Trivial representation

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In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if

• It is defined on a one-dimensional vector space V over a field K.
• All elements g of G act on V as the identity mapping.

Given any such V, this representation always exists, and any two such representations over K are equivalent.

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.

The trivial character is the character that takes the value of one for all group elements.