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

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  • Trivial representation (en)
  • 平凡表示 (zh)
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  • 在數學裡，尤其是在群表示理論裡，一個群的表示若被稱為是一個平凡表示的話，則表示它是被定義在一個體K上的一維向量空間V，且所有於G內的元素g都會以單位映射作用在V上。對於任何一種此類的V，這種表示都會存在著，且在K上的任何兩種此類的表示也都會是等價的。 儘管平凡表示的建構模式使得它看起來像是多餘的，但它確實是這個理論的一個很基本的物件。例如說，當一個子表示會等價於一個平凡表示，即表示其是由不變向量所構成的。因此找尋此類的子表示便成了不變量理論所研究的所有课題了。 平凡特徵是指會將所有群元素的值都取為1的。 (zh)
  • 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. The trivial character is the character that takes the value of one for all group elements. (en)
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  • 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. (en)
  • 在數學裡，尤其是在群表示理論裡，一個群的表示若被稱為是一個平凡表示的話，則表示它是被定義在一個體K上的一維向量空間V，且所有於G內的元素g都會以單位映射作用在V上。對於任何一種此類的V，這種表示都會存在著，且在K上的任何兩種此類的表示也都會是等價的。 儘管平凡表示的建構模式使得它看起來像是多餘的，但它確實是這個理論的一個很基本的物件。例如說，當一個子表示會等價於一個平凡表示，即表示其是由不變向量所構成的。因此找尋此類的子表示便成了不變量理論所研究的所有课題了。 平凡特徵是指會將所有群元素的值都取為1的。 (zh)
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