Trivial representation

Summary

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.

References

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  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..