In the mathematical field of representation theory, a trivial representation is a representation (V, φ) of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (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.
The trivial character is the character that takes the value of one for all group elements.
Famous quotes containing the word trivial:
“In our most trivial walks, we are constantly, though unconsciously, steering like pilots by certain well-known beacons and headlands, and if we go beyond our usual course we still carry in our minds the bearing of some neighboring cape; and not till we are completely lost, or turned round,—for a man needs only to be turned round once with his eyes shut in this world to be lost,—do we appreciate the vastness and strangeness of nature.”
—Henry David Thoreau (1817–1862)