What is trivial representation?

In representation theory, the trivial representation is the simplest and most basic representation of a group. It maps every element of the group to the identity matrix, so it preserves all group operations. In other words, the group acts trivially on the vector space of the representation. The trivial representation is often denoted by the symbol 1 and is of dimension 1.

The trivial representation arises naturally when considering the action of a group on itself. It is also used as a building block for constructing other representations. The direct sum of the trivial representation with another representation is isomorphic to that representation, and the tensor product of the trivial representation with another representation gives the same representation.

The trivial representation is useful in many areas of mathematics, including algebraic geometry, number theory, and the study of Lie groups. In physics, the trivial representation plays an important role in quantum field theory, where its presence indicates the existence of massless particles known as Goldstone bosons.