In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

, a trivial representation is a representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 (V, φ) of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 G on which all elements of G act as the identity mapping of V. A trivial representation

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...

 of an associative

Associative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

 or Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 is a (Lie) algebra representation for which all elements of the algebra act as the zero endomorphism which sends every element of V to the zero vector.

For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations.

Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentations is the whole topic of invariant theory

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

.

The trivial character is the character

Character (mathematics)

In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...

that takes the value of one for all group elements.