In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector spaceV over a field K and (ii) 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.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
According to Schur's lemma, a non-zero morphism between two irreducible complex representations is invertible, and moreover, is given in matrix form as a scalar multiple of the identity matrix.
The dual representation of C[G] as a G×G-representation is equivalent to C[G].
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