5. Representations

Let \(V\) be a finite dimensional vector space over the real or complex numbers. We identify \(V\) respectively with \(\mathbb{R}^D\) or \(\mathbb{C}^D\). Below, we write \(\mathbb{K}\) for either of \(\mathbb{R}\) or \(\mathbb{C}\).

Let \(G\) be a compact group (this family contains finite groups and the continuous groups that can be constructed in RepLAB). A linear representation of a group \(G\) is a map from the group to the group of invertible \(D \times D\) matrices over \(\mathbb{K}\). We direct the reader to Serre [Ser77] for a gentle introduction to the topic.

RepLAB proposes ways to construct and manipulate representations.

  • Some groups have natural representations: permutation groups act naturally on Euclidean space coordinates, matrix groups provide … matrices.

  • Representations of finite groups are defined simply by providing the matrices corresponding to the images of the generators.

  • Representations can be transformed by taking the complex conjugate, the dual representation, changing the underlying field.

  • Representations can be combined by taking tensor products or writing direct sums.

  • One can find and manipulate subrepresentations.

Those operations are described in the following sections.