Editing Trivial representation

{{Short description|Universal representation of a group in terms of its own multiplication}}
{{one source |date=May 2024}}
In the [[mathematics|mathematical]] field of [[representation theory]], a '''trivial representation''' is a [[group representation|representation]] {{math|(''V'', ''φ'')}} of a [[Group (mathematics)|group]] ''G'' on which all elements of ''G'' act as the [[identity mapping]] of ''V''. A trivial [[Representation (mathematics)|representation]] of an [[associative algebra|associative]] or [[Lie algebra]] is a ([[Lie algebra representation|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 representation|irreducible]] trivial representation always exists over any [[field (mathematics)|field]], and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to [[unital algebra]]s 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 (mathematics)|character]] that takes the value of one for all group elements.

==References==
*{{Fulton-Harris}}.

[[Category:Representation theory]]


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