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.

• 5.1 Constructing representations
• 5.2 Representation properties
• 5.3 Representation manipulation
• 5.4 Subrepresentations