# 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.