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.