace , \end{equation}\] which means that the matrix representation \(\rho(g_1 \cdot g_2)\) of the group multiplication of two group elements \(g_1\) and \(g_2\) is the matrix product of their respective matrix representations \(\rho(g_1)\) and \(\rho(g_2)\).